Solving of boundary‐layer equations with transpiration effects,governance on a vertical permeable cylinder using modified differential transform method

The suction and injection effects on the free convection boundary‐layer flow over a vertical cylinder are studied. The main stream velocity and wall temperature are proportional to the axial distance along the surface of the cylinder. Both analytic and numerical solutions of the arising mathematical problem are obtained. An analytic solution is derived by a new analytical method (DTM‐Padé) and numerical solutions have been performed by using a fourth‐order Runge–Kutta and shooting methods. Velocity and temperature profiles are shown graphically. It is shown that the differential transform method (DTM) solutions are only valid for small values of the independent variable but the obtained results by DTM‐Padé are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20366     

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     

MHD boundary layer flow of SWCNT‐water and MWCNT‐water nanofluid over a vertical cone with heat generation/absorption

The analysis is carried out to investigate the magneto hydro dynamics (MHD) boundary layer flow, heat and mass transfer characteristics of two carbon nanotubes, namely, single‐wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs), with water as the base fluid by taking thermal radiation and chemical reaction into consideration. Suitable similarity conversions are employed to reduce nonlinear partial differential equations into the system of ordinary differential equations, and these equations together with boundary conditions are solved numerically using the finite element method. Velocity, temperature, and concentration distributions as well as skin friction coefficient, Nusselt number, and Sherwood number for diverse values of influencing parameters are examined in detail, and the results are displayed graphically and in tabular form. It is found that the rate of heat transfer is remarkably higher in water‐based MWCNTs than the SWCNTs as the value of the nanoparticle volume fraction parameter rises in the boundary layer regime.     

Heat generation/absorption on MHD flow of a micropolar fluid over a heated stretching surface in the presence of the boundary parameter

A model study is reported to examine the effect of magnetic hydrodynamics polar fluid over a semistretched infinite vertical porous surface in the presence of heat source, temperature, magnetic field, and thermal radiation. The governing dimensional partial differential equations are transformed into an ordinary differential equation set by introducing the similarity variables. The reduced model is numerically solved via Runge–Kutta fourth order along with the shooting technique. The effects of various physical parameters on coefficient of skin friction, microrotation coefficient, and Nusselt number are studied whereas the outcomes are explained through a set of graphs. The results obtained are explained in tabular form and graphs. Prandtl and Hartman's numbers enhance the velocity profile while the opposite behavior is noticed for $\phi ,\delta$. Higher values of Pr enlarge the angular velocity near the surface. Improved temperature distribution is noticed for higher values of Ha and ϕ, However, a declined behaviour is observed for $\text{Pr},\delta$, and $fo$.     

3D MHD cross flow over an exponential stretching porous surface

The present study aims to investigate the effects of thermal‐diffusion and diffusion‐thermo onan magnetohydrodynamic convective flow of viscous fluids over an exponentially stretching sheet. Thermal radiation effects are also considered in the study. This analysis is carried out in three dimensions and a similarity transformation is adopted to get a set of ordinary differential equations from a set of partial differential equations. And the Fourth‐order Runge‐Kutta method and shooting technique along with the secant method are employed to find out an iterative solution. We also analyze here the influence of the variable magnetic field, nonuniform permeability and variable chemical reaction on the fluid flow. The impact of various pertinent parameters of interest has extensively been explored through graphs and tables. The major findings of the present study are that resistive Lorentz force diminishes the fluid velocity and uplifts the thermal as well as concentration fields. Inclusion of porous matrix improves the viscous drag force which in turn augments wall shear stresses and peters out the heat and mass transfer rates from the sheet. In addition, the thermal expansion coefficient has an inverse relation with temperature.     

MHD Carreau nanoliquid flow over a nonlinear stretching surface

The Dufour and Soret impacts on magnetohydrodynamic Carreau nanoliquid past a nonlinearly stretching sheet are investigated. Variations in viscosity, heat conductivity, and convective boundary conditions are considered. Suitable similarity conversions are utilized to design the governing equations nondimensional. The Optimal Homotopy Analysis Method is employed to resolve the dimensionless equations. Graphs and tables are utilized to illustrate the impacts of the relevant factors over velocity, temperature, concentration, and streamlines. For the variations of different parameters, numerical values for Nusselt number, Sherwood number, and skin friction are provided in a table. The observed results are in good agreement with the previous literature findings. Furthermore, the current research shows that when the Dufour number increases, the temperature distributions get narrower. However, with increasing Soret number, the concentration distribution has the opposite effect. One of the important outcomes of the current study is that by increasing the Weissenberg number for shear-thinning fluids, one can improve the velocity field.     

MHD heat transfer flow over a moving wedge with convective boundary conditions with the influence of viscous dissipation and internal heat generation/absorption

In this investigation, the problem of the study is the effect of the magnetic field and viscous dissipation on heat transfer flow through a moving wedge in the existence of the internal heat generation/absorption and also suction/injection. The governing equations are changed to some coupled nonlinear differential equations with aid of similarity variables. The numerical calculations of the equations are solved by the MATLAB package solver bvp5c. The changes of the pertinent constraints on the momentum and temperature have been discussed through graphs and numerical values of skin friction and heat transfer factor are listed in the tabular pattern. Although maintaining a constant value for the convection parameter, the Nusselt number is increased for $Q>0$ and decreased for $Q<0$. The temperature rises in conjunction with an increase in $\mathit{Ec}$ and $\mathit{Nc}$ variables.     

MHD free convection flow along a vertical wavy surface with heat generation or absorption effect

We formulate the problem of free convection from a vertical wavy surface embedded in a uniform porous medium in the presence of an external magnetic field and internal heat generation or absorption effects. Using the appropriate transformations, the boundary layer equations are reduced to non-linear partial differential equations. The transformed boundary layer equations are solved numerically using Runge–Kutta integration scheme with the shooting technique. We have focused our attention on the evaluation of the local Nusselt number Nu_x, dimensionless velocity, f′, and temperature, θ. The governing parameters are the amplitude of the waviness of the surface, a, ranging from 0.0 (flat plate) to 0.3, and the heat generation absorption parameter Q ranging from − 0.25 to 0.25, and magnetic parameter Mn, ranging from 0.0 to 2.0. The effect of all these parameters are discussed and plotted.     

Entropy generation on MHD flow of a Casson fluid over a curved stretching surface with exponential space-dependent heat source and nonlinear thermal radiation

The present model concentrates on entropy generation on a steady incompressible flow of a Casson liquid past a permeable stretching curve surface through chemical reaction and magnetic field effects. The exponential space-dependent heat source cum heat and mass convective boundary conditions are accounted for. The resulting nonlinear boundary layer model is simplified by the transformation of similarity. Chebyshev spectral technique is involved for obtaining numerical results of the converted system of the mathematical models. Behavior of the determining thermo-physical parameters on the profiles of velocity, temperature, concentration, skin friction, heat, mass transfer rate, rate of entropy generation, and finally the Bejan number are presented. The major point of the present investigation show that the curvature term weakens the mass transfer profile as the fluid temperature reduces all over the diffusion regime. A decrease in heat generation strengthens the species molecular bond, which prevents free Casson particle diffusion. Furthermore, the mass transfer field diminishes in suction and injection flow medium.     

Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet

In this paper a study is carried out to analyze the unsteady heat transfer effects of viscous dissipation on the steady boundary layer flow past a stretching sheet with prescribed constant surface temperature in the presence of a transverse magnetic field. The sheet is assumed to stretch linearly along the direction of the fluid flow. The assumed initial steady flow and temperature field neglecting dissipation effects becomes transient by accounting dissipation effects when time t′ > 0. The temperature and the Nusselt number are computed numerically using an implicit finite difference method. The obtained steady temperature field with dissipation is of practical importance.     

Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet

Viscoelastic boundary layer flow and heat transfer over an exponential stretching continuous sheet have been examined in this paper. Approximate analytical similarity solution of the highly non-linear momentum equation and confluent hypergeometric similarity solution of the heat transfer equation are obtained. Accuracy of the analytical solution for stream function is verified by numerical solutions obtained by employing Runge-Kutta fourth order method with shooting. These solutions involve an exponential dependent of stretching velocity, prescribed boundary temperature and prescribed boundary heat flux on the flow directional coordinate. The effects of various physical parameters like viscoelastic parameter, Prandtl number, Reynolds number, Nusselt number and Eckert number on various momentum and heat transfer characteristics are discussed in detail in this work.     

Computational Fluid Dynamics model of stratified atmospheric boundary‐layer flow

For wind resource assessment, the wind industry is increasingly relying on computational fluid dynamics models of the neutrally stratified surface‐layer. So far, physical processes that are important to the whole atmospheric boundary‐layer, such as the Coriolis effect, buoyancy forces and heat transport, are mostly ignored. In order to decrease the uncertainty of wind resource assessment, the present work focuses on atmospheric flows that include stability and Coriolis effects. The influence of these effects on the whole atmospheric boundary‐layer are examined using a Reynolds‐averaged Navier–Stokes k‐ ε model. To validate the model implementations, results are compared against measurements from several large‐scale field campaigns, wind tunnel experiments, and previous simulations and are shown to significantly improve the predictions. Copyright © 2013 John Wiley & Sons, Ltd.     

Heat transport on steady MHD flow of copper and alumina nanofluids past a stretching porous surface

An attempt is made to investigate the steady magnetohydrodynamic convective flow of the viscous nanofluid due to a permeable exponentially stretching porous surface. Water is used as a traditional fluid while nanoparticles include copper and alumina. The fluid is electrically conducting, subject to an applied magnetic field with a constant strength. Convective type boundary conditions are employed in modeling the heat transfer process. The nonlinear partial differential equations governing the flow are reduced to an ordinary differential equation by similarity transformations and then solved using the Runge‐Kutta fourth‐order method. A parametric study of the physical parameters is made, and a representative set of numerical results for the velocity and temperature, as well as local shear stress and local Nusselt number, is presented graphically. Hartman number increase diminishes the velocity and has the contrary result in the subjective sense for the mass transfer parameter. An increase in the Prandtl number Pr lessens the temperature and thickness of the thermal boundary layer. The main conclusions have been indicated.     

Heat and mass transfer in MHD boundary layer flow of a second-grade fluid past an infinite vertical permeable surface

Heat and mass transfer of non-Newtonian fluids is increasingly being studied by researchers due to its applications in many branches of science and engineering, such as metallurgical processes, polymer extrusion, glass blowing, crystal growing, and so forth. The present work is mainly concerned with the unsteady laminar magnetohydrodynamic flow of a heat-generating or absorbing second-grade fluid past an infinite vertical porous plate. The nondimensional governing equations are solved for the best analytical solution. Results for various flow characteristics are presented through graphs and tables delineating the effect of various parameters characterizing the flow. For engineering interest, the shear stresses, Nusselt number, and Sherwood number are computed and exchanged of views with reference to the important parameters. Our analysis explored that the influences of a chemical reaction and fluid oscillations reduced the concentration distribution in the entire liquid region. The rotation effect decreases the shear stress, whereas it is augmented through an increase in the permeability of porous medium and second-grade fluid parameters' impact.     

Mono‐diffusive Hadley‐Prats flow in a horizontal porous layer subject to variable gravity and internal heat generation

Analysis of internal heated and gravity effect on the onset of Hadley‐Prats flow in a horizontal porous layer with inclined temperature gradients is investigated using the linear and nonlinear instability analysis. The transformed eigenvalue problem is evaluated numerically to find the eigenvalue, which is treated as a vertical thermal Rayleigh number (R_z). It is evaluated by applying shooting and Runge‐Kutta method. Also, the critical R_z is investigated for different parameters governing the flow. A theoretical study is made to understand the influence of gravity field on the mechanism of mono‐diffusive instability of Hadley‐Prats convection in a fluid saturated horizontal porous layer. Nonlinear stability is evaluated by using energy functional. The comparison between linear and nonlinear instability results are presented and it is noted that linear theory of instability may not be useful to capture the complete picture of stability and instabilities may arise before one attains the linear stability threshold. This subcritical instability region is identified between the linear and energy thresholds in the parameter space of the problem considered.     

Effect of Joule heating on MHD non‐Newtonian fluid flow past an exponentially stretching curved surface

The current endeavor examines the convective heat transfer characteristics on magnetohydrodynamic stagnation point flow of micropolar fluid past an exponential curved surface. The flow is supposed to be laminar and time‐independent. The influence of radiation, irregular heat source/sink, Joule heating, and variable thermal conductivity are supposed. Suitable similarity renovations are considered to transform the original partial differential equations as ordinary ones and then resolved by shooting and fourth‐order Runge–Kutta methods. Graphs are drawn to inspect the impacts of sundry nondimensional parameters on the distributions of velocity, microrotation, and temperature. We detect that there is an escalation in temperature with Eckert number and variable heat source/sink parameters. Also, it is motivating to comment that Biot number is an increasing function of local Nusselt number.     

Influence of undulating wall heat and mass flux on MHD natural convection boundary layer flow from a vertical wall

Current study expounds an unsteady magnetohydrodynamic natural convective flow along a vertical wall in presence of variable transverse magnetic field. Small amplitude undulation in wall heat flux and wall mass flux are imposed at the vertical wall to generate the boundary layer flow. The flow governing equations are divided into sets of steady and unsteady equations and then transformed into the similarity and nonsimilarity equations, respectively, by introducing stream function formulations. The sets of nonsimilarity equations are solved numerically by using three different techniques, namely, perturbation solution technique, asymptotic solution technique and implicit finite difference technique applied, respectively, for lower, higher, and all frequencies (ξ). Results are illustrated in connection with the amplitude and phase angles of shear stress, wall temperature, and concentration against the frequency (ξ) for wide ranges of physically significant parameters. Likening of the results obtained by above mentioned numerical methods are presented in every figure and table. Results reveal that the amplitude of undulating shear stress and wall temperature dwindle and the amplitude of wall concentration increases due to increment in Prandtl number (Pr). Besides, on incrementing Schmidt number (Sc) the amplitude of undulating shear stress and wall concentration dwindle and the amplitude of wall temperature increases. Results also reveal that on incrementing magnetic parameter (M) the amplitude of transient shear stress dwindles while the amplitude of transient wall temperature and concentration increase.