UNSTEADY MIXED CONVECTION WITH SORET AND DUFOUR EFFECTS PAST A POROUS PLATE MOVING THROUGH A BINARY MIXTURE OF CHEMICALLY REACTING FLUID

This study investigates the unsteady mixed convection flow past a vertical porous flat plate moving through a binary mixture in the presence of radiative heat transfer and nth-order Arrhenius type of irreversible chemical reaction by taking into account the diffusion-thermal (Dufour) and thermo-diffusion (Soret) effects. Assuming an optically thin radiating fluid and 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 shooting iteration technique together with fourth-order Runge-Kutta integration scheme. Graphical results for the dimensionless velocity, temperature, and concentration distributions are shown for various values of the thermophysical parameters controlling the flow regime. Finally, numerical values of physical quantities, such as the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are presented in tabular form.     

HEAT AND MASS TRANSFER IN STAGNATION-POINT FLOW OF A POLAR FLUID TOWARDS A STRETCHING SURFACE IN POROUS MEDIA IN THE PRESENCE OF SORET,DUFOUR AND CHEMICAL REACTION EFFECTS

This work is focused on the numerical solution of steady boundary-layer stagnation-point flow of a polar fluid towards a stretching surface embedded in porous media in the presence of the effects of Soret and Dufour numbers and first-order homogeneous chemical reaction. The governing boundary-layer equations of the problem are formulated and transformed into a self-similar form. The obtained equations are solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. Both assisting and opposing flow conditions are considered. Comparisons of the present numerical results with previously published work under limiting cases are performed and found to be in excellent agreement. Representative results for the fluid velocity, angular velocity, temperature, and solute concentration profiles as well as the local heat and mass transfer rates for various values of the physical parameters are displayed in both graphical and tabular forms.     

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.     

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.     

UNSTEADY MHD HEAT AND MASS TRANSFER BY MIXED CONVECTION FLOW IN THE FORWARD STAGNATION REGION OF A ROTATING SPHERE AT DIFFERENT WALL CONDITIONS

This study is focused on the problem of MHD heat and mass transfer by mixed convection flow in the forward stagnation region of a rotating sphere in the presence of heat generation and chemical reaction effects. The surface of the sphere is maintained at constant fluid temperature and species concentration. The governing equations of the problem are converted into ordinary differential equations by using suitable similarity transformations. Two cases are considered, namely, constant wall temperature and mass (CWTM) and constant heat and mass fluxes (CHMF). The obtained self-similar equations for both cases are solved numerically using an efficient iterative implicit finite-difference method. The numerical results are compared with previously published results on special cases of the problem and found to be in excellent agreement. The obtained results are displayed graphically to illustrate the influence of the different physical parameters on the velocity components in x- and y-directions, temperature, and concentration profiles as well as the local surface shear stresses and local heat and mass transfer coefficients.     

MHD MIXED-CONVECTION INTERACTION WITH THERMAL RADIATION AND nTH ORDER CHEMICAL REACTION PAST A VERTICAL POROUS PLATE EMBEDDED IN A POROUS MEDIUM

This article investigates the hydromagnetic mixed convection heat and mass transfer flow of an incompressible Boussinesq fluid past a vertical porous plate with constant heat flux in the presence of radiative heat transfer in an optically thin environment, viscous dissipation, and an nth order homogeneous chemical reaction between the fluid and the diffusing species. The dimensionless governing equations for this investigation are solved numerically by the fourth-order Runge-Kutta integration scheme along with shooting technique. Numerical data for the local skin-friction coefficient, the plate surface temperature, and the local Sherwood number have been tabulated for various values of parametric conditions. Graphical results for velocity, temperature, and concentration profiles based on the numerical solutions are presented and discussed.     

MHD FLOW AND HEAT TRANSFER IN A VISCOUS FLUID OVER A NON-ISOTHERMAL STRETCHING SURFACE WITH THERMAL RADIATION IN SLIP-FLOW REGIME

The present work is concerned with the effects of surface slip conditions and thermal radiation on an electrically conducting fluid over a non-isothermal stretching surface in the presence of a uniform transverse magnetic field. Similarity transformation is used to transform the partial differential equations describing the problem into a system of nonlinear ordinary differential equations, which is solved analytically. The effects of various parameters on the velocity and temperature profiles as well as on the local skin-friction and the local Nusselt number are discussed in detail and displayed through graphs.     

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.     

HYDROMAGNETIC STAGNATION POINT FLOW TOWARDS A POROUS STRETCHING SHEET WITH VARIABLE SURFACE HEAT FLUX IN THE PRESENCE OF HEAT GENERATION

The influence of heat generation or absorption on the steady, two-dimensional flow of an electrically conducting fluid near a stagnation point on a stretching permeable surface with variable surface heat flux in the presence of a magnetic field is investigated. The governing system of partial differential equations describing the problem are converted into highly non-linear ordinary differential equations using similarity transformation. Numerical solutions of these equations are obtained using the fourth-order Runge-Kutta integration scheme with the shooting method. The effects of the heat generation or absorption parameter and the velocity ratio parameter on the velocity and the temperature are displayed graphically and discussed. The numerical values of the local skin-friction coefficient and the local Nusselt number for various values of physical parameters are presented through tables and discussed.     

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.     

Combined effects of internal heat generation and buoyancy force on boundary layer flow over a vertical plate with a convective surface boundary condition

This paper considers the effect of buoyancy force and internal heat generation on laminar thermal boundary layer over a vertical plate with a convective surface boundary condition. We assumed that left surface of the plate is in contact with a hot fluid while a stream of cold fluid flows steadily over the right surface with a heat source that decays exponentially. Using a similarity variable, the steady state governing non‐linear partial differential equations have been transformed into a set of coupled non‐linear ordinary differential equations, which are solved numerically by applying shooting iteration technique together with fourth order Runge–Kutta integration scheme. The effects of Prandtl number, local Biot number, the internal heat generation parameter and the local Grashof number on the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison with previously published results on special case of the problem shows excellent agreement. From our results, an overshoot of fluid velocity within the boundary layer is observed due to combined effect of buoyancy force and internal heat generation, in addition, internal heat generation causes thickening of thermal boundary layer. © 2011 Canadian Society for Chemical Engineering     

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.     

EFFECTS OF HALL CURRENT AND CHEMICAL REACTION ON OSCILLATORY MIXED CONVECTION-RADIATION OF A MICROPOLAR FLUID IN A ROTATING SYSTEM

This article is concerned with the analysis of the effects of thermal radiation on oscillatory mixed convection flow of a micropolar fluid in a rotating frame of reference in the presence of transverse magnetic field and Hall current. The influence of a first-order homogeneous chemical reaction and heat source effects is also analyzed. The governing partial differential equations with the appropriate boundary conditions are reduced to a set of ordinary differential equations using similarity transformations. The dimensionless governing equations for this investigation are solved analytically after using small perturbation approximation. The effects of various parameters on the velocity, temperature, and concentration fields as well as on skin-friction coefficient, Nusselt number, and Sherwood number with their amplitude and phase are discussed in detail. Numerical results are discussed with the help of graphs and tables. Present results are also compared with previously published work.     

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.     

Thermal radiation effect on unsteady MHD free convection flow past a vertical plate with temperature‐dependent viscosity

This article investigates the influence of radiation and temperature‐dependent viscosity on the problem of unsteady MHD flow and heat transfer of an electrically conducting fluid past an infinite vertical porous plate taking into account the effect of viscous dissipation. The governing equations are converted into a system of nonlinear ordinary differential equations via a local similarity parameter which is taken as a function of time. The resulting system of coupled nonlinear ordinary differential equations is solved numerically using the fourth order Runge–Kutta integration scheme with the shooting method. The numerical results for the velocity and the temperature are displayed graphically showing the effects of various parameters. The results show that increasing the Eckert number and decreasing the viscosity of air leads to a rise in the velocity, while increasing in the magnetic or the radiation parameters is associated with a decrease in the velocity. Also, an increase in the Eckert number leads to an increase in the temperature, whereas an increase in radiation parameter leads to a decrease in the temperature.     

A new approach to fluid separation modelling in the columns equipped with structured packings

An analogy between the flow patterns in real separation columns equipped with structured packing and film flow is used to develop a new modelling approach. The packing is represented as a bundle of channels with identical triangular cross section. The dimensions of the channels as well as their number are derived from the packing geometry. The channel inner surface is wetted by a liquid flowing downwards, whereas the rest of the volume is occupied by a countercurrent vapour flow. Both phases are assumed to be totally mixed at regular intervals, determined by the corrugation geometry of the packing. The mathematical model is based on a set of partial differential equations describing hydrodynamics and mass and heat transport phenomena. These equations are complemented by the conjugate boundary conditions at the phase interface. A numerical solution of the model yields velocity profiles as well as concentration and temperature fields throughout the column. The model is verified using experimental data for a binary distillation in a column equipped with Montz-Pak A3-500.     

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.     

RADIATION EFFECT ON FREE CONVECTION LAMINAR FLOW FROM AN ISOTHERMAL SPHERE

Natural convection laminar flow from an isothermal sphere immersed in a viscous incompressible optically dense fluid in the presence of radiation effects has been investigated. The governing boundary layer equations are transformed into a non dimensional form and the resulting nonlinear systems of partial differential equations are reduced to local non-similarity boundary layer equations, which are solved numerically by two distinct, efficient methods, namely, (i) implicit finite difference method together with the Keller box scheme and (ii) local non-similarity method. Numerical results of the velocity and temperature profiles of the fluid are presented. The results of the shearing stress and the heat transfer rate in terms of skin-friction coefficient and Nusselt number respectively are also presented for a wide range of the radiation-conduction parameter or Planck number R _d (=0.0, 1.0, 2.0, 3.0), the surface heating parameter θ _w (=1.1, 1.2, 1.4), and Prandtl number Pr (=7.0, 10.0, 15.0, 20.0).     

THE EFFECTS OF VARIABLE FLUID PROPERTIES ON MHD MAXWELL FLUIDS OVER A STRETCHING SURFACE IN THE PRESENCE OF HEAT GENERATION/ABSORPTION

An analysis is performed to investigate the effects of variable viscosity and thermal conductivity on the two-dimensional steady flow of an electrically conducting, incompressible, upper-convected Maxwell fluid in the presence of a transverse magnetic field and heat generation or absorption. The governing system of partial differential equations is transformed into a system of coupled nonlinear ordinary differential equations, and is solved numerically. Velocity and temperature fields have been computed and shown graphically for various values of the physical parameters. The local skin-friction coefficient and the local Nusselt number have been tabulated. It is found that fluid velocity decreases with an increase in the viscosity parameter and the Deborah number. It is also observed that increasing the magnetic parameter leads to a fall in the velocity and a rise in the temperature. Furthermore, it is shown that the temperature increases due to increasing the values of the thermal conductivity parameter and the heat generation parameter, while it decreases with an increase of both the absolute value of the heat absorption parameter and the Prandtl number.     

NATURAL CONVECTION FLOW OF SECOND-GRADE FLUID ALONG A VERTICAL HEATED SURFACE WITH POWER-LAW TEMPERATURE

In this study we investigated the effects of free convection flow of a viscoelastic second-grade fluid along a vertical flat surface with power-law temperature distribution. The boundary layer equations for the momentum and the energy transport have been reduced to local similarity equations using appropriate transformations. Solutions of the reduced equations are obtained employing the local non-similarity method as well as the implicit finite difference method against ξ (the local Deborah number) in the range [0, 10] for fluids having Prandtl numbers of 10, 50, and 100. The regular perturbation solutions are also been obtained for smaller values of ξ together with the Padé approximation. Results thus obtained are discussed in terms of the local skin friction and local rate of heat transfer for different values of the physical parameters, like n and Pr. Effect of the Deborah number, De, on the velocity and temperature profiles has also been shown graphically. It is observed that both the local skin friction and heat transfer coefficients decrease with increase in the value of De for given values of n and Pr.