UNSTEADY MHD BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A PERMEABLE STRETCHING SHEET WITH SUCTION OR BLOWING

In this study, unsteady MHD boundary layer flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet with suction or blowing and also with power-law variation in wall concentration is investigated. Using similarity transformation, the governing partial differential equations are converted into nonlinear self-similar ordinary differential equations. The transformed equations are then solved by the finite difference method using the quasi-linearization technique. Due to the increase in the unsteadiness parameter, the velocity initially decreases, but after a certain point it increases. A similar effect is also observed in case of concentration distribution. The increase in magnetic parameter causes a decrease in velocity and an increase in concentration. For increasing strength of applied suction both momentum and concentration boundary layer thicknesses decrease. On the other hand, applied blowing has reverse effects. Moreover, the mass transfer from the sheet is enhanced with increasing values of Schmidt number, reaction rate parameter, and also power-law exponent (related to wall concentration distribution). For high negative values of the power-law exponent, mass absorption at the sheet occurs. Moreover, due to increase of unsteadiness, this mass absorption is prevented.     

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.     

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.     

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.     

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.     

EFFECTS OF HALL CURRENT ON UNSTEADY FLOW OF A SECOND GRADE FLUID IN A ROTATING SYSTEM

Rotating flow of a second grade conducting fluid on an infinite oscillating plate is investigated when the fluid is permeated by a transverse magnetic field and the Hall effects are taken into account. It is once again found that an asymptotic solution exists in the presence of both suction and blowing at the plate. For fixed magnetic field parameter the boundary layer thickness increases with the increase in Hall parameter. The present analysis is more general than any previous investigations.     

EFFECTS OF HALL CURRENT ON UNSTEADY FLOW OF A SECOND GRADE FLUID IN A ROTATING SYSTEM

Rotating flow of a second grade conducting fluid on an infinite oscillating plate is investigated when the fluid is permeated by a transverse magnetic field and the Hall effects are taken into account. It is once again found that an asymptotic solution exists in the presence of both suction and blowing at the plate. For fixed magnetic field parameter the boundary layer thickness increases with the increase in Hall parameter. The present analysis is more general than any previous investigations.     

Combined effects of internal heat generation and higher order chemical reaction on the non‐darcian forced convective flow of a viscous incompressible fluid with variable viscosity and thermal conductivity over a stretching surface embedded in a porous medium

In this paper, we study the combined effects of internal heat generation and higher order chemical reaction on a steady two‐dimensional non‐Darcian forced convective flow of a viscous incompressible fluid with variable dynamic viscosity and thermal conductivity in a fluid saturated porous medium passing over a linear stretching sheet. Using similarity transformations, the governing nonlinear‐coupled partial differential equations are made dimensionless and solved numerically for similarity solutions using very robust computer algebra software Maple 8. The non‐dimensional velocity, temperature and concentration distributions are presented graphically for various pertinent parameters such as relative temperature difference parameter, Darcy number, porosity parameter, reaction rate parameter and the order of the chemical reaction. The variations of Prandtl number and Schmidt number within the boundary layer are also displayed graphically when the fluid dynamic viscosity and thermal conductivity are temperature dependent. From the present numerical computations it is found that Prandtl number as well as Schmidt number must be taken as variables within the flow domain when the fluid's dynamic viscosity and thermal conductivity are variable. In the presence of internal heat generation, dynamic viscosity and thermal conductivity of the fluid are found to be higher than when it is absent. Increasing Darcy number reduces dynamic viscosity as well as thermal conductivity whereas increasing pore size reduces the Schmidt number and increases the Prandtl number within the boundary layer. For higher order reaction the rate of increase in mass transfer function is less compared to the rate of increase for the lower order reaction. © 2011 Canadian Society for Chemical Engineering     

Chemically reactive solute transfer in a boundary layer slip flow along a stretching cylinder

This paper presents the distribution of a solute undergoing a first order chemical reaction in an axisymmetric laminar boundary layer flow along a stretching cylinder. Velocity slip condition at the boundary is used instead of no-slip condition. Similarity transformations are used to convert the partial differential equations corresponding to momentum and concentration into highly nonlinear ordinary differential equations. Numerical solutions of these equations are obtained by the shooting method. The velocity decreases with increasing slip parameter. The skin friction as well as the mass transfer rate at the surface is larger for a cylinder than for a flat plate.     

THE EFFECT OF VARIABLE THERMAL CONDUCTIVITY ON MICRO-POLAR FLUID FLOW BY CHEBYSHEV COLLOCATION METHOD

In this article, the authors analyzed the effect of thermal conductivity on unsteady magnetohydrodynamic (MHD) free convection in a micro-polar fluid past a semi-infinite vertical porous plate. The fluid thermal conductivity is assumed to vary as a linear function of temperature. By using the Chebyshev collocation method in the spatial direction and the Crank-Nicolson method in the time direction, the boundary layer equations are transformed into a linear algebraic system. There are several material parameters whose affect on the flow have been studied, for instance, thermal conductivity, radiation, magnetic, micro-polar, suction (or injection) parameters, and Prandtl number. Boundary layer and Boussineq approximations have been introduced together to describe the flow field. The domain of the problem is discretized according to the Chebyshev collocation scheme. The numerical results show that, the values of velocity, angular velocity and temperature profiles approach to the steady state when the time reach to infinity. However, the friction factor has been found to increase as micro-polar and thermal conductivity parameters increase. But it decreases as magnetic parameter increases. Meanwhile, Nusselt number increases as thermal conductivity parameter increases, and vice versa with the micro-polar parameter. Moreover, the local couple stress has been found to decrease as micro-polar and thermal conductivity parameters increase. On the other hand, it increases as magnetic parameter increases.     

Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretched sheet with variable concentration in a porous medium

An analysis has been carried out to obtain the effects of higher order chemical reaction on flow and mass transfer characteristics of micropolar fluids past a nonlinear permeable stretching sheet immersed in a porous medium with variable concentration of the reactant. The local similarity solutions for the flow, microrotation and mass transfer are obtained numerically and are illustrated graphically for various material parameters. Comparisons carried out with results from previously published work present excellent agreement. The results show that rate of mass transfer from the sheet to the surrounding fluid decreases significantly with the increase of nonlinear stretching parameter and increases with the increase of concentration parameter. The mass transfer of the reactive species strongly depends on the reaction rate parameter as well as order of the chemical reaction. It is stronger for the first‐order reaction than that for the higher order reaction. The rate of surface mass transfer decreases with the increase of the Darcy parameter. The results also show that the effect of these parameters on the micropolar fluids are less compared to the Newtonian fluids.     

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.     

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.     

RADIATION EFFECTS ON MHD AXISYMMETRIC STAGNATION POINT FLOW TOWARDS A HEATED SHRINKING SHEET

In this article, a comprehensive numerical study of MHD axisymmetric stagnation point flow with radiation effects towards a heated shrinking sheet immersed in an electrically conducting incompressible viscous fluid in the presence of a transverse magnetic field is analyzed. The governing continuity, momentum, and heat equations together with the associated boundary conditions are first transformed to a set of self-similar nonlinear ordinary differential equations and are then solved by a method based on finite difference discretization. Some significant features of the flow and heat transfer in terms of normal and horizontal velocities and temperature field for various values of the governing parameters are analyzed, discussed, and presented through tables and graphs. The present investigations predict that the shear stresses increase and the thermal boundary layer becomes thinner by applying a strong magnetic field. The heat loss per unit area from the sheet decreases with an increase in the shrinking parameter. The thermal boundary layer thickness decreases with increasing values of the radiation parameter. The present results may be beneficial in flow and thermal control of polymeric processing.     

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.     

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.     

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.     

SLIP FLOW AND HEAT TRANSFER OF A SECOND-GRADE FLUID IN A POROUS MEDIUM OVER A STRETCHING SHEET WITH POWER-LAW SURFACE TEMPERATURE OR HEAT FLUX

This article deals with the study of boundary layer flow of a second-grade fluid in a porous medium past a stretching sheet and heat transfer characteristics with power-law surface temperature or heat flux. The flow in the boundary layer is considered to be generated solely by the linear stretching of the boundary sheet adjacent to a porous medium, and boundary wall slip condition is assumed. In the energy equation effects of viscous dissipation, work done due to deformation and internal heat generation/absorption is taken into account. Closed form solutions are obtained for this problem.     

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.     

Numerical Simulation of Quenched Combustion Model for AP/HTPB Propellant under Transient Depressurization

In order to study the out‐of‐phase blowing effect of ammonium perchlorate/hydroxyl terminated polybutadiene (AP/HTPB) under transient depressurization, a two‐dimensional periodic sandwich model of AP/HTPB sandwich unsteady combustion was established. The gas‐solid two‐step total reaction was used to couple the gas‐solid boundary layer, and the AP/HTPB step‐down temperature criterion was applied. The numerical simulation comparative analysis under the initial combustion pressure 3.5 MPa∼10 MPa and the initial depressurization rate of 1000 MPa/s∼2000 MPa/s was experimented. The results show that under the condition of initial combustion pressure of 3.5 MPa and pressure reduction rate of 1000 MPa/s, the formation of narrow diffusion chemical reaction zone appears in the initial stage of depressurization; when the combustion pressure drops to about 1.7 MPa, the flame appears dual characteristics: diffusion and premixed combustion; when the pressure dropped to 0.1 MPa, the flame is premixed combustion. AP(g) is the main factor causing the out‐of‐phase blowing effect. During the initial stage of combustion, AP(g) decomposes rapidly and its decomposition product is close to the combustion surface; with the rapid decrease of pressure, the gas‐phase heat feedback decreases, leading to the decrease of AP(g) decomposition rate and AP(g) diffusion to the gas phase, which is the initial phase of out‐of‐phase blowing effect; when the gas phase heat feedback reduced and the temperature is below 750 K, the decomposition of AP(g) is stopped and the undecomposed AP(g) is further diffused to the gas phase region, which exacerbates the out‐of‐phase blowing effect, leading to the extinguishment of AP/HTPB. The unsteady combustion flame extinction time increases with the initial combustion pressure, and decreases with the initial depressurization rate.