A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-Padé

In this study, the problem of mixed convection about an inclined flat plate embedded in a porous medium is performed. The similarity transformations are applied to reduce governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. An efficient mathematical technique, called the differential transform method (DTM), is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Then, Padé approximant is applied to the solutions to increase the convergence of given series. It has been attempted to show the reliability and performance of the DTM in comparison with the numerical method (fourth-order Runge–Kutta) in solving this problem. The obtained solutions, in comparison with the numerical solutions admit a remarkable accuracy.     

Quartic autocatalysis of homogeneous and heterogeneous reactions in the bioconvective flow of radiating micropolar nanofluid between parallel plates

This study deals with the quartic autocatalysis of homogeneous–heterogeneous chemical reaction that occurs in the bioconvective flow of micropolar nanofluid between two horizontally parallel plates. The quartic autocatalysis is found to be more effective than cubic autocatalysis since the concentration of the homogeneous species is substantially high. The upper plate is assumed to be in motion and the lower plate is kept stationary. Such a flow of micropolar fluid finds application in the pharmaceutical industry, microbial enhanced oil recovery, hydrodynamical machines, chemical processing, and so forth. The governing equations for this flow are in the form of the partial differential equation and their corresponding similarity transformation is obtained through Lie group analysis. The governing equations are further transformed to coupled nonlinear differential equations that are linearized through the Successive linearization method and are solved using the Chebyshev Collocation method. The effects of various parameters, such as micropolar coupling parameter, spin gradient parameter, reaction rates, and so forth, are analyzed. It is observed that the fluid flows with a greater velocity away from the channel walls, whereas near the channel walls the velocity decreases with an increase in the coupling parameter. Furthermore, the spin parameter increases the spin gradient viscosity that reduces the microrotation of particles that further decreases the microrotation profile.     

Optimal Homotopy Asymptotic Solutions for Nonlinear Ordinary Differential Equations Arising in Flow and Heat Transfer due to Nonlinear Stretching Sheet

Optimal homotopy asymptotic method (OHAM) is used to obtain solutions for nonlinear ordinary differential equations (ODEs) arising in fluid flow and heat transfer at a nonlinear stretching sheet. The solutions for skin friction and temperature gradient for some special cases are tabulated and compared with the available numerical results in the literature. Moreover, OHAM is found to be very easy to use and the technique could be used for solving coupled nonlinear systems of ordinary differential equations arising in science and engineering.     

A novel analytical solution of heat transfer of a micropolar fluid through a porous medium with radiation by DTM‐Padé

In this paper, the differential transform method (DTM) was applied to heat transfer of a micropolar fluid through a porous medium with radiation. The governing equations can be written as a system of nonlinear ordinary differential equations. The approximate solutions of these equations were obtained in the form of series with easily computable terms. Then, Padé approximant was applied to increase the convergence rate of the series. The results obtained in this study were compared with the numerical results (fourth‐order Runge–Kutta method). © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res, 39(8), 575–589, 2010; Published online 26 July 2010 in Wiley Online Library ( wileyonlinelibrary.com ). DOI 10.1002/htj.20317     

Bioconvection in a squeezing flow of a micropolar fluid in a horizontal channel

The bioconvection flow of an incompressible micropolar fluid containing microorganisms between two infinite stretchable parallel plates is considered. A mathematical model, with a fully coupled nonlinear system of equations describing the total mass, momentum, thermal energy, mass diffusion, and microorganisms is presented. The governing equations are reduced to a set of nonlinear ordinary differential equations with the help of suitable transformations. The resulting nonlinear ordinary differential equations are linearized using successive linearization method, and the resulting system of linear equations is solved using the Chebyshev collocation method. The detailed analysis illustrating the influences of various physical parameters, such as the micropolar coupling number, squeezing parameter, the bioconvection Schmidt number, Prandtl numbers, Lewis number, and bioconvection Peclet number on the velocity, microrotation, temperature, concentration and motile microorganism distributions, skin friction coefficient, Nusselt number, Sherwood number, and density number of motile microorganism, is examined. The influence of the squeezing parameter is to increase the dimensionless velocities and temperature and to decrease the local Nusselt number and local Sherwood number. The density number of motile microorganism is decreasing with squeezing parameter, bioconvection Lewis number, bioconvection Peclet number, and bioconvection Schmidt number.     

Analytical modeling of heat convection in magnetized micropolar fluid by using modified differential transform method

In this study, a new analytical method (DTM‐Padé) and the numerical method (by using a fourth‐order RungeBKutta and shooting method) were compared to solve convective heat transfer for a micropolar fluid in the presence of uniform magnetic field. It was shown that the differential transform method (DTM) solutions are only valid for small values of independent variables; therefore the DTM is not applicable for solving magnetohydrodynamic (MHD) boundary‐layer equations. The new method (DTM‐Padé) has removed this problem. Numerical comparisons between the DTM‐Padé and the numerical method revealed that the new method is a powerful method for solving MHD boundary‐layer equations. Finally, the analytical and numerical solutions of the problem for different values of the dimensionless parameters are shown simultaneously. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com ). DOI 10.1002/htj.20337     

Interaction of Cu and Ag nanoparticles on MHD water‐based nanofluids past a stretching sheet

The study explores the MHD flow of water‐based nanofluids past a stretching sheet that melts at a constant rate. Cu and Ag nanoparticles are considered to merge into the base fluid to discuss the flow, heat and mass transfer characteristics. Suitable transformation is employed to transform the governing partial differential equations to a system of nonlinear coupled ordinary differential equations (ODEs). A semi‐analytical technique, that is, in particular, the Adomian decomposition method is implemented to tackle this system of ODEs. The influences of characterizing parameters for the flow phenomena are determined via graphs and displayed. Furthermore, the computed values of the quantities of engineering interest are exhibited through tables and discussed. The main findings of the results are laid down as follows: the Cu‐water nanofluid momentum is more pronounced than that of Ag‐water due to the heavier density of the Ag nanoparticles and an increasing melting parameter is favorable to decrease the fluid temperature, which is useful for the cooling of the substances at the final stage of production in industries.     

Lie symmetry analysis and numerical solutions for thermosolutal chemically reacting radiative micropolar flow from an inclined porous surface

Steady, laminar, incompressible thermosolutal natural convection flow of micropolar fluid from an inclined perforated surface with convective boundary conditions is studied. Thermal radiative flux and chemical reaction effects are included to represent phenomena encountered in high-temperature materials synthesis operations. Rosseland's diffusion approximation is used to describe the radiative heat flux in the energy equation. A Lie scaling group transformation is implemented to derive a self-similar form of the partial differential conservation equations. The resulting coupled nonlinear boundary value problem is solved with Runge-Kutta fourth order numerical quadrature (shooting technique). Validation of solutions with an optimized Adomian decomposition method algorithm is included. Verification of the accuracy of shooting is also conducted as a particular case of nonreactive micropolar flow from a vertical permeable surface. The evolution of velocity, angular velocity (microrotation component), temperature, and concentration are examined for a variety of parameters including coupling number, plate inclination angle, suction/injection parameter, radiation-conduction parameter, Biot number, and reaction parameter. Numerical results for steady-state skin friction coefficient, couple stress coefficient, Nusselt number, and Sherwood number are tabulated and discussed. Interesting features of the hydrodynamic, heat and mass transfer characteristics are examined.     

Application of Hermite wavelet method for heat transfer in a porous media

In this article, the flow and heat transfer for non-Newtonian viscoelastic fluid in an axisymmetric channel with a porous wall is investigated. Convective boundary conditions have been used in the problem formulation. We obtain coupled, highly nonlinear ordinary differential equations from the fundamental governing equations via appropriate similarity variables. The solution for velocity and temperature are computed by applying the Hermite wavelet method (HWM). The comparison between the results from the HWM, differential transform method, and numerical method are well in agreement which proves the capacity of HWM for solving such problems. The effects of Reynolds number and Prandtl number on the velocity and temperature are illustrated through graphs and tables for different values of an independent variable.     

Micropolar flow in a porous channel with high mass transfer

In this research the injective micropolar flow in a porous channel is investigated. The flow is driven by suction or injection on the channel walls, and the micropolar model is used to describe the working fluid. This problem is mapped into the system of nonlinear coupled differential equations by using Berman's similarity transformation. These are solved for large mass transfer via Optimal Homotopy Asymptotic Method (OHAM). Also the numerical method is used for the validity of this analytical method and excellent agreement is observed between the solutions obtained from OHAM and numerical results. Trusting this validity, effects of some other parameters are discussed.     

Magnetohydrodynamic micropolar slip flow due to an exponentially stretching sheet with chemical reaction and nonuniform heat generation

The main focus of the current study is to examine the impact of melting heat transfer and chemical reaction on magnetohydrodynamic micropolar fluid flow over a sheet that is exponentially stretching and immersed in a porous medium. A nonuniform heat source is placed within this flow system. Other impacts like slip phenomena and thermal radiation are also taken into consideration. The governing partial differential equations are converted to a system of ordinary differential equations (ODEs) via similarity transformation and we also get the corresponding necessary boundary conditions. These nonlinear ODEs are resolved with the help of shooting technique and an Runge-Kutta fourth order (RK-4) iterative strategy. Also, we solve this problem using the Bvp4c approach for validating the results of the RK-4 method. Both outcomes are consistent with previously published data. With the help of tables and graphs, we examine the influence of multiple physical parameters on velocity, thermal, microrotation, concentration, Nusselt number, Sherwood number, coefficient of skin friction, and wall couple stress. We see that the temperature distribution and velocity profiles decrease when the melting parameter increases. The temperature profile boosts when the heat source parameter is increased.     

A comparative series solutions of Japanese encephalitis model using differential transform method and variational iteration method

In this study, a deterministic mathematical model involving the transmission dynamics of Japanese encephalitis (JE) is presented and studied. The biologically feasible equilibria and their stability properties have been discussed. This study investigates a series of solutions to the system of ordinary differential equations (ODEs) in the transmission dynamics of JE. To get approximate series solutions of the JE model, we employed the differential transform method (DTM) and variational iteration method (VIM). DTM utilizes the transformed function of the original JE model, while VIM uses the general Lagrange multiplier to develop the correction functional for the JE model. The results show that the VIM solution is more accurate than the DTM solution for short intervals of time. In addition, the fractional compartmental model of JE is briefly discussed. We illustrated the profiles of the solutions of each of the compartments, from which we found that the fourth-order Runge–Kutta method solutions are more accurate than the DTM and VIM solutions for long intervals of time.     

The Couette flow of a conducting Jeffrey fluid when the walls are lined with deformable porous material

The paper deals with the flow, past a deformable porous channel bounded by finite deformable porous layer with moving rigid parallel plates. Transverse magnetic field is also applied and incorporated in the momentum equation. The coupled nonlinear equations are transformed to ordinary differential equations (ODEs) with suitable choice of similarity transformation. Further, these sets of nonlinear ODEs are solved analytically and are used to get results for the flow phenomena. The effects of the porous layer thickness and the drag on the flow phenomena are discussed graphically. It is observed that rigid velocity decreases with increasing in the drag, whereas the decrease in the deformable is noted. It is clear to see that the retards in solid displacement are shown with enhancing viscosity parameter η.     

Numerical simulation of heat and mass transfers radiative–convective fluid flow on a stretching porous surface with temperature-dependent thermophysical properties

This paper is focused on the analysis of heat and mass transfer radiative–convective fluid flow using quadratic multiple regression and numerical approach. The physical phenomenon is analyzed by utilizing partial differential equations (PDEs). Thermophysical properties, such as viscosity, thermal conductivity, and mass diffusivity, are varied and temperature-dependent. This study is unique because of its applications in magnetohydrodynamic power accelerators, drilling operators, and bioengineering. The governing PDEs are transformed into coupled nonlinear ordinary differential equations (ODEs). The transformed ODEs are solved numerically using the spectral homotopy analysis method. Also, a quadratic multiple regression analysis is performed on quantities of engineering interest to show the significance of the flow parameters. It is observed that the heat and mass transfer process is affected by nonlinear buoyancy impact. The Lorentz force produced by the imposed magnetic field decline the thickness of the hydrodynamic boundary layer. Findings revealed that the nonlinear convective parameter and variable thermophysical properties are greatly affected by the rate of heat and mass transfer. Previously published work was used to validate the present one, which conformed with it. The slope of linear regression through data points is adopted to show the rate of change in skin friction, Nusselt, and Sherwood numbers during the flow phenomenon.     

Second-Law Analysis for an Inclined Channel Containing Porous-Clear Fluid Layers by Using the Differential Transform Method

In this study, convection in a porous medium for a laminar, incompressible, non-Darcy model flow in an inclined channel has been investigated. The flow field considered is composed of porous and clear viscous layers. The solutions are carried out for both clear fluid and porous regions by using the differential transform method (DTM). For the solutions of governing equations, constant values for some parameters such as angle of inclination (φ), porous parameter (σ), and the ratio of the heights of two layers (h) are assigned. In order to verify the applied solution technique, the results obtained are compared to the already existing ones evaluated by perturbation method. It is noticed that the results by two methods are in agreement for small values of Brinkman number (Br). However, for higher values of Br, the solutions carried out by perturbation method lose accuracy but the results of the DTM are still valid. The entropy generation number (N _s ) is derived and plotted by using dimensionless velocity and temperature profiles. One of the advantages of this study to similar studies is to give an open form series solution, which gives a tractable and easily applicable recurative form of nonlinear field equations. In similar studies, it is said that the equations are solved; however, neither solution technique nor accuracy or applicability of given technique are clear. In this work, these are well documented.     

A study on heat transfer in a second‐grade fluid through a porous medium with the modified differential transform method

This study investigates the boundary‐layer flow and heat transfer characteristics in a second‐grade fluid through a porous medium. The similarity transformation for the governing equations gives a system of nonlinear ordinary differential equations which are analytically solved by the differential transform method (DTM) and the DTM‐Padé. The DTM‐Padé is a combination of the DTM and the Padé approximant. The convergence analysis elucidates that the DTM does not give accurate results for large values of independent variables. Hence the DTM is not applicable for the solution of boundary‐layer flow problems having boundary conditions at infinity. Comparison between the solutions obtained by the DTM and the DTM‐Padé with numerical solution (fourth‐order Runge–Kutta with shooting method) illustrates that the DTM‐Padé is the most effective method for solving the problems that have boundary conditions at infinity. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21030     

Natural convection heat and mass transfer from a sphere in micropolar fluids with constant wall temperature and concentration

This work examines the natural convection heat and mass transfer near a sphere with constant wall temperature and concentration in a micropolar fluid. A coordinate transformation is used to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. For micropolar fluids, higher viscosity tends to retard the flow and thus decreases the natural convection heat and mass transfer rates from the sphere with constant wall temperature and concentration. Moreover, the natural convection heat and mass transfer rates from a sphere in Newtonian fluids are higher than those in micropolar fluids.     

Micropolar fluid flow with temperature‐dependent transport properties

In the present article, the heat transfer rate and the fluid flow of a micropolar fluid along with temperature‐dependent transport properties are scrutinized in the presence of heat generation. The variability in transport properties leads to a rise in the heat transfer and decreases the skin friction. Furthermore, Fourier's heat flux model is implemented in the analysis of heat transfer, employing a suitable transformation to convert the flow model into nonlinear ordinary differential equations. Numerical solutions are obtained by using the shooting method/bvp4c technique. Physical quantities of interest, such as local skin friction and Nusselt number, are discussed and computed. Skin friction decreases with the micropolar parameter but the Nusselt number shows the opposite behavior for the micropolar parameter.     

Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet

This computation reports the mixed convection flow of Williamson fluid past a radially stretching surface with nanoparticles under the effect of first‐order slip and convective boundary conditions. The coupled nonlinear ordinary differential equations (ODEs) are obtained from the partial differential equations, which are derived from the conservation of momentum, energy, and species. Then, utilizing suitable resemblance transformation, these ODEs were changed into dimensionless form and then solved numerically by means of a powerful numerical technique called the Galerkin finite element method. The effect of different parameters on velocity, temperature, and concentration profiles is inspected and thrashed out in depth by graphs and tables. The upshots exhibit that the velocity profile augments as the values of concentration buoyancy and mixed convection parameters are engorged. Also, the results demonstrated that both temperature and concentration profiles are improved with an enhancement in values of thermophoresis parameters. The outcomes also indicate that for finer values of magnetic field parameter and thermophoresis parameter, the numerical value of local Nusselt and Sherwood numbers is reduced.     

Influence of variable viscosity and thermal conductivity,hydrodynamic, and thermal slips on magnetohydrodynamic micropolar flow: A numerical study

Thermophysical and wall‐slip effects arise in many areas of nuclear technology. Motivated by such applications, in this article, the collective influence of variable‐viscosity, thermal conductivity, velocity and thermal slip effects on a steady two‐dimensional magnetohydrodynamic micropolar fluid over a stretching sheet is analyzed numerically. The governing nonlinear partial differential equations have been converted into a system of nonlinear ordinary differential equations using suitable coordinate transformations. The numerical solutions of the problem are expressed in the form of nondimensional velocity and temperature profiles and discussed from their graphical representations. The Nachtsheim‐Swigert shooting iteration technique together with the sixth‐order Runge‐Kutta integration scheme has been applied for the numerical solution. A comparison with the existing results has been done, and an excellent agreement is found. Further validation with the Adomian decomposition method is included for the general model. Interesting features in the heat and momentum characteristics are explored. It is found that a greater thermal slip and thermal conductivity elevate thermal boundary layer thickness. Increasing Prandtl number enhances the Nusselt number at the wall but reduces wall couple stress (microrotation gradient). Temperatures are enhanced with both the magnetic field and viscosity parameter. Increasing momentum (hydrodynamic) slip is found to accelerate the flow and elevate temperatures.