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.     

MHD flow of a micropolar fluid towards a vertical permeable plate with prescribed surface heat flux

The problem of a steady mixed convection stagnation point flow towards a permeable vertical plate with prescribed surface heat flux immersed in an incompressible micropolar fluid is studied numerically. The governing partial differential equations are first transformed into a system of ordinary differential equations using a similarity transformation, before being solved numerically by a finite-difference scheme known as the Keller-box method and the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter, buoyancy parameter, suction/injection parameter and the Prandtl number on the fluid flow and heat transfer characteristics are discussed. It is found that dual solutions exist for both assisting and opposing flows. The skin friction coefficient and the local Nusselt number increase in the presence of suction and magnetic field. Moreover, suction as well as fluids with larger Prandtl number widens the range of the buoyancy parameter for which the solution exists.     

Similarity Solutions of Marangoni Convection Boundary Layer Flow with Gravity and External Pressure

This study is focused on a steady dissipative layer, which is generated by Marangoni convection flow over the surface resulted from an imposed temperature gradient, coupled with buoyancy effects due to gravity and external pressure. A model is proposed with Marangoni condition in the boundary conditions at the interface. The similarity equations are determined and approximate analytical solutions are obtained by an efficient transformation, asymptotic expansion and Pade approximant technique. For the cases that buoyancy force is favorable or unfavor-able to Marangoni flow, the features of flow and temperature fields are investigated in terms of Marangoni mixed convection parameter and Prantl number.     

SORET AND DUFOUR EFFECTS IN A FREE CONVECTIVE DOUBLY STRATIFIED FLOW OVER A VERTICAL PLATE WITH CHEMICAL REACTION

An investigation was performed to study the influence of thermo-diffusion and diffusion-thermo effects in the transient, free convective flow of a viscous, incompressible, and doubly stratified fluid past an isothermal vertical plate in the presence of first-order chemical reaction. The governing boundary layer equations were solved numerically using an implicit finite difference scheme of the Crank-Nicolson type. The effects of the Soret number, Dufour number, thermal stratification parameter, mass stratification parameter, and chemical reaction parameter are analyzed and presented graphically. Also, the influence of the parameters on local as well the average skin-friction coefficient and the rate of heat and mass transfer are analyzed and discussed. The results are compared with particular solutions available in the literature. The present results are found to be in good agreement with the existing solution.     

THE EFFECT OF THERMAL RADIATION ON THE HEAT AND MASS TRANSFER FLOW OF A VARIABLE VISCOSITY FLUID PAST A VERTICAL POROUS PLATE PERMEATED BY A TRANSVERSE MAGNETIC FIELD

The effect of temperature-dependent viscosity on free convective flow past a vertical porous plate is studied in the presence of a magnetic field, thermal radiation, and a first-order homogeneous chemical reaction. Boundary layer equations are derived and the resulting approximate nonlinear ordinary differential equations are solved numerically by the shooting method. A parametric study of all parameters involved is conducted, and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter and the Nusselt and Sherwood numbers is illustrated graphically to show typical trends of the solutions. The dynamic viscosity in this study is taken as a function of the temperature although the Prandtl number is considered constant.     

SIMILARITY SOLUTION FOR UNSTEADY HEAT AND MASS TRANSFER FROM A STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM WITH SUCTION/INJECTION AND CHEMICAL REACTION EFFECTS

An analysis is presented to investigate the effects of chemical reaction on unsteady free convective heat and mass transfer on a stretching surface in a porous medium. The governing partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically using an efficient tri-diagonal implicit finite-difference method. The results obtained show that the flow field is influenced appreciably by the presence of unsteadiness parameter, chemical reaction parameter, permeability parameter, and suction/injection parameter.     

Mixed convection flow of a power-law fluid in horizontal ducts

The laminar mixed convection heat transfer of a non-Newtonian fluid modelled by a power-law constitutive equation is studied. The equations of motion subject to the Boussinesq approximation and an axially uniform heat flux and peripherally uniform wall temperature condition are discretized by finite difference approximation and are solved by the successive relaxation method. Dual solutions with two-cell and four-cell patterns have been observed in certain regions of the parameter space. The secondary flow caused by the buoyancy force has a shear thinning effect for pseudoplastic fluids. The Nusselt number, however, always increases with increasing Grashof number.     

Radiation-absorption,chemical reaction,Hall and ion slip impacts on magnetohydrodynamic free convective flow over semi-infinite moving absorbent surface

The investigation of radiation-absorption,chemical reaction,Hall and ion-slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame has been premeditated.The plate is assumed to be moving with a constant velocity in the direction of fluid movement.A uniform transverse magnetic field is applied at right angles to the porous surface,which is absorbing the fluid with a suction velocity changing with time.The non-dimensional governing equations for present inves-tigation are solved analytically making use of two term harmonic and non-harmonic functions.The graphical results of velocity,temperature and concentration distributions on the analytical solutions are displayed and discussed with reference to pertinent parameters.It is found that the velocity profiles decreased with an increasing in Hartmann number,rotation parameter,the Schmidt number,heat source parameter,while it increased due to an increase in permeability parameter,radiation-absorption param-eter,Hall and ion slip parameters.However,the temperature profile is an increasing function of radiation-absorption parameter,whereas an increase in chemical reaction parameter,the Schmidt num-ber Sc or frequency of oscillations decrease the temperature profile on cooling.Also,it is found that the concentration profile is decreased with an escalating in the Schmidt number or the chemical reaction parameter.     

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.     

NUMERICAL MODELING OF MHD CONVECTIVE HEAT AND MASS TRANSFER IN PRESENCE OF FIRST-ORDER CHEMICAL REACTION AND THERMAL RADIATION

An analysis was carried out numerically to study unsteady heat and mass transfer by free convection flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate under the action of transverse magnetic field taking into account thermal radiation as well as homogeneous chemical reaction of first order. The fluid considered here is an optically thin gray gas, absorbing-emitting radiation, but a non-scattering medium. The porous plate was subjected to a constant suction velocity with variable surface temperature and concentration. The dimensionless governing coupled, nonlinear boundary layer partial differential equations were solved by an efficient, accurate, extensively validated, and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields were studied for the effects of Hartmann number (M), radiation parameter (R), chemical reaction (K), and Schmidt number (Sc). The local skin friction, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is found that velocity is reduced considerably with a rise in the magnetic body parameter (M), whereas the temperature and concentration are found to be markedly boosted with an increase in the magnetic body parameter (M). An increase in the conduction-radiation parameter (R) is found to escalate the local skin friction (τ), Nusselt number, and concentration, whereas an increase in the conduction-radiation parameter (R) is shown to exert the opposite effect on either velocity or temperature field. Similarly, the local skin friction and the Sherwood number are both considerably increased with an increase in the chemical reaction parameter. Possible applications of the present study include laminar magneto-aerodynamics, materials processing, and MHD propulsion thermo-fluid dynamics.     

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     

Modeling and numerical analysis of nanoliquid (titanium oxide,graphene oxide) flow viscous fluid with second order velocity slip and entropy generation

The prime objective of the present communication is to examine the entropy-optimized second order velocity slip Darcy–Forchheimer hybrid nanofluid flow of viscous material between two rotating disks.Electrical conducting flow is considered and saturated through Darcy–Forchheimer relation. Both the disks are rotating with different angular frequencies and stretches with different rates. Here graphene oxide and titanium dioxide are considered for hybrid nanoparticles and water as a continuous phase liquid. Joule heating, heat generation/absorption and viscous dissipation effects are incorporated in the mathematical modeling of energy expression. Furthermore, binary chemical reaction with activation energy is considered. The total entropy rate is calculated in the presence of heat transfer irreversibility, fluid friction irreversibility,Joule heating irreversibility, porosity irreversibility and chemical reaction irreversibility through thermodynamics second law. The nonlinear governing equations are first converted into ordinary differential equations through implementation of appropriate similarity transformations and then numerical solutions are calculated through Built-in-Shooting method. Characteristics of sundry flow variables on the entropy generation rate, velocity, concentration, Bejan number, temperature are discussed graphically for both graphene oxide and titanium dioxide hybrid nanoparticles. The engineering interest like skin friction coefficient and Nusselt number are computed numerically and presented through tables. It is noticed from the obtained results that entropy generation rate and Bejan number have similar effects versus diffusion parameter. Also entropy generation rate is more against the higher Brinkman number.     

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.     

MHD‐Buoyancy Aiding and Opposing Flows with Viscous Dissipation Effects from Radiate Vertical Surfaces

The combined effects of forced and natural convection heat transfer in the presence of transverse magnetic field form a vertical surfaces with radiation heat transfer is studied. The buoyancy aided flow and the buoyancy opposing flows are investigated with viscous dissipation effects included in the governing equations. It is found that four parameters can describe the problem under consideration, the mixed parameter, χ, the radiation‐conduction parameter, R_d, the magnetic field parameter, Ha²_x/Re_x, and the Eckert number, Ec. The local wall shear stresses and the local Nusselt number variations are drawn for different dimensionless groups.     

NETWORK NUMERICAL SIMULATION OF HYDROMAGNETIC MARANGONI MIXED CONVECTION BOUNDARY LAYERS

The study of a steady coupled dissipative layer, known as the Mangaroni mixed convection boundary layer, in the presence of a magnetic field is presented. The mixed convection boundary layer is generated when in addition to Marangoni (thermocapillary) effects there are also buoyancy effects due to gravity and external pressure gradient effects. In the model considered the Marangoni coupling condition has been included in the boundary conditions at the interface. Similarity transformations are utilized to transform the governing partial differential conservation equations into nondimensional ordinary differential equations in a single independent space variable (η) and solved using the network simulation method (NSM) using an electronic circuit simulator, Pspice. NSM is founded on the classical thermoelectric analogy between thermal and electrical variables. A set of finite-differential equations, one for each control volume, was obtained by spatial discretization of the transformed equations. The solutions obtained are compared with earlier computations using other numerical techniques, showing excellent agreement. The influence of the Marangoni mixed parameter and Hartmann number on the velocity and temperature functions are studied in detail. The effectiveness of utilizing magnetic fields to control heat transfer in Marangoni convection boundary layers is identified. An increase in Hartmann hydromagnetic number (M) is found to strongly decelerate the flow but increase temperatures. An increase in Marangoni mixed convection parameter (λ) for the scenario opposing Marangoni flow (Γ > 0) considerably accelerates the flow but decreases temperatures in the boundary layer. Conversely, an increase in Marangoni mixed convection parameter (λ) for the case favorable to the Marangoni flow (Γ < 0) decelerates the flow but enhances temperatures in the boundary layer. Applications of the model include semiconductor crystal hydromagnetic heat transfer control.     

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.     

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.     

COMBINED FORCED AND FREE CONVECTION IN BOUNDARY LAYER FLOW OF NON-NEWTONIAN FLUID ON A HORIZONTAL PLATE

An analysis is presented to study the effects of buoyancy-induced streamwise pressure gradients on laminar forced convection heat transfer to non-Newtonian fluids from a horizontal plate. Numerical solution of the transformed boundary layer equations has been carried out for different values of the flow behavior index and the buoyancy parameter by means of the local nonsimilarity method. A discussion is provided for the effect of the buoyancy force on the friction factor, the dimensionless heat transfer rate and the details of the velocity as well as temperature fields.     

UNSTEADY HEAT AND MASS TRANSFER BY MHD MIXED CONVECTION FLOW OVER AN IMPULSIVELY STRETCHED VERTICAL SURFACE WITH CHEMICAL REACTION AND SORET AND DUFOUR EFFECTS

This work considers unsteady, laminar, and coupled heat and mass transfer by MHD mixed convective boundary-layer flow of an electrically conducting fluid over an impulsively stretched vertical surface in an unbounded quiescent fluid with aiding external flow in the presence of a transverse magnetic field, homogeneous chemical reaction, and Soret and Dufour effects. The stretching velocity and surface temperature and concentration are assumed to vary linearly with the distance along the surface. The flow is impulsively set into motion and both the temperature and concentration at the surface are also suddenly changed from those of the ambient fluid. The governing partial differential equations are transformed into a set of nonsimilar equations and solved numerically by an efficient implicit, iterative, finite-difference method. A parametric study illustrating the influence of various physical parameters is performed. Numerical results for the steady-state velocity, temperature, and concentration profiles as well as the time histories of the skin-friction coefficient, local Nusselt number, and local Sherwood number are presented graphically and discussed.     

Mass transfer with axial diffusion and chemical reaction

Mass transfer with axial diffusion and chemical reaction is studied for three different cases. The first one is plug flow mass transfer in semi-infinite region. The second one is laminar flow mass transfer in a semi-infinite region. The third one is laminar flow convective diffusion in the region of x ≤ 0 while mass transfer with chemical reaction in the region of x ≥ 0. The first problem is solved in terms of Bessel functions while the second in terms of confluent hypergeometric functions. In order to solve the third problem, orthonormal functions are constructed by linear combination of the eigenfunctions of the two semi-infinite regions respectively. The series expansion coefficients of the solutions are determined by solving a set of forty simultaneous equations obtained through matching boundary conditions at x = 0.Solutions are generally obtained for low Peclet number (Pe = 1, 5, 10) and high reaction rate parameters. Effects of axial diffusion and chemical reaction on the concentration field are discussed.