State space approach to unsteady free convection flow of micropolar fluid

Summary The equations of unsteady free convection flow of a micropolar fluid are cast into a matrix form using the state space and Laplace-transform techniques. The results obtained are used to generate solutions in the Laplace-transform domain to a broad class of problems in free convection flow. The technique is applied to a heated vertical plate problem and to a problem pertaining to a plate under uniform heating. The inversion of Laplace-transform is performed using a numerical approach. Numerical results concerning velocity, microrotation and temperature are given and illustrated graphically for both problems.     

Unsteady free convective flow through a porous medium

Unsteady 2-dimensional free-convective flow through a porous medium bounded by an infinite vertical plate is considered, when the temperature of the plate is oscillating with the time about a constant non-zero mean value. An analytical solution for the velocity field is derived and the effects of K (permeability parameter) and ω (parameter of frequency) on the velocity field are discussed.     

Free convection effects on the oscillatory flow of a couple stress fluid through a porous medium

Summary Effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. Analytical expressions for the velocity and the angular velocity fields have been obtained, using the regular perturbation technique. The effects of Grashof numberG; material parameters and ; Prandtl numberP; permeability parameterK and frequency parametern on the velocity and the angular velocity are discussed. The effects of cooling and heating of a polar fluid compared to a Newtonian fluid have also been discussed. The velocity of a polar fluid is found to decrease as compared to the Newtonian fluid.List of symbols C _p specific heat at constant pressure - g acceleration due to gravity - G Grashof number - K ⁺ permeability of the porous medium - K dimensionless permeability - P Prandtl number - t ⁺ time - t dimensionless time - T _w ⁺ mean temperature of the surface - T ⁺ temperature of the fluid - T ⁺ temperature of the fluid away from the surface - density of the fluid - viscosity - _r rotational viscosity - C _a ,C _d coefficients of couple stress viscosities - I a scalar constant of dimension equal to that of the moment of inertia of unit mass - x ⁺,y ⁺ coordinate system - u ⁺,v ⁺ velocity components in thex ⁺ andy ⁺ directions - u dimensionless velocity in thex ⁺-direction - ⁺ angular velocity component - dimensionless angular velocity - n ⁺ frequency of oscillations - n dimensionless frequency - perturbation parameter - U a constant velocity - u ₀ mean velocity - u ₁ fluctuating part of the velocity - ⁰ mean angular velocity - ¹ fluctuating part of the angular velocity - T ₀ mean temperature - T ₁ fluctuating part of the temperature - ⁰ coefficient of the volume expansion - kinematic viscosity - ^r rotational kinematic viscosity - , material parameters characterizing the polarity of the fluid - v ₀ suction velocity - density of the fluid far from the surface - y dimensionless coordinate normal to the surface     

Effects of chemical reaction on free convection flow through a porous medium bounded by a vertical surface

The effect of a chemical reaction on a free convection flow through a porous medium bounded by a vertical infinite surface has been studied. Velocity, temperature, and concentration profiles have been obtained for different values of parameters like the Grashof number, Prandtl number, and the chemical reaction parameter in the presence of homogeneous chemical reaction of first order. It is observed that the velocity and concentration increase during a generative reaction and decrease in a destructive reaction. The same is true for the behavior of the fluid temperature. The presence of the porous media diminishes the temperature.     

Hydromagnetic convection flow through a porous medium in axially varying pipe

The hydromagnetic mixed convection flow through a porous medium in a pipe of varying radius in a uniform axial magnetic field is analyzed. The pipe wall is maintained at a prescribed nonuniform temperature. The governing equations are solved analytically to obtain the velocity, temperature, and induced magnetic field. Their behaviors are evaluated for different variations in the governing parameters. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 79, No. 4, pp. 96–104, July–August, 2006.     

Heat transfer in a viscoelastic boundary layer flow through a porous medium

This paper examines the viscoelastic fluid flow and heat transfer characteristics in a saturated porous medium over an impermeable stretching surface with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for two different heating processes, namely (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). The governing equations for the boundary layer flow problem result similar solutions. For the specified five boundary conditions, it is not possible to solve directly the resulting sixth-order nonlinear ordinary differential equation. For the present incompressible boundary layer flow problem with constant physical parameters, the momentum equation is decoupled from the energy equation. Two closed–form solutions for the momentum equation are obtained and identified the realistic solution of the physical problem. Exact solution for the velocity field and the skin-friction are obtained. Also, the solution for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Asymptotic results for the temperature function for large Prandtl numbers are presented. The work due to deformation in the energy equation, which is essential and escaped from the attention of researchers while formulating the visco-elastic boundary layer flow problems, is considered. Drastic variation in the values of heat transfer coefficient is observed when the work due to deformation is ignored.The authors would like to thank the reviewers for their valuable comments/ suggestions to improve the clarity of the paper.     

Effects of free convection on the oscillatory flow of a polar fluid through a porous medium in the presence of variable wall heat flux

The purpose of this work is to study a laminar two-dimensional free convective oscillatory flow of an incompressible polar fluid through a saturated porous medium occupying a semi-infinite region of the space which is bounded by an infinite vertical permeable plate in the presence of oscillating suction and variable wall heat flux. The governing equations are based on the local volume averaging technique. The generalized Darcy equation accounting for polar effects is employed. Analytical expressions for the linear momentum, angular momentum, and energy fields are obtained by using the two-term perturbation technique. The numerically computed results are shown graphically and compared with the corresponding ones for a viscous (Newtonian) fluid. Distribution of the mean velocity of a polar fluid is found to increase as compared to the Newtonian fluid. The analysis reveals multiple boundary-layer structure for velocity. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 5, pp. 868–884, September–October, 2008.     

Oscillatory flow through a porous medium by the presence of free convective flow

The purpose of this work is to study the effect of non-constant 2-dimensional free convective flow during the motion of a viscous incompressible fluid through a highly porous medium. The porous medium is bounded by a vertical plane surface of constant temperature. This surface absorbs the fluid with a constant velocity and the free stream velocity of the fluid vibrates about a mean constant value. Analytical expressions for the velocity of the fluid are given. The effects of Grashof number and the permeability parameter upon the velocity field are also shown in a graphic representation.     

Polar fluid through a porous medium

Summary An analysis of the steady flow of a polar fluid through a porous medium bounded by an infinite plate by using the generalized Forchheimer's model is considered. The velocity profiles are shown for different values of the permeability parameter.     

Incompressible fluid flow and heat transfer through a nonsaturated porous medium

This work studies a nonsaturated flow and the heat transfer associated phenomenon of a newtonian fluid through a rigid porous matrix, using a mixture theory approach in its modelling. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid and an inert gas, included to account for the compressibility of the system as a whole. A set of four nonlinear partial differential equations describe the problem whose hydrodynamical part is approximated by means of a Glimm's scheme combined with an operator splitting technique.     

Integral method to free convection in thermally stratified porous medium

Summary Present investigation deals with natural convection heat transfer from a vertical plate embedded in a fluid saturated porous medium whose ambient temperature is not uniform. The problem is solved by integral method of von-Karman type and the closed form solution is obtained. The effects of stratification parameter on local and overall heat transfer coefficients, velocity and temperature profiles are analysed.     

The flow of a viscoelastic fluid past a porous plate

Summary The flow of a second order viscoelastic fluid past a porous plate is considered. It is characterized by a boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. The boundary value problem is solved by making a plausible assumption, namely that the variation of the normal derivative of the velocity at the plate withk is sufficiently smooth, wherek is the viscoelastic fluid parameter. Under this assumption it is shown that dual solutions exist for values ofk less than a critical value. Beyond this value, no solution exists.     

MHD combined free and forced convection flow through two parallel porous walls

Summary The flow of a viscous conducting fluid between two parallel plates of infinite length (one of which is at rest and the other moving parallel to itself with a linear axial temperature variation) under the influence of a uniform transverse magnetic field is considered. It is assumed that the fluid is injected into the channel through the lower wall and sucked from the channel through the upper wall with the same velocityv ₀. The velocity and temperature distributions, the coefficient of skin friction and the rate of heat transfer coefficient are evaluated. The effects of magnetic parameterM, the suction Reynolds numberR, the Grashoff numberG and Binkmann numberB on the above mentioned physical quantities are investigated.With 21 Figures     

Magnetohydrodynamic poiseuille flow through a porous medium

Abstract

This paper is concerned with formulating equations for the flow of an electrically conducting fluid through a non‐conducting porous medium with non‐porous and non‐conducting boundaries. Equations are developed for the general case of the flow of a solid‐fluid suspensions; flow through a porous medium is treated as a special case by letting the velocity of the particle phase goes to zero. Two cases are considered. Exact solution is obtained for the case of flow between parallel plates, but for flow in pipes of square and circular cross sections, the equations have to be solved numerically. The numerical technique developed can treat elliptical cross sections as well. The flow in all cases is assumed to be steady, laminar, incompressible, viscous, and fully developed. The results are presented in terms of a parameter which measures the resistance of the porous medium.     

Numerical solution of free convection micropolar fluid flow between two parallel porous vertical plates

The fully developed free convection micropolar fluid flow between two vertical porous parallel plates is studied in the presence of temperature dependent heat sources including the effect of frictional heating. The basic equations are solved using quasi-linearization finite difference technique with an error of order 0.5 × 10₋₆. The velocity, microrotation and temperature are displayed in graphs whereas the skin friction, couple stress and Nusselt numbers at the plates are shown in tables. It is noted that the couple stress on either plates increases numerically with increase in micropolar parameter. Also the Nusselt number follows the same pattern for a negative suction velocity.     

Numerical solution of free convection MHD micropolar fluid flow between two parallel porous vertical plates

The fully developed electrically conducting micropolar fluid flow between two vertical porous parallel plates is studied in the presence of temperature dependent heat sources including the effect of frictional heating and in the presence of a magnetic field. Profiles for velocity, microrotation and temperature are presented for a wide range of Hartmann numbers and the micropolar parameter. The skin friction, couple stress and Nusselt numbers at the plates are shown in the tables.     

Transient free convection about a vertical flat plate embedded in a porous medium

The problems of transient free convection in a porous medium adjacent to a vertical semi-infinite flat plate with a step increase in wall temperature and surface heat flux are considered in this paper. By assuming a temperature profile in each case, the governing equation for the boundary layer thickness is obtained by an integral method. These governing equations are first-order partial differential equations of the hyperbolic type that can be solved exactly by the method of characteristics and approximately by the method of integral relations. The results based on the method of characteristics clearly indicate that during the initial stage when the leading edge effect is not being felt, heat is transferred as if by transient 1-dimensional heat conduction. At a later time, depending on the vertical location, the heat transfer characteristics change from transient 1-dimensional heat conduction to steady 2-dimensional convection. The thickness of the boundary layer is shown to be increasing with time until it reaches steady state where its value remains constant thereafter. The growth rate of the boundary layer thickness exhibits a discontinuity at the end of the transient period and the beginning of the steady state period. On the other hand, the results based on the method of integral relations show that the boundary layer thickness grows continuously with time and approaches the steady state value asymptotically; the growth rate of the boundary layer thickness decreases from a finite value to zero continuously as the steady state is approached. Except between the end of the transient period and the beginning of the steady state period, the results based on the method of integral relations are in good agreement with those based on the method of characteristics.     

Nonsteady flow of non-Newtonian fluids through a porous medium

In this paper, the nonsteady flow of power-law fluids with a yield stress through a porous medium is investigated. In order to illustrate the rheological behaviour effects on pressure and flow rate distributions, a flow system of practical interest was analysed. The approximate solutions in a closed form were obtained by using the integral method. For a short time, these have been formulated in terms of a moving boundary problem. The flow deviation from Newtonian behaviour was emphasized by means of several dimensionless groups. These have been found to be relevant in evaluating the rheological effects on steady and nonsteady flow behaviour through a porous medium.     

Non-Darcian convection flow in a circular duct partially filled with porous medium

The flow and heat transfer in a circular duct bounded by a porous bed is considered. The entire flow region is divided into two zones. The clean fluid region is described by the Navier–Stokes equations, while the Brinkman-extended Darcy model is used in the flow through a porous bed. In either zone the momentum and temperature equations are coupled. In order to obtain a better insight into this complex problem, the Galerkin finite element analysis with quadratic polynomial approximations is applied. The behavior of the velocity and temperature is analyzed. The shear stress and the rate of heat transfer are also obtained for various governing parameters.     

Mixed convection flow in a porous medium bounded by two vertical walls

The aim of the investigation is to study the fully developed mixed convection flow in a porous medium bounded by two vertical walls having a linear axial temperature variation. Solution of the governing second order simultaneous differential equations, derived by non-dimensionalising the momentum and energy equations, has been obtained by converting them into a fourth order differential equation. It has been found that the fourth order differential equation has five different solutions depending upon the values of the Darcy number, Rayleigh number and ratio of the effective viscosity of the porous domain to the viscosity of the fluid. Graphical representation of the analytical solution shows that the flow is of parabolic type only for higher Darcy number whereas there is a reverse type of motion for lower Darcy numbers. The effect of the ratio of the effective viscosity of the porous matrix to the viscosity of the fluid is to decrease the velocity field.