SLIP EFFECT ON DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A VERTICAL STRETCHING SHEET WITH SUCTION OR BLOWING

In the present investigation, we study the effects of slip boundary condition on the diffusion of chemically reactive species in steady boundary layer flow of viscous incompressible fluid over a vertical stretching sheet with suction or blowing. The first-order chemical reaction is considered and wall concentration varies linearly along the sheet. The self-similar equations are obtained using similarity transformations and are solved numerically using shooting method. Our study reveals that due to the increase of diffusion parameter and blowing, the velocity increases, and it decreases with suction, Schmidt number, and reaction rate parameter. Importantly, for increase of slip parameter, the boundary layer thickness increases. In contrast, the concentration at a point increases only for increasing slip and blowing, while it decreases for increase of all other parameters.     

UNSTEADY FLUID AND HEAT FLOW INDUCED BY A STRETCHING SHEET WITH MASS TRANSFER AND CHEMICAL REACTION

The effect of chemical reaction on the flow, heat, and mass transfer within a viscous fluid on an unsteady stretching sheet is examined. The stretching rate, temperature and concentration of the sheet, and the chemical reaction rate are assumed to vary with time. The time-dependent boundary layer equations governing the flow are reduced through a convenient similarity transformation to a set of ordinary differential equations, which are numerically solved by applying the fourth-order Runge-Kutta-Fehlberg scheme with the shooting technique. Results for the velocity, temperature, and concentration distributions as well as the wall temperature and concentration gradients are presented graphically for various values of the unsteadiness parameter A, Prandtl number Pr, Schmidt number Sc, and chemical reaction parameter γ.     

Effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction/injection

In this paper, an investigation is made to study the effects of radiation and heat source/sink on the unsteady boundary layer flow and heat transfer past a shrinking sheet with suction/injection. The flow is permeated by an externally applied magnetic field normal to the plane of flow. The self-similar equations corresponding to the velocity and temperature fields are obtained, and then solved numerically by finite difference method using quasilinearization technique. The study reveals that the momentum boundary layer thickness increases with increasing unsteadiness and decreases with magnetic field. The thermal boundary layer thickness decreases with Prandtl number, radiation parameter and heat sink parameter, but it increases with heat source parameter. Moreover, increasing unsteadiness, magnetic field strength, radiation and heat sink strength boost the heat transfer.     

MHD flow and heat transfer of a micropolar fluid over a nonlinear stretching surface with variable surface heat flux and heat generation

An analysis has been carried out to study magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a nonlinear stretching surface with variable wall heat flux in the presence of heat generation/absorption and a non‐uniform transverse magnetic field. The governing system of partial differential equations is first transformed into a system of ordinary differential equations using similarity transformation. The transformed equations are solved numerically. Results for the dimensionless velocity, micro‐rotation, and temperature profiles are displayed graphically delineating the effects of various parameters characterising the flow. The results show that the velocity profile decreases as the magnetic parameter and the velocity exponent increase, while it increases as the material parameter increases. The results show also that the temperature profile increases as the magnetic parameter, the velocity exponent, and the heat generation parameter increase. Furthermore, the temperature profile decreases as the material parameter, the heat absorption parameter, and the Prandtl number increase.     

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.     

Unsteady flow and heat transfer on an accelerating surface with blowing or suction in the absence and presence of a heat source or sink

The problem of unsteady flow and heat transfer in the laminar boundary layer on a linearly accelerating surface with suction or blowing in the absence and presence of a heat source or sink is considered. The governing partial differential equations for this investigation are transformed into the non-dimensional equations by using pseudo-similarity time and pseudo-similarity coordinate. The resulting two points boundary-value problem is solved numerically by the central finite difference method associated with Newton's iteration from the initial stage (ξ=0) to a steady state (ξ=1) completely. A parametric study is performed to illustrate the effects of Prandtl number, power-law surface temperature (PLST) or power-law heat flux (PLHF), heat sink or heat source, and suction or blowing parameter on the dynamic velocity and temperature fields as well as the transient development of the skin-friction coefficients and the Nusselt number. These results are depicted graphically to display special aspects of unsteady flow and heat transfer characteristics in all time.     

DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A POROUS PLATE IN POROUS MEDIUM

Analysis was made to study the effect of diffusion of chemically reactive species in boundary layer flow of an incompressible fluid over a porous flat plate in porous medium. The first-order chemical reaction was considered and the reaction rate of the reactive species was taken in such a manner that it varied inversely along the plate. Self-similar equations were obtained using similarity transformations and were then solved by the shooting technique using the fourth-order Runge-Kutta method. This analysis revealed that at a fixed point, with increase of the permeability of the porous medium the velocity increases, but the concentration decreases. The suction reduces the thicknesses of momentum and concentration boundary layers but due to blowing the thicknesses become larger. With increase of both the Schmidt number and the reaction rate parameter, the reactive concentration profiles decrease. For a destructive chemical reaction, the concentration profiles exhibit negative value when the Schmidt number is large. It is very significant to note that in some constructive chemical reaction cases the mass absorption at the plate occurs for small Schmidt number.     

MASS TRANSFER OVER AN EXPONENTIALLY STRETCHING POROUS SHEET EMBEDDED IN A STRATIFIED MEDIUM

The boundary layer flow and mass transfer towards an exponentially stretching porous sheet embedded in a stratified medium is presented in this analysis. A first-order constructive/destructive chemical reaction is also considered. Similarity transformations were used to convert the partial differential equations corresponding to the momentum and concentration into highly nonlinear ordinary differential equations. Numerical solutions of these equations were obtained by the shooting method. Mass absorption at the surface was found in the case of a stratified medium, and it increased with an increase of stratification parameter. Due to increasing reaction rate parameter the concentration decreased. It is important to note that concentration overshoot was observed in the case of a stratified medium.     

Laminar film boiling on a porous vertical wall with uniform suction velocity

Laminar free convection film boiling on a porous vertical wall with uniform suction or blowing is analysed using boundary layer theory. The solutions are obtained assuming suction or blowing to be a disturbance superposed on the isothermal, impermeable wall case. Using a parameter involving the suction or blowing velocity, universal functions are derived for various values of Prandtl Number and c_p(T_w — T_sat)/h_fgPr. These universal functions can be used to estimate the heat transfer rate in the presence of suction or blowing. As expected, suction increases the heat transfer rate while blowing decreases the heat-transfer. Even small velocities of suction or blowing could significantly affect the heat transfer. It is also found that the effects of suction or blowing are more pronounced at lower wall superheats.     

Magnetohydrodynamic slip flow and diffusion of a reactive solute past a permeable flat plate with suction/injection

The magnetohydrodynamic (MHD) boundary layer slip flow and solute transfer over a porous plate in the presence of a chemical reaction are investigated. The governing equations were transformed into self-similar ordinary differential equations by adopting the similarity transformation technique. Then the numerical solutions are obtained by a shooting technique using the fourth order Runge-Kutta method. The study reveals that due to the increase in the boundary slip, the concentration decreases and the velocity increases. On the other hand, with an increase in the magnetic field and mass suction, both boundary layer thicknesses decreased. As the Schmidt number and the reaction rate parameter increases, the concentration decreases and the mass transfer increases.     

Thermal Diffusion and MHD Effects on Combined Free‐Forced Convection and Mass Transfer of a Viscous Fluid Flow Through a Porous Medium with Heat Generation

The problem of thermal diffusion and magnetic field effects on combined free‐forced convection and mass transfer flow past a vertical porous flat plate, in the presence of heat generation is studied numerically. The governing momentum, energy and concentration equations are converted into a system of nonlinear ordinary differential equations by means of similarity transformations. The resulting system of coupled nonlinear ordinary differential equations is solved numerically by using the Shooting method. Numerical results are presented for velocity, temperature and concentration profiles within the boundary layer for different parameters of the problem including suction parameter, heat generation parameter, Soret number, Dufour number, magnetic parameter, etc. In addition, the effects of the pertinent parameters on the skin friction and the rates of heat and mass transfer are discussed numerically and illustrated graphically.     

NON-NEWTONIAN POWER-LAW FLUID FLOW PAST A SHRINKING SHEET WITH SUCTION

The steady boundary-layer flow of a non-Newtonian fluid, represented by a power-law model, over a shrinking sheet is investigated. The transformed boundary-layer equation is solved numerically for some values of the power-law index n and suction parameter s. The effects of these parameters on the skin friction coefficient are analyzed and discussed. Different from those of a stretching sheet, the solutions are not unique and exist only if adequate suction on the boundary is imposed.     

TRANSIENT MIXED CONVECTION ALONG A VERTICAL PLATE EMBEDDED IN POROUS MEDIA WITH INTERNAL HEAT GENERATION AND OSCILLATING TEMPERATURE

The effects of oscillating plate temperature on transient mixed convection heat transfer from a porous vertical surface embedded in a saturated porous medium with internal heat generation or absorption are studied. The governing equations are transformed into dimenionless form by a set of variables and solved using the Galerkine finite element method. As the energy generation increases, the temperature near the wall will be higher than the wall temperature, thus increasing buoyancy forces inside the boundary layer and consequently increasing the velocity. The increase of energy absorption term for either space or temperature dependence will decrease the velocity inside the boundary layer and increase heat transfer rates. Different temperature and velocity profiles are drawn for different dimensionless groups. Numerical values for Nusselt numbers as well as local skin friction coefficient are also tabulated.     

TRANSIENT MIXED CONVECTION ALONG A VERTICAL PLATE EMBEDDED IN POROUS MEDIA WITH INTERNAL HEAT GENERATION AND OSCILLATING TEMPERATURE

The effects of oscillating plate temperature on transient mixed convection heat transfer from a porous vertical surface embedded in a saturated porous medium with internal heat generation or absorption are studied. The governing equations are transformed into dimenionless form by a set of variables and solved using the Galerkine finite element method. As the energy generation increases, the temperature near the wall will be higher than the wall temperature, thus increasing buoyancy forces inside the boundary layer and consequently increasing the velocity. The increase of energy absorption term for either space or temperature dependence will decrease the velocity inside the boundary layer and increase heat transfer rates. Different temperature and velocity profiles are drawn for different dimensionless groups. Numerical values for Nusselt numbers as well as local skin friction coefficient are also tabulated.     

HEAT TRANSFER DUE TO MHD SLIP FLOW OF A SECOND-GRADE LIQUID OVER A STRETCHING SHEET THROUGH A POROUS MEDIUM WITH NONUNIFORM HEAT SOURCE/SINK

An analysis is carried out to study the heat transfer characteristics of a second-grade non-Newtonian liquid due to a stretching sheet through a porous medium under the influence of external magnetic field. The stretching sheet is assumed to be impermeable. Partial slip condition is used to study the flow behavior of the liquid. The effects of viscous dissipation, nonuniform heat source/sink on the heat transfer are addressed. The nonlinear partial differential equations governing momentum and heat transfer in the boundary layer are converted into nonlinear ordinary differential equations using similarity transformation. Analytical solutions are obtained for the resulting boundary value problems in the case of two types of boundary heating, namely, constant surface temperature (CST) and prescribed surface temperature (PST). The effects of slip parameter, second-grade liquid parameter, combined (magnetic and porous) parameter, Prandtl number, Eckert number, and nonuniform heat source/sink parameters on the heat transfer are shown in several plots. Analytical expressions for the wall frictional drag coefficient and wall temperature gradient are obtained.     

A NEW CLASS OF SIMILARITY SOLUTIONS OF AN UNSTEADY ELECTRICALLY CONDUCTING FREE-FORCED CONVECTIVE FLOW IN A VERTICAL POROUS SURFACE WITH DUFOUR AND SORET EFFECTS

The 2-D unsteady magnetohydrodynamic free-forced convective boundary layer flow of a viscous incompressible fluid is studied numerically taking into account heat and mass transfer. The fluid is subjected to uniform heat and mass fluxes embedded in a porous medium by the presence of coupled Dufour and Soret effects. A new class of similarity equations has been obtained by introducing a time-dependent length scale and a corresponding similarity variable. The resulting equations are then integrated numerically using the Nachtsheim-Swigert shooting iteration technique along with the sixth-order Runge-Kutta integration scheme. By developing locally similar solutions of the fluid flow, the behavior of the velocity, temperature, and concentration fields as well as the rate of heat transfer, wall temperature gradient, rate of mass transfer, and skin friction coefficient have been investigated. The effects of Grashof number (Gr), modified Grashof number (Gm), combined effects of the porous and magnetic parameter (S), suction/injection parameter Fw, Brinkman number (Br), Soret number (Sr), and Dufour number (D_f) have been observed on the flow field and discussed.     

ON BOUNDARY LAYER STAGNATION POINT FLOW OF A NANOFLUID OVER A PERMEABLE FLAT SURFACE WITH NEWTONIAN HEATING

This study investigates the boundary layer stagnation point flow of a nanofluid past a permeable flat surface with Newtonian heating. The model used for the nanofluid is the one that incorporates the combined effects of Brownian motion and thermophoresis. Using a local similarity variable, the governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations, which are solved numerically by applying the shooting iteration technique together with a fourth-order Runge-Kutta integration scheme. Graphical results for the dimensionless velocity, temperature, and nanoparticle concentration distributions are shown for various values of the six thermophysical parameters controlling the flow regime: Prandtl number Pr, Lewis number Le, convection Biot number Bi, the Brownian motion parameter Nb, the thermophoresis parameter Nt, and the suction/injection parameter β. The expressions for the local skin friction, reduced Nusselt number, and reduced Sherwood number were obtained numerically and are discussed quantitatively.     

A STUDY OF NON-NEWTONIAN FLOW AND HEAT TRANSFER OVER A NON-ISOTHERMAL WEDGE USING THE HOMOTOPY ANALYSIS METHOD

The thermoconvective boundary layer flow of a generalized third-grade viscoelastic power-law non-Newtonian fluid over a porous wedge is studied theoretically. The free stream velocity, the surface temperature variations, and the injection velocity at the surface are assumed variables. A similarity transformation is applied to reduce the governing partial differential equations for mass, momentum, and energy conservation to dimensionless, nonlinear, coupled, ordinary differential equations. The homotopy analysis method (HAM) is employed to generate approximate analytical solutions for the transformed nonlinear equations under the prescribed boundary conditions. The HAM solutions, in comparison with numerical solutions (fourth-order Runge-Kutta shooting quadrature), admit excellent accuracy. The residual errors for dimensionless velocity and dimensionless temperature are also computed. The influence of the “power-law” index on flow characteristics is also studied. The mathematical model finds important applications in polymeric processing and biotechnological manufacture. HAM holds significant promise as an analytical tool for chemical engineering fluid dynamics researchers, providing a robust benchmark for conventional numerical methods.     

HYDROMAGNETIC BOUNDARY LAYER FLOW AND HEAT TRANSFER IN VISCOELASTIC FLUID OVER A CONTINUOUSLY MOVING PERMEABLE STRETCHING SURFACE WITH NONUNIFORM HEAT SOURCE/SINK EMBEDDED IN FLUID-SATURATED POROUS MEDIUM

Analytical study for the problem of flow and heat transfer of electrically conducting viscoelastic fluid over a continuously moving permeable stretching surface with nonuniform heat source/sink in a fluid-saturated porous medium has been undertaken. The momentum and thermal boundary layer equations, which are partial differential equations, are converted into ordinary differential equations, by using suitable similarity transformation. The resulting nonlinear ordinary differential equations of momentum are solved analytically assuming exponential solution, and similarly thermal boundary layer equations are solved exactly by using power series method, with the solution obtained in terms of Kummer's function. The results are shown with graphs and tables. The effect of various physical parameters like viscoelastic parameter, porosity parameter, Eckert number, space, and temperature-dependent heat source/sink parameters enhances the temperature profile, whereas increasing the values of the suction parameter and Prandtl number decreases the temperature profile. The results have technological applications in liquid-based system involving stretchable materials.     

FURTHER RESULTS ON HYDROMAGNETIC BOUNDARY-LAYER FLOW OF A NON-NEWTONIAN POWER-LAW FLUID OVER A CONTINUOUSLY MOVING SURFACE WITH SUCTION

The flow of a non-Newtonian, power-law conducting fluid under the effect of a constant transverse magnetic field is considered. The flow is produced by a plate moving with constant velocity in a calm fluid. The plate is porous and fluid can either be sucked or injected through it. The boundary layer equations are transformed into a nondimensional form and are solved with a finite difference method. Part of this problem has been investigated in the past but only for suction and pseudo-plastic fluids. However, all flows of the present work reach an asymptotic state and exact analytical solutions exist for Newtonian fluids. In the present work we extend the investigation to both pseudo-plastic Newtonian and dilatant fluids in both suction and injection cases.