Hall current effect on MHD mixed convection flow from an inclined continuously stretching surface with blowing/suction and internal heat generation/absorption

Hydromagnetic heat transfer by mixed convection along an inclined continuously stretching surface, with power-law variation in the surface temperature or heat flux, in the presence of Hall current and internal heat generation/absorption has been studied. The surface is considered to be permeable to allow fluid suction or blowing, and stretching with a surface velocity varied according to a power-law. Two cases of the temperature boundary conditions were considered at the surface. The governing equations have been transformed into non-similar partial differential equations which have been integrated by the forth-order Runge–Kutta method. The effect of Hall parameter, magnetic parameter, dimensionless blowing/suction parameter, space and temperature dependent internal heat generation/absorption parameters and buoyancy force parameters on the temperature, primary and secondary flow velocity have been studied parametrically. All parameters involved in the problem affect the flow and thermal distributions except the temperature-dependent internal heat generation/absorption in the case of prescribed heat flux (PHF). Numerical values of the local skin-friction and the local Nusselt numbers for various parametric conditions have been tabulated.     

Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation

A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet

This article presents a numerical solution for the magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a semi-infinite non-isothermal stretching sheet with internal heat generation/absorption. The flow is caused by linear stretching of a sheet from an impermeable wall. Thermal conductivity is assumed to vary linearly with temperature. The governing partial differential equations of momentum and energy are converted into ordinary differential equations by using a classical similarity transformation along with appropriate boundary conditions. The intricate coupled non-linear boundary value problem has been solved by Keller box method. It is important to note that the momentum and thermal boundary layer thickness decrease with increase in the power-law index in presence/absence of variable thermal conductivity.     

MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

In this work, the effects of slip velocity on the flow and heat transfer for an electrically conducting micropolar fluid over a permeable stretching surface with variable heat flux in the presence of heat generation (absorption) and a transverse magnetic field are investigated. The governing partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformation, which is solved numerically using the Chebyshev spectral method. The effects of the slip parameter on the flow, micro-rotation and temperature profiles as well as on the local skin-friction coefficient, the wall couple stress and the local Nusselt number are presented graphically. The numerical results of the local skin-friction coefficient, the wall couple stress and the local Nusselt number are given in a tabular form and discussed.     

Unsteady heat and mass transfer over a vertical stretching sheet in a parallel free stream with variable wall temperature and concentration

Our aim in this article is to investigate numerically the unsteady two‐dimensional mixed convection flow along a vertical semi‐infinite stretching sheet in a parallel free stream with a power‐law wall temperature and concentration distributions of the form T _w (x) = T_∞ + Ax^2m?1 and C_w (x) = C_∞ + Bx^2m?1, where A, B and m are constants. The unsteadiness in the flow is caused by the time dependent stretching sheet as well as by the free stream velocity. The governing nonlinear partial differential equations in the velocity, temperature and concentration fields are written in nondimensional form using suitable transformations. The final set of resulting coupled nonlinear partial differential equations is solved using an implicit finite‐difference scheme in combination with a quasi‐linearization technique. The effects of various governing parameters on the velocity, temperature and concentration profiles as well as on the skin friction coefficient, local Nusseltnumber and local Sherwood number are presented and discussed in details. The computed numerically results are compared with previously reported work and are found to be in excellent agreement. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions

In this paper, we investigate the heat transfer of a viscous fluid flow over a stretching/shrinking sheet with a convective boundary condition. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameter on the temperature profiles and wall heat flux are presented and discussed. The solution is given as an incomplete Gamma function. It is found the convective boundary conditions results in temperature slip at the wall and this temperature slip is greatly affected by the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameters. The temperature profiles in the fluid are also quite different from the prescribed wall temperature cases.     

The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions

This article examines the magnetohydrodynamic (MHD) flow of non-Newtonian nanofluid in a pipe. The temperature of the pipe is assumed to be higher than the temperature of the fluid. In particular two temperature dependent viscosity models, have been considered. The nonlinear partial differential equations along with the boundary conditions are first cast into a dimensionless form and then the equations are solved by homotopy analysis method (HAM). Explicit analytical expressions for the velocity field, the temperature distribution and nano concentration have been derived analytically. The effects of various physical parameters on velocity, temperature and nano concentration are discussed by using graphical approach.     

Viscous flow over a shrinking sheet with an arbitrary surface velocity

In this paper, an analytical solution in a closed form for the boundary layer flow over a shrinking sheet is presented when arbitrary velocity distributions are applied on the shrinking sheet. The solutions with seven typical velocity profiles are derived based on a general closed form expression. Such flow is usually not self-similar and the solution can only be implemented when the mass transfer at the wall is prescribed and determined by the moving velocity of the wall. The characteristics of the flows with the typical velocity distributions are discussed and compared with previous similarity solutions. The flow is observed to have quite different behavior from that of the self-similar flow reported in the literature and the results demonstrate distinctive momentum and energy transport characteristics. Some plots of the stream functions are also illustrated to show the difference in flow field between the shrinking sheet and the stretching sheet. An integral approach to solve boundary layer flow over a shrinking or stretching sheet with uncoupled arbitrary surface velocity and wall mass transfer velocity is outlined and the effectiveness of this approach is discussed.     

Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: A Lie group analysis

In this paper, heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing is investigated. The governing partial differential equation and auxiliary conditions are converted to ordinary differential equations with the corresponding auxiliary conditions via Lie group analysis. The boundary layer temperature, concentration and nanoparticle volume fraction profiles are then determined numerically. The influences of various relevant parameters, namely, thermophoresis parameter Nt, Brownian motion parameter Nb, Lewis number Le, suction/injection parameter S, permeability parameter k₁, source/sink parameter λ and Prandtl parameter Pr on temperature and concentration as well as wall heat flux and wall mass flux are discussed. Comparison with published results is presented.     

A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Lie group analysis for the effect of temperature-dependent fluid viscosity with thermophoresis and chemical reaction on MHD free convective heat and mass transfer over a porous stretching surface in the presence of heat source/sink

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. Impact of thermophoresis particle deposition with chemical reaction in the presence of heat source/sink plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.     

GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid,assuming that it is in the thermodynamically sense perfect and polytropic.The fluid is between a static solid wall and a free boundary connected to a vacuum state.We take the homogeneous boundary conditions for velocity,microrotation and heat flux on the solid border and that the normal stress,heat flux and microrotation are equal to zero on the free boundary.The proof of the global existence of the solution is based on a limit procedure.We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.     

Numerical study of a mathematical model of a catalytic fuel processor with cocurrent oxidation and conversion flows

The results are presented of the numerical study of a mathematical model in the form of a nonlinear boundary value problem describing the stationary regimes in a catalytic fuel processor. We study a two-dimensional model for the endoblock, with the longitudinal heat and mass transfer by the gas and the transversal heat conductivity along the catalyst in the two-temperature approximation. For the exochannel, a model is considered with the longitudinal heat and mass transfer by the gas flow and the longitudinal heat transfer along the catalytic wall. These two blocks are related to each other through the equality of the temperature and heat flux on the boundary. The results obtained are in good agreement with experimental data.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink

An analysis has been carried out to study the flow and heat transfer characteristics for MHD viscoelastic boundary layer flow over an impermeable stretching sheet with space and temperature dependent internal heat generation/absorption (non-uniform heat source/sink), viscous dissipation, thermal radiation and magnetic field due to frictional heating. The flow is generated due to linear stretching of the sheet and influenced by uniform magnetic field, which is applied vertically in the flow region. The governing partial differential equations for the flow and heat transfer are transformed into ordinary differential equations by a suitable similarity transformation. The governing equations with the appropriate conditions are solved exactly. The effects of viscoelastic parameter and magnetic parameter on skin friction and the effects of viscous dissipation, non-uniform heat source/sink and the thermal radiation on heat transfer characteristics for two general cases namely, the prescribed surface temperature (PST) case and the prescribed wall heat flux (PHF) case are presented graphically and discussed. The numerical results for the wall temperature gradient (the Nusselt number) are presented in tables and are discussed.     

Cattaneo–Christov heat flux model for three-dimensional magnetohydrodynamic flow of an Eyring Powell fluid over an exponentially stretching surface with convective boundary condition

The present model concentrates on three-dimensional steady incompressible flow of an Eyring-Powell nanofluid past an exponentially stretching sheet with magnetic field. The Cattaneo–Christov heat flux with convective boundary condition is accounted for. Shooting method is the instrumental for obtaining numerical solution of the transformed-converted system of the mathematical models. Behavior of the determining thermo-physical parameters on the velocity, temperature, skin friction, heat transfer rate, and finally isotherms are considered. The major relevant outcomes of the current investigation are that increment in Eyring-Powell parameter uplifts flow velocity, while that peters out the fluid temperature. Enhanced values of the mixed convection parameter weakened the skin friction coefficient while it slightly strengthened the rate of heat transfer.     

Momentum and heat transfer of the Falkner–Skan flow with algebraic decay: An analytical solution

In this paper, an analytical solution of the Falkner–Skan equation with mass transfer and wall movement is obtained for a special case, namely a velocity power index of ?1/3, with an algebraically decaying velocity profile. The solution is given in a closed form. Under different values of the mass transfer parameter, the wall can be fixed, moving in the same direction as the free stream, or opposite to the free stream (reversal flow near the wall). The thermal boundary layer solution is also presented with a closed form for a prescribed power-law wall temperature, which is expressed by the confluent hypergeometric function of the second kind. The temperature profiles and the wall temperature gradients are plotted. Interesting but complicated variation trends for certain combinations of controlling parameters are observed. Under certain parameter combinations, there exist singular points or poles for the wall temperature gradients, namely wall heat flux. With poles, the temperature profiles can cross the zero temperature line and become negative. From the results, it is also found empirically that under certain given values of the Prandtl number (Pr) and flow controlling parameter (b), the number of times for the temperature profiles crossing the zero line is the same as the number of poles when the wall temperature power index varies between zero and a given value n. The current result provides a new analytical solution for the Falkner–Skan equation with algebraic decay and greatly enriches the understanding of this important equation as well as the heat transfer characteristics for this flow.     

A common error made in investigation of boundary layer flows

The concept of boundary layer flow, introduced in 1904 by Prandtl, is a popular field in Fluid Mechanics for engineers, physicists and mathematicians. The present work is a critique to many papers published in the last 15 years in the field of boundary layer flow. The critique concerns the shape of velocity, temperature and concentration profiles which are truncated due to small calculation domain used during the numerical solution procedure. These truncated profiles are not compatible with the boundary layer theory and introduce errors in wall shear stress and wall heat transfer values.