The objective of this numerical study is to examine how different Reynolds numbers impact heat and mass transfer in an unsteady forced convective two-dimensional flow within a right-angle triangular cavity. The lowest surface of the enclosure is held at a fixed temperature and concentration, whereas the slanted surface is taken to be a cool surface. Furthermore, the cavity's left wall is adiabatically positioned to move in two directions: upwards (aiding flow) and downwards (opposing flow), with a constant speed being maintained. The partial differential equations that govern the system are converted into a non-dimensional form through a straightforward transformation. The finite-element scheme is employed to solve these dimensionless equations. The analysis facilitates the investigation of the belongings of the Reynolds number on the heat and mass transfer appearances by using streamlines, isotherms, and isoconcentration lines. It is initiated that the temperature spreading as well as concentration within the cavity depends strongly on the Reynolds number. Moreover, the motion of the moving wall influences the patterns of fluid flow, temperature, and concentration fields. This study provides a comprehensive investigation into heat and mass transfer behavior occurring within a lid-driven right-angled triangular cavity moving in two opposite directions for aiding flow and opposing flow respectively.

The main objective of the study is to investigate the fluid flow pattern and thermal behavior of time-dependent 2-dimensional lid-driven cavity flow in the presence of mixed convection. The square-shaped lid-driven chamber was filled with a nanofluid consisting of carbon nanotubes (CNT) dispersed in kerosene oil. A semicircular heater is positioned along the lower wall. The research also incorporates the examination of magnetohydrodynamics and radiation. The use of the Galerkin residual approach inside the finite element method is utilized to derive nonlinear dimensionless governing equations. The phenomenon of Brownian motion in nanoparticles is included into models that describe thermal conductivity and dynamic viscosity. A constant magnetic field with a magnitude of Ha = 10, a constant Reynolds number of Re = 100, and a constant radiation parameter of Rd = 1 were selected for the study. Additionally, a nanofluid concentration of 5 % was chosen. The study aims to investigate the impact of mixed convection by using variable Richardson numbers within the range of 0.1 ≤ Ri ≤ 10. The results are presented in the form of streamlines, isotherms, 2D and 3D charts, illustrating the temporal variations of diverse thermophysical parameters. The investigation reveals that the magnitudes of the drag force and pressure gradient are comparatively greater in the scenario of natural convection. The drag force is 2.8 times greater in the Ri=10 case compared to the Ri=0.1 case. The velocity magnitude is found to be 18 times higher in the Ri= 10 case than in the Ri= 1 case. The Nusselt number has an elevated value as the Richardson number is augmented in conjunction with the dimensionless time (τ).

The objective of this research is to examine the thermophysical features of magnetic parameter (Ha) and time step (τ) in a lid-driven cavity using a water-based Al2O3 nanofluid and the efficacy of ANN models in accurately predicting the average heat transfer rate. The Galerkin weighted residual approach is used to solve a set of dimensionless nonlinear governing equations. The Levenberg-Marquardt back propagation technique is used for training ANN using sparse simulated data. The findings of the investigation about the flow and thermal fields are shown. Furthermore, a comparative study and prediction have been conducted on the impact of manipulating factors on the average Nusselt number derived from the numerical heat transfer analysis. The findings of the research indicate that, in the absence of magnetohydrodynamics, a rise in the Hartmann number resulted in a drop in both the fluid velocity profile and magnitude. Conversely, it was observed that the temperature and Nusselt number exhibited an increase under these conditions. The mean temperature of the fluid rises as the Hartmann number drops, reaching a peak value of 0.114 when Ha = 0. The scenario where Ha = 0, representing the lack of magnetohydrodynamics, shows the highest average Nusselt number, whereas the instance with Ha = 45 presents the lowest Nusselt number. The ANN model has a high level of accuracy, as seen by an MSE value of 0.00069 and a MAE value of 0.0175, resulting in a 99% accuracy rate.

This work explores diverse novel soliton solutions due to fractional derivative, dispersive, and nonlinearity effects for the nonlinear time M-fractional paraxial wave equation. The advanced exp [-φ(ξ)] expansion method integrates the nonlinear M-fractional Paraxial wave equation for achieving creative solitonic and traveling wave envelopes to reconnoiter such dynamics. As a result, trigonometric and hyperbolic solutions have been found via the proposed method. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. For any chosen set of the allowed parameters 3D, 2D and density plots illustrate, this inquisition achieved kink shape, the collision of kink type and rogue wave, periodic rogue wave, some distinct singular periodic soliton waves for time M-fractional Paraxial wave equation. As certain nonlinear effects cancel out dispersion effects, optical solitons typically can travel great distances without dissipating. We have constructed reasonable soliton solutions and managed the actual meaning of the acquired solutions of action by characterizing the particular advantages of the summarized parameters by the portrayal of figures and by interpreting the physical occurrences. New precise voyaging wave configurations are obtained using symbolic computation and the previously described methodologies. However, the movement role of the waves is explored, and the modulation instability analysis is used to describe the stability of waves in a dispersive fashion of the obtained solutions, confirming that all created solutions are precise and stable.

This work explores diverse novel soliton solutions of two fractional nonlinear models, namely the truncated time M-fractional Chafee-Infante (tM-fCI) and truncated time M-fractional Landau-Ginzburg-Higgs (tM-fLGH) models. The several soliton waves of time M-fractional Chafee-Infante model describe the stability of waves in a dispersive fashion, homogeneous medium and gas diffusion, and the solitary waves of time M-fractional Landau-Ginzburg-Higgs model are used to characterize the drift cyclotron movement for coherent ion-cyclotrons in a geometrically chaotic plasma. A confirmed unified technique exploits soliton solutions of considered fractional models. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. Keeping special values of the constraint, this inquisition achieved kink shape, the collision of kink type and lump wave, the collision of lump and bell type, periodic lump wave, bell shape, some periodic soliton waves for time M-fractional Chafee-Infante and periodic lump, and some diverse periodic and solitary waves for time M-fractional Landau-Ginzburg-Higgs model successfully. The required solutions in this work have many constructive descriptions, and corporal behaviors have been incorporated through some abundant 3D figures with density plots. We compare the m-fractional derivative with the beta fractional derivative and the classical form of these models in two-dimensional plots. Comparisons with others’ results are given likewise.

This work focuses on the study of the paraxial wave model with a space–time fractional form. This model has more importance for describing light propagation in nonlinear optical fibers and telecommunication lines. The main aim of this work is to observe the effect of the fractional parameters and compare the truncated -fraction with the beta-fraction, conformable fraction form, and classical form of the PW model. For this observation, we applied the Simplest equation technique to acquire analytical solutions to the space–time (spatiotemporal) -fractional paraxial wave model. We are able to acquire several new optical soliton solutions, including periodic waves, kink-type waves, rogue-type waves, and several novel periodic waves, by providing the appropriate fractional parametric values. These solutions have significance for shedding light on a number of physical phenomena in the realms of optical fiber and communication sciences. The diverse values of fractional parameters and the three-dimensional and contour plot graphs of certain chosen solutions are depicted, which are the most accurate physical characterizations of the outcomes. We also sketch the comparative graph of diverse fractional forms and the classical form of the paraxial wave equation in two-dimensional plots. Consequently, our findings represent an important breakthrough in this complex area and help further develop our comprehension of the behavior of solitons. Similar content being viewed by others

The study of optical soliton solutions plays a vital role in nonlinear optics. The foremost area of optical solitons research encompasses around optical fiber, telecommunication, meta-surfaces and others related technologies. The aim of this work is to integrate optical soliton solutions of the complex Ginzburg-Landau (CGL) model with Kerr law nonlinearity, also showing the effect of diverse fraction derivative and comparing it with the classical form. Here, the local derivative is used as the conformable wisdom known as the truncated M-fractional derivative, beta fractional derivative, and conformable fraction derivative. We also deliberated on some assets satisfied by the derivative. The CGL model is useful to describe the light propagation in optical communications, optical transmission, and nonlinear optical fiber. Under the right circumstances, the affectionate unified scheme is implemented for the complex Ginzburg-Landau model to generate the optical wave pattern. For α1 = 2α2, the unified scheme generates the solution of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme provides some novel optical solitons such as periodic waves, periodic with rogue waves, breather waves, different types of periodic rogue waves, a singular soliton solution, and rogue with the periodic wave for the special value of the free parameters. For α2 = − ρ+6α1 8 , the unified scheme generates the solutions of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme offers some fresh optical solitons such as periodic rogue waves, multi-rogue waves, double periodic waves, and periodic waves. In numerical argument, the wave patterns are offered with 3-D and density plots. To test the stability of the obtained solutions, we show diverse fractional forms such as beta time fractional, and conformable time fractional derivative and compare these fractional derivatives with their classical form in 2D plots. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of the obtained solutions, emphasizing their distinct features and depictions using unified technique. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations.

Rotating cylinder movement in a cavity flow is an exciting field of study in heat transfer. Considerable research has been carried out on rotating cylinders under MHD mixed convection in various types of enclosures. However, considering partially heated square enclosure and magnetic field using CNT-water nanofluid is very limited. This study's goal is to assess the hydrothermal phenomena in a square enclosure with a rotating cylinder. Simulation has been conducted for different rotational speeds (Ω) and dimensionless times (τ) to observe the thermal and fluid flow behaviour. The Galerkin Residual based finite element method has been used to conduct numerical calculations. The results are shown as isotherms, streamlines, and average Nusselt number at the cylinder wall. Moreover, the drag force at the moving wall, and the fluid properties such as the root mean square (rms) of velocity, the temperature, the vorticity functions, and the average fluid temperature are also presented. The heat transfer rate, drag force, rms velocity, and temperature increase with the rise of rotational speed and dimensionless time rise. Maximum vorticity occurs at Ω = 8 and τ = 1. The maximum vorticity function increases 12 times with the increasing rotational speed. Higher rotational speed leads to increased average fluid temperature. The case of Ω = 8, τ = 1 shows the most temperature variance, while Ω = 1, τ = 0.1 has the least. Increasing rotational speed results in higher drag force on the cylinder's surface. At Ω = 4, the drag force is 2.8 times greater than at Ω = 2. Overall, the fluid flow and thermal performance boost up while the rotating speed of the cylinder is higher.

The goal of the paper is to evaluate the characteristics of the fluid flow, heat, and contaminant transfer in an indoor air environment. A two-dimensional ventilated space having a discrete heat and contaminant source at their mutually perpendicular position is constructed to conduct a numerical study. The governing equations with apprupirate boundary conditions of the model are evaluated by Galerkin weighted residual method. The transport characteristics are discussed for varying Richardson number 0.1 ≤ Ri ≤ 10 and dimension less time 0.1 ≤ τ ≤ 1 by the corresponding stream function, heat function, and mass function. Priority is given to analyzing the necessary factors – dependence on average Nusselt number, average Sherwood number, the average fluid temperature and mass concentration in the enclosure, along with the average temperature of the exit port to visualize the ventilation technique. Numerical output illustrates that – the attributes of the transport particles are largely determined by the interaction of the natural convection of heat and contaminant source with the mechanically forced flow of cool air. Results showed that mixed convection method is a better way to showcase efficient ventilation of the transport particles from the enclosure through a practical yet non-complex approach. This study will be a guide for design consideration for roof-based double-diffusive systems with partial heat and mass generation.

This study is aimed to perform a numerical time-dependent investigation thermal conductivity effect of the annular cylinder within a vented cavity using CNT based-water nanofluid. For demonstrating the effect of thermal conductivity, four distinct hollow cylinder materials such as Ks = 0.5(Plastic tiles), Ks = 0.84(Clay tiles), Ks = 1.1(Concrete tiles), and Ks = 2(Slate tiles) are introduced together with a suitable variation of dimensionless time (0 ≤ τ ≤ 1). The governing equations of the model with associated boundary conditions is solved using finite element based Galerkin's weighted residual method. Different contour plots for thermal and flow field transformation and mean Nusselt number, mean fluid temperature, bulk convective field temperature, temperature gradient, pressure gradient, vortices, and fluid velocity magnitude are presented for qualitative and quantitative thermal performance analysis. With the decrease of solid thermal conductivity, 27.3% thermal transport enhancement is noted from the heated surface of the cylinder. However, a 16.3% increase in the bulk fluid temperature has been recorded with the increase in cylinder conductivity. The numerical outcomes from this investigation propose a better thermo-fluid efficiency compared to the existing methodology which can be suggestive to engineers and researchers for designing heat exchangers, heat pipes, and other thermal systems.

The Biswas-Arshed (BA) model is an essential wave propagation model in nonlinear optics and optical fibers. In this manuscript, the enhanced (G'/G) -expansion method is taken into consideration. The BA model is used with this technique to locate visual soliton solutions. The solutions to the hyperbolic and trigonometric functions are determined using this technique. Each solution includes a large number of model and method parameters. The wave solutions are defined by the waves' character, which is determined by the three-dimensional (3D) and two-dimensional (2D) wave patterns with varying parameter values. The structure of the soliton shifts depending on the specific values of the parameters. These shifts are reflected in the 2D pictures that demonstrate the various soliton configurations. Finally, the obtained soliton solutions can be applied to high-speed data transmission to describe propagation pulses and to transmit light between two ends in optical fibers and communication systems.

In this study, we solve fractional order PDEs with free parameters that regulate wave motion in a low-pass electrical transmission line equation (LPET) using the unified technique. To convert the governing equation into an ordinary differential equation (ODE), we have used a simple linear fractional transformation. The unified technique can therefore be used to find various single and extracting wave solutions to the equation of attention. Using Maple-18 software, all of the established solutions are validated and verified. By selecting appropriate values for the fractional order and free parameters, the two dimensional (2D) and three dimensional (3D) profiles of some of the assimilation solutions demonstrating kink, singular kink, cross kink, bright-dark kink, lump type kink, anti-kink, dark-anti kink, singular periodic, periodic, singular soliton solutions are displayed. The effect of free parameters and fractionality on the dynamical properties of the precise and exploited solutions are graphically illustrated and discussed in detail. We have also compared our results to those found in the literature review. Wherever possible, the judgment demonstrates no significant differences between our realized and published solutions. The results of this investigation show that the adopting technique is both productive and powerful in extracting wave solutions to nonlinear evolution problems that arise in the areas of applied mathematics, mathematical physics, and engineering.

Medical scientists, clinicians, and bioengineers are fascinated in the detection of blood flow pattern through cardiovascular system because it is employed to diagnose the human circulatory disorders such as atherosclerosis. This work is dedicated to the mathematical modeling for the blood flow through human circulatory regimen with electroosmotic force, tri-hybrid nanoparticles, swimming microbes and activation energy. A novel ternary fluid is prepared by dispersing titania, silica, and alumina nanoparticles in pure blood. It is a well-known fact that blood is regarded as incompressible non-Newtonian fluid, due to which Sutterby fluid paradigm is regarded to detect the shear-thinning characteristics of blood precisely. To describe the electroosmotic flow, Poisson-Boltzmann expression is used. The novelty of the appraisal is the incorporation of electroosmotic force on blood flow containing with microbes and three distinct nanoparticles. Appropriate dimension-free variables are utilized to modify the dimensional system into dimension-free version. An unraveller approach, finite-difference is assigned to evaluate the numerical solution of dimensionless system accompanied by stability and convergence. The computed numerical results are sketched versus crucial factors and explored through graphs and tables. Electroosmotic variable enhances the motion of ternary blood nanofluid. Higher exponential heat source variable strengthens the thermal borderline wideness. Heat transport rate is weakened via higher estimation of thermo-migration and random motion variables. Decay in mass transport rate is noticed for thermo-migration factor. Peclet number weakens the motile microbe's density. Bio-active Lewis number strengthens the density number of motile microbes. This study is extremely worthwhile in medical domain such as cancer treatment, heart surgeries, tumor treatment, microbial fuel cells, and microfluidic systems.

Nanomaterial flow has fascinated the concern of scientists across the globe due to its innovative applications in various manufacturing, industrial, and engineering domains. Bearing aforementioned uses in mind, the focal point of this study is to examine the Carreau nanofluid flow configured by the Riga surface with Arrhenius catalysts. Microorganisms are also suspended in nanofluid to strengthen the density of the regular fluid. Time-dependent coupled partial differential equations that represent the flow dynamics are modified into dimensionless patterns via appropriate non-dimensional variables, and handled through an explicit finite difference approach with stability appraisal. The performances of multiple flow variables are examined graphically and numerically. Representation of 3D surface and contour plots for heat transportation and entropy generation are also epitomized. The findings express that the modified Hartmann number strengthens the motion of nanomaterial. Reverse outcomes for heat transport rate and entropy are seen for the radiation variable. Concentration diminishes for chemical reaction variable. Activation energy enhances the concentration of nanomaterial, whereas reduction happens in the movement of microbes for bio-Lewis number. Greater Brinkman variable heightens the entropy.

The modified extended tanh technique is used to investigate the conformable time fractional Drinfel'd-Sokolov-Wilson (DSW) equation and integrate some precise and explicit solutions in this survey. The DSW equation was invented in fluid dynamics. The modified extended tanh technique executes to integrate the nonlinear DSW equation for achieve diverse solitonic and traveling wave envelops. Because of this, trigonometric, hyperbolic and rational solutions have been found with a few acceptable parameters. The dynamical behaviors of the obtained solutions in the pattern of the kink, bell, multi-wave, kinky lump, periodic lump, interaction lump, and kink wave types have been illustrated with 3D and density plots for arbitrary chose of the permitted parameters. By characterizing the particular benefits of the exemplified boundaries by the portrayal of sketches and by deciphering the actual events, we have laid out acceptable soliton plans and managed the actual significance of the acquired courses of action. New precise voyaging wave arrangements are unambiguously gained with the aid of symbolic computation using the procedures that have been announced. Therefore, the obtained outcomes expose that the projected schemes are very operative, easier and efficient on realizing natures of waves and also introducing new wave strategies to a diversity of NLEEs that occur within the engineering sector.

The (2 + 1)-dimensional Konopelchenko-Dubrovsky (KD) model and the modified version of the new Kudryashov (MVNK) technique are chosen in the current research to obtain the traveling wave solutions (TWSs). The obtained solutions represent the rich range of explicit solutions to the studied model. As a result, TWSs to the stated model are expressed as the different types of wave profiles such as the kink shape, bell shape, anti-bell shape, and W-shape wave profiles. The Hamiltonian function is found from the stated model and shown it as three dimensional, contour and phase plane in this manuscript. The effects of wave velocity and other parameters on the wave profile are also discussed. The obtained wave profiles are typically useful in applications how waves interact with high-dimensional systems in new, specialized structures. Additionally, the direction and position of solitons for changing other parameters can offer a clear-cut explanation of all the different features of wind and water waves. It is seen that the mentioned scheme is effective, potential and easy in mathematical physics. Finally, this study may be opened up brand-new avenues for further study and application in the fields of mathematical physics.

This ongoing study is carried out to analyze the thermal performance with sensitivity study of a mixed convective hexagonal heat exchanger containing TiO2–H2O nanofluid. Magnetic force is regarded to be horizontal while all surrounding walls are adiabatic. The finite element method is used to simulate the regulatory equations. For the first time, response surface methodology is used to analyze the sensitivity of independent factors on a hexagonal heat exchanger. The findings are depicted for four parameters, Reynolds number (10 ≤ Re ≤ 200), Richardson number (0.01 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 100), and nanoparticle volume fraction (0 ≤ φ ≤ 0.1) against velocity distribution, average Nusselt number (Nuav), streamlines, isotherm lines, and heatlines. The results indicate that the growing value of Re and φ strengthen the thermal performance of nanofluid whereas increasing Ha causes it to decrease. Moreover, φ and have positive sensitivity to the Nuav while Ha has negative sensitivity. When Ha is maintained at 0, the optimal value of Nuav reaches when Re = 200 and φ = 0.1. The use of TiO2–H2O nanofluid improves the water's heat transmission ability to 17.69%. Finally, the results of this study may offer advice for creating an effective mixed convective heat exchanger.

This study integrates the (1 + 1)-dimensional time M-fractional Lonngren-wave (tM-fLW) equation. This equation is useful to describe the phenomenon of the electric signals in telegraph lines at the source of the tunnel diode. To extract some soliton solutions, two reliable analytical techniques, the new form of modified Kudryashov’s and the simplest equation, are used to accomplish this job. By implementing these techniques, the obtained solutions are expressed as rational, exponential, trigonometric, and hyperbolic functions with some free constraints. For the special values of the constraints, we obtained w-shape wave, bell-shaped wave, linked rogue wave, and multi-bell wave solutions by the NMK method, and dark bell and bright bell-shaped wave, linked rogue wave, periodic rogue wave, and w-shape wave by the SE method. The graphical depiction analyzes these traits, demonstrating the validity and efficacy of the suggested techniques. The obtained solutions, which have never been done before and show a good balance between the nonlinear physical components, are what make this research novel. It is remarkable to perceive that the simplest equation technique and the new form of modified Kudryashov’s technique are relaxed, companionable, and authoritative mathematical tools to elucidate a non-linear model. The movement role of the waves is explored and the modulation instability analysis is used to discuss the stability analysis of the obtained solutions, confirming that all created solutions are accurate and stable.

In this study, we have considered the (2+1)-dimensional paraxial nonlinear Schrödinger (NLS) equation in Kerr media and used the (w/g) -expansion method. The g' and (g'/g2)-expansion techniques have been customized from the (w/g) -expansion method. We applied these two techniques to the paraxial NLS equation and found the optical soliton solutions. The optical soliton solutions are attained as the flat kink, kink, singular kink, peakon, anti-parabolic, W-shape, M-shape, bell, and periodic wave solitons in terms of free parameters. We have presented three-dimensional (3D), two-dimensional (2D) and contour plots of the obtained results and discussed the effect of the free parameters and nonlinearity of the equation by determining different parametric values, which have not been discussed in the previous literature. We have studied the impact of the Kerr nonlinearity and wavenumber on the travelling wave solutions. Moreover, we also analyze the streamlines pattern and instantaneous local directions of the wave profile. All wave phenomena are applied to signal transmission, magneto-acoustic waves in plasma, optical fiber art, coastal engineering, quantum mechanics, hydro-magnetic waves, nonlinear optics and so on. The achieved solutions prove that the proposed methods are very powerful and effective for modern science and engineering for scrutinizing nonlinear evolutionary equations.

This study retrieves some new exact solutions to the (2+1)-dimensional Calogero-Bogoyavlenskii-Schilf (CSB) equation and regularized long wave (RLW) equation in the context of nonlinear traveling wave phenomena. In this regard, the advanced -expansion method is imposed to the -dimensional CBS and RLW equation, and consequently, rogue, kink, singular kink, periodic, singular, and multiple soliton solutions are exhibited in terms of trigonometric, hyperbolic, and rational function solutions. To enucleate the underlying nonlocalized traveling wave features, accomplished exact solutions are established by making their dynamic behavior of the exact solutions exhibited in three-dimensional (3D) and two-dimensional (2D) combined chart with the help of computational software Maple 18. All of our accomplished solutions are claimed to be new in the sense of conformable derivative, chosen a unique fractional type wave transformation, dynamical behavior of fractionally and free variable, and the imposed method on our preferred equations.

Double diffusive flow under mixed convection has become a significant area of research. This study aims to evaluate the Lewis number effect in such a double-diffusive problem in a roof-based air-ventilated system. The Lewis number 0.01 ≤ Le ≤ 5 was chosen for the study, considering other parameters as constant to observe the impact of different heat and mass transfer properties varying the dimensionless time 0.1 ≤ τ ≤ 1. The Galerkin residual technique was chosen to generate the governing equations. The outcomes have been plotted in streamlines, isotherms, isoconcentration, heat and mass transfer, average fluid temperature and mass concentration. The major findings elucidate that heat and mass transfer rate augments when Lewis number goes higher and vice versa. Moreover, when the thermal diffusivity is higher than the mass diffusivity, the average fluid temperature and mass concentration become lower. Overall, this study will be a guide on double-diffusive system design, such as roof-based air ventilation systems.

This numerical study was conducted to investigate the ventilation and airflow in a confined enclosure undergoing rapid inclusion of heat and contaminants representing air quality changes in working environments. A flow of air is passing inside a square enclosure with a heat source and a pollutant source on the right and bottom wall, creating double-diffusion, as well as two ventilation ducts on the upper sides asserting mixed convection. The Galerkin weighted residual method is implemented, which is based on the finite element method. For fixed values of Lewis number, Le = 0.5, Richardson number, Ri = 1, Reynolds number, Re = 100, and Prandtl number, Pr = 1.01, simulations were done with Buoyancy Ratio, Br = −10, 1, 10 and 20 over dimensionless time, τ = 0.1, 0.5, and 1.0. The results have been shown with the streamline, isothermal lines, iso-concentration lines, average Nusselt and Sherwood number, average fluid temperature, average mass concentration, and the average temperature at the square cavity's exit port plots. The findings have been discussed to ensure understanding of the changes in parameters, and an enhancement in heat as well as mass transfer was seen with the rapid change in dimensionless time. This specific study could be a guide for designing air conditioning and ventilation systems.

This article discusses the dynamical structures of unique traveling wave solutions to the unidirectional time-fractional Dullin–Gottwald–Holm equation, as well as the modulation instability analysis of solitary wave prorogations in shallow water mechanics. Different sorts of explicit solutions are obtained which are expressed as the kink, singular kink, singular soliton, multiple solitons and other forms of soliton with specified parameters by utilizing the unified method. Three-dimensional plots, density plots, and their two-dimensional combined line plots of the obtained novel solutions satisfying the corresponding equation of interest are given to comprehend the underlying mechanisms of the identified family. The obtained novel wave solutions can motivate applied scientists to refine their theories to the best of their abilities and can be utilized to verify the outcomes of numerical simulations for wave propagation in shallow water and other nonlinear cases. The implemented methods are shown to be straightforward and effective for approximating the considered equation, and it may be utilized to resolve numerous classes of nonlinear partial differential equations that arise in engineering and mathematical physics.

Nanoparticle is highly used in enhancing thermal performance, especially in lid-driven cavity flow in powder-related applications. Many researchers considered different conditions, such as MHD, radiation, and mixed convection incorporating nano liquid in such flow. However, no study was carried out with CNT-water nano liquid in a triangular cavity, heated sinusoidally under natural convection to determine the Rayleigh number (Ra) and nano-powder liquid concentration effect on heat transfer. As a result, this study attempts to find the Rayleigh number and particle concentration effects in such transient cavity flow. The governing equations were employed with the Galerkin residual method. While calculating the thermal conductivity and dynamic viscosity of the nano liquid, Brownian motion was taken into consideration. The four solid volume fractions (0.01, 0.05, 0.1, 0.15) and the Ra (104 ≤ Ra ≤ 106) were chosen to evaluate the impact within dimensionless time (0.1 ≤ τ ≤ 1). The findings have been shown by plotting the streamlines, isotherms, heat transfer variation, and pressure gradient. It is found that when the Rayleigh number goes up, the velocity, vorticity, and pressure gradient magnitude become high. But, the fluid flow vortices decrease with dimensionless time. Due to the sinusoidal heat flux conditions, the variation of the result was found to be sinusoidal as well. When the nanoparticle concentration rises, the heat transfer rate and average fluid temperature also increase. Therefore, the highest heat transfer rate is found using the nano liquid with a15% concentration.

The Kundu–Mukherjee–Naskar (KMN) model is considered for describing the pulse propagation in optical fibres and communication systems. Two efficient approaches via the exp -expansion and the improved -expansion schemes are applied to the KMN model and in order to convert the KMN equation to an ODE, we test the complex wave conversions and present some optical soliton solutions, which are expressed in terms of hyperbolic, rational and trigonometric functions. We create three dimensional and two-dimensional images to explain the underlying optical wave properties of the KMN equation and compare our solutions with the ones in the literature. Our example shows that the two adopted methods provide an efficient mathematical tool for constructing new optical wave solutions to nonlinear partial differential equations arising in fibre optic communication systems.

This work conducts a numerical analysis of thermal radiation with a semicircular heater on the middle part of the bottom wall and a sliding lid on the top in a square enclosure. Magnetohydrodynamic (MHD) unsteady mixed convection for kerosene oil-based CNT nanofluid is studied. The finite element-based Galerkin residual technique is used to obtain nonlinear dimensionless governing equations. The thermal conductivity and dynamic viscosity models incorporate nanoparticle Brownian motion. Simulations were conducted for Reynolds number = 100, Hartmann number = 10, Richardson number = 1, and particle concentration = 0.05. The effects of fluid velocity magnitude, pressure gradient, temperature gradient magnitude, average temperature, bulk temperature, drag force of the sliding lid, and Nusselt number of the semicircular heater were investigated for different radiation parameters and dimensionless time. Results showed that increasing radiation intensity and dimensionless time improves fluid velocity, pressure gradient, and temperature gradient but decreases the heat transfer rate of the semicircular heater. Furthermore, the drag force of the moving lid is likewise found to be substantially dependent on the radiation parameter as well as dimensionless time. It is worth noted that after a while, the considered parameter exhibits consistent behavior.

This manuscript employs an exact and explicit soliton solution of the Drinfel'd–Sokolov–Wilson (DSW) equation. The wave phenomena of the obtained solutions are applied to water wave mechanics, fluid dynamics, ocean engineering and science. The new auxiliary equation (NAE) method was not applied to the DSW model. As a result, we applied this model to construct a lot of analytic solutions in the expression of hyperbolic, trigonometric, and explicit functions that included free parameters. The obtained solutions are explained in some plots in three-dimensional (3D) and two-dimensional (2D) graphs in this paper, which also discusses the effect of the wave phenomena in 2D diagrams. However, we have made comparisons between our solution and other solutions in the previous literature. Therefore, it is revealed that the mentioned method may be a potential tool for creating new precise soliton solutions for various NLEEs, which play an important role in applied science and engineering.

An enclosure with a moving lid is commonly used in heat and mass transmission. Also, many investigations have been done so far on a mixed convective flow in a cavity, but no research has been done to observe the low Reynolds number effect in the presence of MHD and radiation in a cavity with a semi-circular heater incorporated kerosine oil-based CNT nanofluid. In addition, oil-based nanofluid makes the working fluid more stable at a higher temperature. This study numerically investigates the time-dependent effect of low Reynolds number on Kerosene oil-based CNT nanofluid with magnetic field and radiation. The governing equations were employed with the finite element method based Galerkin residual technique. Brownian motion of nanoparticles was considered to determine the thermal conductivity and dynamic viscosity. Lower values of Reynolds number are taken, such as 50 to 200 with fixed values of radiation parameter, the particle concentration, the Hartmann number, and the Richardson number. The results were illustrated as heat transfer and fluid flow for three dimensionless time conditions. Results indicate that increasing the fluid velocity improves the Nusselt number and drag force. The vorticity rises to 54% while increasing the fluid velocity, however, the pressure gradient and average temperature become lower. It is also found that the average fluid velocity is 2.2 times higher in the Re = 50 than in the Re = 100. For the time dependency of this study, the thermo-hydrodynamics behavior changes with dimensionless time. Finally, this study would be a guide for designing thermal devices related to heat transfer, especially using the Kerosene oil-based CNT nanofluid under different conditions.

This research examines the natural convection phenomena in a triangular enclosure, assumed as a mechanical chamber. Water-based CNT-nanofluid was used as the fluid. The nature of the flow is unsteady, and a sinusoidal heat source is situated at the bottom of the triangle. The inclined two walls of the triangle are assumed as cold temperature. Based on the finite element method, dimensionless nonlinear governing equations were obtained employing the Galerkin weighted residual method. Brownian motion of nanoparticles was considered to determine the thermal conductivity and dynamic viscosity. Rayleigh number (Ra), oscillation period τp, and dimensionless time τ are assumed as primary controlling factors, and nanofluid concentration is considered as constant φ = 0.04. The result indicates that the heat transfer rate displays varied patterns varying the Rayleigh number, and it rises as the oscillation period increases. The fluid temperature and flow fields exhibit periodic behavior due to the sinusoidal heat source. The findings of this work can be utilized to build an effective cooling or heating system for the mechanical chamber, ensuring that temperature distributions are effective and consistent.

Oil-based nanofluids are used to strengthen the stability of nanofluids as well as their thermophysical properties when they are exposed to high temperatures. The time-dependent thermophysical characteristics of multiwalet carbon nanotubes for kerosine oil-based nanofluid are numerically explored in this study. Considering a constant magnetic field, the radiative domain is examined in a lid-driven squared shape cavity with a semicircular heater on the middle part of the bottom wall. The Galerkin residual technique based on finite elements is used to obtain nonlinear dimensionless governing equations. Thermal conductivity along with dynamic viscosity models integrate Brownian motion of nanoparticles. The Reynolds number, the Hartmann number, the Radiation Parameter and the Richardson number are assumed to be constant to account for the fluctuation in solid volume fraction (ϕ = 0% to 10%). The results demonstrated that particle concentration improves the nanofluid’s thermophysical properties from 1 to 9 times than the 0% concentration and the heat transfer rate from 1 to 3 times. In addition, dimensionless time enhances all except the heat transfer rate and drag force of the sliding lid. It is worth noting that the considered parameter exhibits consistent behavior after a while.

In this script, we consider the modified Oskolkov equation in incompressible viscoelastic Kelvin–Voigt fluid and fluid dynamics. A dominant direct algebraic method namely modified simple equation method (MSE) uses to retrieve various dynamical structural solutions of the nonlinear models. As a results, we can derive a disguise version of analytic structural solutions in exponential, hyperbolic, trigonometric and Jacobi elliptic functions with some appreciate parameters. In this work, we depict the physical explanation of cross periodic lump wave, interaction of lump and kink, lump and anti-kink shape solution, double periodic wave, kink, anti-kink shape solution, disguise version of soliton solutions with 3D contour plot and 2D plot.

In this article, we have developed a deterministic Susceptible-Latent-Infectious-Recovered (SLIR) model for diphtheria outbreaks. Here, we have studied a case of the diphtheria outbreak in the Rohingya refugee camp in Bangladesh to trace the disease dynamics and find out the peak value of the infection. Both analytical and numerical investigations have been performed on the model to find several remarkable behaviors like the positive and bounded solution, basic reproductive ratio, and equilibria such as disease extinction equilibrium and disease persistence equilibrium which are characterized depending on the basic reproductive ratio and global stability of the model using Lyapunov function for both equilibria. Parameter estimation has been performed to determine the values of the parameter from the daily case data using numerical technique and determined the value of the basic reproductive number for the outbreak as .

The numerical investigation of mixed convection in a heat exchanger with double-pipe, where the upper lid is given a constant velocity, is presented in this paper. The governing equations, as well as the boundary conditions, define the physical problem. The boundary conditions and governing equations are translated into non-dimensional form and solved by means of a finite-element methodology based on the Galerkin weighted residual. The investigation is conducted for three different governing parameters; namely, Prandtl number (Pr), Richardson number (Ri) and Reynolds number (Re). The outcomes are presented in terms of streamlines, isotherms and average heat transfer rate. Computations are done for a range of Ri (0.01 ≤ Ri ≤ 10), Re (50 ≤ Re ≤ 500), and Pr (1 ≤ Pr ≤ 10) for different dimensionless time (τ = 0.1, 0.5, 1). It is found that the influence of Ri on streamlines, isothermal lines, and heat transfer rate is significant and the maximum heat transfer is gained when values of Ri and Re are higher and Pr is lower. It is also found that the flow strength increases substantially with time and multiple vortices emerge as the convection rate increases.

Double-diffusive mixed convection problem had been a notable topic of research in the last decade. Lack of study is noticed under mixed convection using this double-diffusive in an enclosure specifically for roof-based ventilation systems with unsteady, partially heated conditions. This study investigates the Prandtl number effect in a double-diffusive unsteady flow in a square-shaped room where the upper channel was used for ventilation, and inside the enclosure, the walls are partially heated and mass concentrated. Galerkin residual method was implemented to get the governing equations. The Prandtl number 0.071 ≤ Pr ≤ 7.1 was chosen for the study, changing the dimensionless time 0.1 ≤ τ ≤ 1. The results have been shown in the mode of graphical representation and contour plots to observe the Prandtl number effect in such problem. The velocity streamline, isotherm, isoconcentration plots, heat and mass transfer rate, average fluid temperature and average mass concentration, etc., are shown. Results showed that heat and mass transfer rise with the growth of the Prandtl number. This research would be the guide for the design consideration in such double-diffusive systems like roof-based air ventilation systems with partial heat and mass generation.

This article deals with a numerical analysis of magneto hydrodynamic natural convection phenomena in a prismatic heat exchanger containing Cu-nanoparticles with water as a base fluid. The nature of this fluid flow is steady, and a heat exchanger is created by involving two uniform cylinders with the same shape in a prismatic cavity. To create the heat exchanger and the buoyancy force, a hot cylinder and another cool cylinder are taken a left and right sides respectively. All other walls of this prismatic cavity are kept as heat insulation. For solving governing equations, the finite element technique is applied by involving Galerkin weighted residual method. The impact of the magnetic field, the buoyancy force, and the size of nanoparticles are explained with the help of involving parameters Rayleigh number (Ra), Hartmann number (Ha), and the nanoparticles volume fraction (ϕ). These parameters are graphically and physically explained by using streamline, isotherm, and heatline contours. The Nusselt number is also calculated and explained in detail to show the validation of this investigation. The outcomes indicate that the natural convective heat and energy transfer are improved by increasing the Rayleigh number, and the nanoparticle volume fraction. Reverse behavior is noticed for the upsurge of Hartmann number. The findings of this heat exchanger model can be pragmatic to construct an operative cooling system for numerous shape mechanical chambers.

This study presents a modification form of modified simple equation method, namely new modified simple equation method. Multiple waves and interaction of soliton solutions of the Phi-4 and Klein-Gordon models are investigated via the scheme. Consequently, we derive various novels and more general interaction, and multiple wave solutions in term of exponential, hyperbolic, and trigonometric, rational function solutions combining with some free parameters. Taking special values of the free parameters, interaction of two dark bells, interaction of two bright bells, two kinks, two periodic waves, kink and soliton, kink-rogue wave solutions are obtained which is the key significance of this method. Properties of the achieved solutions have many useful descriptions of physical behavior, correlated to the solutions are attained in this work through plentiful 3D figures, density plot and 2D contour plots. The results derived may increase the prospect of performing significant experimentations and carry out probable applications.

Extensive research has focused on mixed convection in a lid-driven cavity. However, the thermal characteristics of a hollow cylinder made of low-conducting materials such as wood, high-density plastic, bricks, and concrete are not explored, even though they have a lot of real-world applications in engineering and technology. Thus the thermal performance of a hollow cylinder with several low conductive materials such as wood, high-density plastic, bricks, and concrete is investigated in this study. A square enclosure with a moving lid and partially cooled right vertical wall is considered. A heated hollow cylinder is placed in the center of the enclosure as a heat source. A set of governing equations with the appropriate initial and boundary conditions of this model is solved using finite element approach based on Galerkin weighted residual method. The numerical study is performed by varying the thermal conductivity (K) of the cylinder’s materials with transient conditions, and outcomes are depicted in terms of streamlines, isotherms, and different thermo-hydrodynamic properties such as average shear rate, drag force, average rate of heat transfer and heat absorption rate. The results indicated that the heat transfer rate of wood is initially greatest at the top but gradually decreases to the rock-bottom. Furthermore, the heat absorption rate of all materials decreases significantly over time; on average, wood absorbs 31% more heat than concrete.

In this study, heat transfer in a tall, rectangular permeable cavity with active thermal walls is investigated. Inside the enclosure, the two side walls' central portions are partially cooled at a fixed temperature. The central portion of the footwall is heated. Additionally, the top wall, the remaining portion of the footwall, and the side walls are insulated. The controlling equations are obtained from the Brinkman–Forchheimer-extended Darcy prototypical model using Boussinesq calculations. The leading equations are explained numerically by the finite element Galerkin method of weighted residuals. The computations are executed for some governing and physical parameters. The isotherms, streamlines and average heat transfer rate along with the partially active hot wall are shown for different groupings of governing parameters with respect to dimensionless time (τ). The outcomes indicated that the stream and thermal fields are strongly dependent on the considered parameters. It is also established that the average heat transfer rate is a function of these governing parameters.

This study’s prime goal is to explore new and general analytic solutions to the conformable time-fractional Phi-four equation by utilizing the advanced -expansion approach with the conformable derivative principle’s help. This approach is essentially a modified particular case of the generalized form of -expansion method. Concerning hyperbolic and trigonometric function solutions, an enormous class of modern precise moving wave solutions is revealed. Some interaction solutions obtained, such as lump-periodic and lump-periodic-kink structure and periodic wave, are seen by showing their three-dimensional (3D) and two-dimensional (2D) combination plots with the aid of computational software Maple. We have portrayed the figures of the estimated solutions to understand the physical phenomena.

An analytical investigation is performed on soliton, lump wave solution, and rogue waves in the Klein–Gordon with quadratic nonlinearity through the extended tanh approach, which possesses complicated wave propagation arising in the field of nonlinear optics, theory of quantum field and solid state physics. As a result, an advanced form of interacting analytical solutions is achieved with some unrestricted parameters. Different conditions on the existing parameters of these solutions are found after analyzing its dynamic behavior. Based on the conditions, different type of rogue wave, bright bell and dark bell shape nature of the solutions are considered. The dynamics nonlinear wave solutions are showed in 3D plots with specific values of the existing parameters. Moreover, it is shown that nonlinear wave packets are localized in two dimensions with characteristics of rogue wave profiles.

A numerical analysis has been investigated for unsteady natural convection heat transfer problem in diamond shaped cavity. Carbon nanotube (CNT)-water nanofluid is used to analyze the heat transfer and compared with the pure water as working fluid. The lower right wall is kept at heated surface with constant heat flux and upper left wall is kept cold. Both the bottom left and upper right wall are kept adiabatic. The considered physical problem is characterized mathematically by a set of governing equations along with the corresponding boundary conditions. Galerkin weighted residual method of finite element analysis is used to solve the governing equations. Calculations are done for pertinent parameters such as the solid volume fraction (ϕ = 0.0–0.12) and Rayleigh number (Ra = 104–108) at dimensionless times (τ = 0.01 and 1.0) and fixed Pr = 6.2. Obtained results are presented in terms of average Nusselt number and maximum fluid temperature, streamlines and isotherms in graphs, charts and related contours. It has been observed that increase of Rayleigh number and solid volume fraction enhance the heat transfer rate and reduce the maximum fluid temperature. In this investigation the highest amount of overall heat transfer reaches up to 115%.

A computational study has been done to investigate the heat and fluid flow in an open channel with a semicircular heater on the bottom wall under the effect of magnetic field. The walls of the channel are adiabatic while the semicircular heater in the bottom wall is kept at a constant temperature. The inlet and outlet are fixed at the left and right side of the channel. The governing equations are solved by using Galerkin weighted residual finite element technique. In this investigation the involved parameters are Reynolds number, Prandtl number, Hartmann number and joule heating parameter. The effect of Reynolds number, Prandtl number and joule heating parameter for different Rayleigh numbers are investigated while the magnetic parameter Ha (Hartmann number) is kept fixed at 10. The results show that at higher Rayleigh number, joule effect parameter can be utilized to control heat and fluid flow fields. In addition, the effect of Reynolds number on the heat and flow fields becomes insignificant at higher values of Rayleigh number. Finally, Prandtl number is found to have a positive effect on heat transfer rate.

In the present study, natural convection flows in a porous enclosure with a heater on the bottom wall have been investigated numerically. To change the heat transfer in the cavity, a heater is placed at different locations on the bottom wall of the cavity, while the top wall is considered to be cold and the vertical walls are kept adiabatic. The governing equations are obtained by applying the Brinkman extended Darcy flow Model and Boussinesq approximation to characterize heat flow paths along with the heat transfer rate. Finite element method is used to solve the dimensionless governing equations with the specified boundary conditions. The parameters leading the problem are the Rayleigh number (Ra), Darcy number (Da), Thermal conductivity ratio of porous media (k), Prandlt number (Pr), length and location of the heater. To observe the effects of the heater locations at various length of heater on natural convection in the cavity, three different locations of heater at bottom wall for various heater length with different values of Ra varying in the range 104 to 106 are considered. Simulated results are presented in terms of streamlines, isotherms and average Nusselt number at the hot wall in the cavity for the mentioned parameters. The results show that the length, locations of the heater and Rayleigh number have significant effect on the flow and thermal fields as well as the rate of heat transfer from the heated wall.

Natural convection heat transfer and fluid flow through a two-dimensional rectangular porous enclosure with partially active thermal walls are investigated numerically in this study. A small part of two sidewalls are partially cooled at a constant temperature θc and a small part of the bottom wall is kept at constant temperature θh (θh > θc), where the top and the remaining part of the bottom wall and the side walls of the enclosure are assumed to be insulated. The physical problems are presented mathematically by different sets of governing equations (such as the mass, momentum and energy balance equations) along with the corresponding initial and boundary conditions by using Brinkman-Forchheimer-extended Darcy flow model and the Boussinesq approximation. Galerkin weighted residual method of finite element analysis is implemented to discretise the governing equations. The analysis has been carried out for the cavity Aspect ratio (Ar = 1, 2, 3), the three positions of heating source: left position (Lp), middle position (Mp), right position (Rp) and the three positions of cooling source: top cold (Tc), middle cold (Mc), and bottom cold (Bc) with various heater length and fixed Darcy number (Da), Porosity (ε), Prandtl number (Pr) for two values of Grashof number (Gr = 105 , 106 ) with respect to time (τ). Results are presented in terms of isotherms, streamlines, average Nusselt number along the partially active thermal wall for different combinations of the governing parameters. The results indicate that both the flow and the thermal fields strongly depend on the above-mentioned parameters. The computational results also indicate that the average Nusselt number at the heated part of active wall with respect to time are depending on the aforementioned parameters. The results in terms of average Nusselt number are also shown in tabular form.

Heat transfer enhancement in a two-dimensional inclined lid-driven triangular enclosure utilizing cu-water nanofluids is investigated for various relevant parameters. A model is developed to analyze heat transfer performance of nanofluids inside an enclosure taking into account the force convection parameter, namely Reynolds number, Re. The transport equations are solved numerically using the Galerkin finite element method. Comparisons with previously published work on the basis of special cases are performed and found to be in excellent agreement. Results are obtained for a wide range of parameters such as the Richardson number, Ri, and Reynolds number, Re. Copper-water nanofluids are used with Prandtl number, Pr = 6.2 and Reynolds number, Re is varied from 100 to 500. The streamlines, isotherm plots and the variation of the average Nusselt number at the hot surface as well as average fluid temperature in the enclosure are offered and discussed in detailed. It is observed that the force convection parameter strongly influenced the fluid flow and heat transfer in the enclosure at the considered three convective regimes. Furthermore, the variation of the average Nusselt number at the heated surface is found to increase when Re increases and average fluid temperature in the cavity decreases with the raise of Re.

In existence of concerning magnetic field, heat together with mass transfer features on mixed convective copper-water nanofluid flow through inclined plate is investigated in surrounding porous medium together with viscous dissipation. A proper set of useful similarity transforms is considered as to transform the desired governing equations into a system as ordinary differential equations which are nonlinear. The transformed equations for nanofluid flow include interrelated boundary conditions which are resolved numerically applying Runge-Kutta integration process of sixth-order together with Nach-tsheim and Swigert technique. The numerical consequences are compared together with literature which was published previously and acceptable comparisons are found. The influence of significant parameters like as magnetic parameter, angle for inclination, Eckert number, fluid suction parameter, nanoparticles volume fraction, Schmidt number and permeability parameter on concerning velocity, temperature along with concentration boundary layers remains examined and calculated. Numerical consequences are presented graphically. Moreover, the impact regarding these physical parameters for engineering significance in expressions of local skin friction coefficient in addition to local Nusselt together with Sherwood numbers is correspondingly examine

Ferrofluid convective heat transfer in a square cavity with sinusoidal temperature boundary condition at the bottom wall under the influence of magneto hydrodynamics effect was investigated numerically in the present study. Top wall is maintained at comparatively low temperature and Vertical walls are assumed to be adiabatic. The working fluid is Fe3O4-water nanofluid. Single phase model is used to estimate the behavior of ferrofluid and the Maxwell-Garnet model is used for modeling the effective thermal conductivity and viscosity for the ferrofluid. The Galerkin weighted residual method of finite element analysis is implemented to discretize the governing equations. The analysis has been carried out for solid volume fraction (φ = 0- 0.1) and the Hartmann number (Ha = 0-20) for two values of Rayleigh number (Ra = 105 , 106 ) with respect to dimensionless time (τ = 0.1, 1.0). The results are presented both in terms of streamlines, isotherms and Local Nusselt number through the active thermal wall for different combinations of the governing parameters. The results indicate that both the flow and the thermal fields strongly depend on the above-mentioned parameters. The computational results also indicate that the Local Nusselt number at the heated wall for different time are depending on the aforementioned parameters. The results in terms of local Nusselt number are also shown in tabular form. The present investigation lays down a solid foundation for the investigation of time-varying sinusoidal boundary conditions in MHD convection.

A numerical study of the unsteady mixed convection heat transfer characteristics of an Ag–water nanofluid confined within a square shape lid-driven cavity has been carried out. The Galerkin weighted residual of the finite element method has been employed to investigate the effects of the periodicity of sinusoidal boundary condition for a wide range of Grashof numbers (Gr) (105 to 107) with the parametric variation of sinusoidal even and odd frequency, N, from 1 to 6 at different instants (for τ = 0.1 and 1). It has been observed that both the Grashof number and the sinusoidal even and odd frequency have a significant influence on the streamlines and isotherms inside the cavity. The heat transfer rate enhanced by 90% from the heated surface as the Grashof number (Gr) increased from 105 to 107 at sinusoidal frequency N = 1 and τ = 1.

Purpose The purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically. Design/methodology/approach Non-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations. Findings Both the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer. Originality/value Calculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

A lid-driven L-shaped cavity filled with a porous medium is analyzed. The Galerkin weighted residual method is applied to obtain numerical solutions. The effect of the Reynolds number (Re = 1–100), Grashof number (Gr = 103–105) and Darcy number (Da = 10−5–10−3) on the velocity and temperature fields is examined. For the vertical wall, a higher heat transfer rate is observed when a low Grashof number, higher Darcy number and higher Reynolds number are applied, but the opposite characteristic is found in the horizontal wall. It is evident that heat transfer decreases up to 63% in the horizontal wall when the flow has a high Reynolds number (Re = 100).

In this paper, the influence of Joule heating and magneto-hydrodynamics on mixed convection in a lid-driven cavity along with a heated hollow circular plate placed at the centre of the square cavity is investigated. The governing equations which are derived by considering the effects of both Joule heating and magneto-hydrodynamics are solved via the penalty finite-element method with the Galerkin-weighted residual technique. The effects of the Richardson number and Hartmann number arising from the MHD and Joule heating on the flow and heat transfer characteristics have been examined. The results show that the flow behavior, temperature distribution and heat transfer inside the cavity are strongly affected by the presence of the magnetic field. On the other hand, only the temperature distribution and heat transfer inside the cavity are strongly affected by the Joule heating parameter. The results also show that if the Hartmann number is increased from 5 to 100 then the heat transfer detraction is 20%, and if the Joule heating parameter is increased from 1 to 5 then the heat transfer detraction is 58%. In addition, multiple regressions among the various parameters are obtained.

Mixed convection has been a center point of attraction to the heat transfer engineers for many years. Here, pure mixed convection analysis in cavity is carried out for two different geometric heater configurations under externally applied magnetic field. Ferrofluid (Fe3O4–water) is considered as working fluid and modeled as single phase fluid. The heaters at the bottom wall are kept at constant high temperature while vertical side walls are adiabatic. The top wall is moving at a constant velocity in both geometric configurations and is kept at constant low temperature. Galerkin weighted residuals method of finite element analysis is implemented to solve the governing equations. The analysis has been carried out for a wide range of Richardson number (Ri = 0.1–10), Reynolds number (Re = 100–500), Hartmann number (Ha = 0–100) and solid volume fraction (φ = 0–0.15) of ferrofluid. The overall heat transfer performance for both the configurations is quantitatively investigated by average Nusselt number at the heated boundary wall. It is observed that higher Ri enhances the heat transfer rate, although higher Ha decreases heat transfer rate. Moreover, at higher Ri and lower Ha, semi-circular notched cavity shows significantly better (more than 30%) heat transfer rate.

This paper presents unsteady numerical results of double diffusive mixed convection flow in a trapezoidal enclosure with the uniform magnetic field effect applied in negative horizontal direction. At the bottom wall, the uniform and non-uniform heat and mass are applied while the heat and mass absorbed uniformly at the top wall. Other side walls are impermeable and adiabatic. The top wall moves along x-axis direction with a constant velocity. The transport phenomenon of this problem can be expressed by the coupled governing equation derived from the conservation of mass and momentum along with the energy equation for temperature and concentration. The finite element method (FEM) based on Galerkin weighted residual technique is used to compute the numerical result from these governing equations. The numerical computation is carried out for Lewis number (Le = 0.1–50) and Richardson’s number (Ri = 0.1–100). Computed numerical results of mass, temperature and velocity distribution are expressed graphically as iso-concentration lines, isotherm lines and streamlines respectively. Average Sherwood and Nusselt number values are used to show the mass and heat transfer rate from the heated and concentrated surface of the enclosure. It is found from the analysis that mass transfer strongly depends on Lewis number. Heat and mass transfer for uniformly heated and concentrated bottom wall is larger than the non-uniformly heated and concentrated bottom wall. Finally, a correlation has been done for average Nusselt and Sherwood numbers for both of the cases.

A finite element solution has been performed in this work to solve unsteady governing equations of natural convection in a carbon nanotube–water-filled cavity with inclined heater. The temperature of ceiling and left vertical walls is lower than that of the heater while the other walls are adiabatic. The main governing parameters are nanofluid volume fraction and Rayleigh number (Ra). It is found that the heat transfer rate shows different trends based on Rayleigh number and it increases with increase in nanoparticle volume fraction. It has been estimated that average Nusselt number (Nu) is dependent onRa through power regression models with strong positive linear correlation between ln (Nu) and ln (Ra). In particular, for the maximum time, when the solid volume fraction is varied from 0 to 0.1 the dependence between average Nu and linear Ra, on a logarithmic scale, is very high.

Purpose The purpose of this paper is to make a numerical analysis on unsteady analysis of natural convection heat and mass transfer to obtain flow field, temperature distribution, and concentration distribution. Design/methodology/approach A finite element method is applied to solve governing equations of natural convection in curvilinear-shaped system for different parameters as thermal Rayleigh numbers (103=RaT=106), inclination angle (0°=φ=60°) and Hartmann numbers (0=Ha=100). Findings Both magnetic field and inclination angle can be used as control parameter on heat and mass transfer. Flow strength decreases almost 100 percent between Ha=0 and Ha=100 on behalf of the higher values of thermal Rayleigh number. Originality/value The originality of this work is to application of magnetic field on time-dependent natural convection flow, heat and mass transfer for curvilinear geometry.

In this paper unsteady magnetohydrodynamic convection has been analyzed using numerical and statistical techniques for a semicircular-shaped enclosure filled with ferrofluid. Cobalt–kerosene ferrofluid is considered for the present investigation. Galerkin weighted residuals method of finite element ananlysis is adopted for the numerical simulation. The effects of Rayleigh number (Ra), solid volume fraction (ϕ) of ferrofluid and Hartmann number (Ha) are considered as pertinent parameters and varied for a wide range of values (Ra = 105–107, ϕ = 0–0.15, Ha = 0–50) to capture the flow and thermal interaction phenomena for unsteady situation. It is observed that higher ferrofluid solid volume fraction escalates the heat transfer rate. Enhancing the intensity of external magnetic field (by increasing Ha) retards the heat transfer rate while increment of Ra exhibits improvement on the heat transfer. It is also found that there exists a strong interacion between ferromagnetic particle (cobalt) and base fluid (kerosene) in the presence of magnetic field which can be utilized properly for desired heat transfer augmentation. If ϕ = 0 (or ϕ = 0.15) and Ra is changed from 105 to 107, the linear dependence of Ψmax on time changes significantly from positive to negative.

In this paper we concentrate to investigate the double diffusive unsteady mixed convection flow in a trapezoidal enclosure in presence of magnetic field. The bottom wall of the enclosure is heated and concentrated uniformly (case-I) and non-uniformly (case-II) while the top wall is cooled and moved uniformly with a constant velocity. Both side walls are adiabatic and impermeable. The coupled governing equations for this phenomenon is solved numerically using weighted residual based Galerkin technique of finite element method (FEM) for Richardson’s number (Ri = 0.1–100) and Buoyancy ratio (Br = −10 to 10) at time τ = 1.0. Reynolds number, Prandtl number and Lewis number are fixed at 100, 0.71 and 10 respectively. Streamlines, isotherm lines and iso-concentration lines are used to show the result graphically for velocity, temperature and mass distribution respectively. Nusselt and Sherwood number values are presented graphically to show the heat and mass transfer rate from the bottom surface of the cavity.

A computational study has been done in this work to investigate the effects of CNT–water nanofluids in a cavity with non-isothermal heating for higher Rayleigh numbers. The cavity is heated non-isothermally from the bottom wall, it is cooled from the top and vertical walls are adiabatic. Unsteady governing equations of heat and fluid flow are solved via finite element method for different parameters as Ra and ϕ. A correlation analysis shows that, when controlling for ϕ and τ there was a significant correlation between Ra and Num. When controlling for Ra and ϕ there was a negative significant correlation between τ and Num. It is also found that there is an optimum value for nanofluid volume fraction to control heat transfer, temperature distribution and flow field.

This article focuses on exploring the effect of the magnetic field on double-diffusive mixed convection in a lid-driven trapezoidal enclosure with a uniformly (case I) and nonuniformly (case II) heated and concentrated bottom wall for unsteady flow. Numerical calculation is carried out for different values of Hartmann number (0–50) and Richardson number (0.1–100) at time τ = 1 using the finite-element method (FEM). Temperature, mass, and velocity distribution are expressed graphically as heatlines, masslines, and stream function, respectively. Nusselt and Sherwood numbers are used to demonstrate heat and mass transfer graphically. Better heat and mass transfer were found for case I than case II.

Numerical study of heat transfer phenomena has become a major field of research nowadays. In engineering applications, different boundary conditions arise which have various effects on heat transfer characteristics. For the present work, a triangular-shape cavity has been analyzed for the sine-squared thermal boundary condition which is common in practical cases. The augmentation of heat transfer has been done by introducing a nanofluid inside the cavity. Different solid volume fractions (φ = 0, 0.05, 0.1, 0.2) of water-CuO, water-Al2O3, and water-TiO2 nanofluid have been tested for the cavity with a wide range of Rayleigh number (Ra = 105–108) and for dimensionless time (τ = 0.1 to 1). The Galerkin weighted residual finite-element method has been applied for the numerical solution, and numerical accuracy has been checked by code validation. The heat transfer augmentation for different nanofluids has been done in the light of local (NuL) and overall Nusselt number (Nuav), and the results have been presented with streamline, isotherm, and related contours, in graphs and charts. It has been found that variable boundary condition has significant effect on flow and thermal fields and increase of solid volume fraction enhances the heat transfer.

A computational work is performed to investigate the transient heat and mass transfer inside a ventilated enclosure. The enclosure has two ventilation ports as inlet and outlet. Three different configurations are tested according to location of outlet ports while location of inlet port is fixed. In case 1, the outlet port is located on the top of the left vertical wall, in case 2 at the right and case 3 at the middle of the ceiling. Finite element method is employed to solve the governing equations of flow, heat and mass transfer. Also, the heatline and massline techniques are used to visualize the heat and mass transfer patterns. Obtained results show the evolution of various contours of stream function, isotherms and iso-concentrations as well as various parameters such as Nu and Sc numbers. It is found in particular that in order to reach highest heat and mass transfer rates for Gr = 107, the outlet port should be located near the top of the left vertical wall. On the other hand, the effect of outlet location is insignificant for the lower values of Gr.

A computational study has been performed on natural convection heat transfer and fluid flow in a porous media filled enclosure with semi-circular heaters by using finite element method. The ceiling of the cavity moves with a constant velocity and it is insulated. Vertical walls temperature is lower than that of heaters. Results are presented via streamlines, isotherms, average Nusselt numbers, and cross-sectional velocity for different governing parameters such as Richardson and Darcy numbers, and dimensionless time. It is observed that both circulation of the flow and heat transfer is strongly affected with time increment and Darcy number inside the cavity.