Viscous fluid flow between moving parallel plates

The problem of two-dimensional time-dependent viscous fluid flow in a clearance between transversely and longitudinally moving rigid planes is considered. Non-self-similar solutions of this problem are found within the framework of the Hiemenz class of exact solutions of hydrodynamic equations and the admissible laws of motion of a movable plane are described.     

LAMINAR DISSIPATIVE FLOW IN A POROUS CHANNEL BOUNDED BY ISOTHERMAL PARALLEL PLATES

IntroductionTheproblemofforcedconvectioninaporousmediumchannelorductisaclassicalone (atleastforthecaseofslugflow (Darcymodel) .Therehasrecentlybeenrenewedinterestintheproblembecauseoftheuseofhyperporousmediainthecoolingofelectronicequipment.Recently ,NieldandBejan[1]refertomorethan 3 0papersonthetopic ,butnoneofthemdealsexplicitlywiththecaseofthermaldevelopment.ThisgapintheliteraturehasbeenpartlyfilledbyNieldetal.[2 - 4 ].Lahjomrietal.[5 ,6 ]havesolvedmathematicallysimilarproblemsbyusingthe…     

Heat and mass transfer in laminar flow between parallel porous plates

Summary The problem of heat transfer in a two-dimensional porous channel has been discussed by Terrill [6] for small suction at the walls. In [6] the heat transfer problem of a discontinuous change in wall temperature was solved. In the present paper the solution of Terrill for small suction at the walls is revised and the whole problem is extended to the cases of large suction and large injection at the walls. It is found that, for all values of the Reynolds number R, the limiting Nusselt number Nu increases with increasing R.Nomenclature stream function - 2h channel width - x, y distances measured parallel and perpendicular to the channel walls respectively - U velocity of fluid at x=0 - V constant velocity of fluid at the wall - =y/h nondimensional distance perpendicular to the channel walls - f() function defined in equation (1) - coefficient of kinematic viscosity - R=Vh/ suction Reynolds number - density of fluid - C _p specific heat at constant pressure - K thermal conductivity - T temperature - x=x ₀ position where temperature of walls changes - T ₀, T ₁ temperature of walls for x<x ₀, x>x ₀ respectively - = (T – T ₁)/T ₀ – T ₁) nondimensional temperature - =x/h nondimensional distance along channel - R ^* = Uh/v channel Reynolds number - Pr = C _p/K Prandtl number - _n eigenvalues - B _n() eigenfunctions - B _n ⁽⁰⁾ , () eigenfunctions for R=0 - B ₀ ⁽ⁱ⁾ , B ₀ ⁽ⁱⁱ⁾ ... change in eigenfunctions when R0 and small - K _n constants given by equation (13) - h heat transfer coefficient - Nu Nusselt number - _m mean temperature - C _n constants given by equation (18) - perturbation parameter - B _0i () perturbation approximations to B ₀() - Q = B ₀/ ₀ derivative of eigenfunction with respect to eigenvalue - z nondimensional distance perpendicular to the channel walls - F(z) function defined by (54)     

Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium

Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.Nomenclature C empirical constant for the Forchheimer term - f velocity function for the small time solution - F velocity function for the large time solution - g acceleration due to gravity - Gr* micro-scale Grashof number - H a half distance between two infinite plates - K permeability - Nu Nusselt number - Pr Prandtl number - t time - T temperature - u, v Darcian velocity components - x, y Cartesian coordinates - effective thermal diffusivity - coefficient of thermal expansion - porosity - dimensionless time - similarity variable - dimensionless temperature - viscosity - kinematic viscosity - density - the ratio of heat capacities     

Two-phase fluid flow in a porous tube: a model for blood flow in capillaries

A theoretical study of blood flow, under the influence of a body force, in a capillary is presented. Blood is modeled as a two-phase fluid consisting of a core region of suspension of all erythrocytes, represented by a micropolar fluid and a plasma layer free from cells modeled as a Newtonian fluid. The capillary is modeled as a porous tube consisting of a thin transition Brinkman layer overlying a porous Darcy region. Analytical expressions for the pressure, microrotation, and velocities for the different regions are given. Plots of pressure, microrotation, and velocities for varying micropolar parameters, hydraulic resistivity, and Newtonian fluid layer thickness are presented. The overall system was found to be sensitive to variations in micropolar coupling number. It was also discovered that high values of hydraulic resistivity result in an overall slower velocity of the micropolar and Newtonian fluid.     

Peristaltic flow of a Johnson-Segalman fluid through a deformable tube

To understand theoretically the flow properties of physiological fluids we have considered as a model the peristaltic motion of a Johnson–Segalman fluid in a tube with a sinusoidal wave traveling down its wall. The perturbation solution for the stream function is obtained for large wavelength and small Weissenberg number. The expressions for the axial velocity, pressure gradient, and pressure rise per wavelength are also constructed. The general solution of the governing nonlinear partial differential equation is given using a transformation method. The numerical solution is also obtained and is compared with the perturbation solution. Numerical results are demonstrated for various values of the physical parameters of interest.      

Nonisothermal viscoplastic fluid flow through porous media

The Alishaev model [1] is extended to the case of nonisothermal flow. Neglecting conductive heat transfer, it is shown that for the model in question in the plane of the complex potential not only are the problems linear but the decoupling of the thermal and hydrodynamic problems is also allowed. The latter is reduced to a mixed problem for an analytic function. This makes it possible to use the wellknown methods and results of the theory of limiting equilibrium pillars for isothermal flow [2–5]. It is also established that the solutions of the unsteady problems tend asymptotically to the solutions of the corresponding steady-state problems and can be obtained from the latter by simpler conversion. The effectiveness of the approach proposed is illustrated with reference to the problem of a source-sink system [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–122, July–August, 1990.     

Unsteady dissipative structures in non-Newtonian fluid flow through a porous medium

The nonuniform space-time pressure and velocity distributions in an initially nonempty stratum with constant initial pressure created by pumping a non-Newtonian fluid through the boundary of the stratum are investigated. The injected fluid and the fluid present in the stratum before injection have identical physical properties. The conditions of formation of traveling fronts and localized structures are analyzed as functions of the nonlinearity of the rheological law of the fluid and the injection regime.Baku. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–112, November–December, 1994.     

MHD stagnation flow of a micropolar fluid through a porous medium

The unsteady MHD boundary layer flow of a micropolar fluid near the forward stagnation point of a two dimensional plane surface is investigated by using similarity transformations. The transformed nonlinear differential equations are solved by an analytic method, namely homotopy analysis method (HAM). The solution is valid for all values of time. The effect of MHD and porous medium, non dimensional velocity and the microrotation are presented graphically and discussed. The coefficient of skin friction is also presented graphically.     

Unsteady natural convection Couette flow of heat generating/absorbing fluid between vertical parallel plates filled with porous material

The extended Brinkman Darcy model for momentum equations and an energy equation is used to calculate the unsteady natural convection Couette flow of a viscous incompressible heat generating/absorbing fluid in a vertical channel(formed by two infinite vertical and parallel plates) filled with the fluid-saturated porous medium.The flow is triggered by the asymmetric heating and the accelerated motion of one of the bounding plates.The governing equations are simplified by the reasonable dimensionless parameters and solved analytically by the Laplace transform techniques to obtain the closed form solutions of the velocity and temperature profiles.Then,the skin friction and the rate of heat transfer are consequently derived.It is noticed that,at different sections within the vertical channel,the fluid flow and the temperature profiles increase with time,which are both higher near the moving plate.In particular,increasing the gap between the plates increases the velocity and the temperature of the fluid,however,reduces the skin friction and the rate of heat transfer.     

Analytic solution for flow of Sisko fluid through a porous medium

This paper presents the analytic solution for flow of a magnetohydrodynamic (MHD) Sisko fluid through a porous medium. The non-linear flow problem in a porous medium is formulated by introducing the modified Darcy’s law for Sisko fluid to discuss the flow in a porous medium. The analytic solutions are obtained using homotopy analysis method (HAM). The obtained analytic solutions are explicitly expressed by the recurrence relations and can give results for all the appropriate values of material parameters of the examined fluid. Moreover, the well-known solutions for a Newtonian fluid in non-porous and porous medium are the limiting cases of our solutions.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.     

Dynamic fluid flow behaviour of a tank draining through a vertical tube

This paper describes the unsteady draining of a sealed tank partially filled with water. The water discharges via a vertical tube into an open tank at atmospheric conditions. The air inflow, compensating for the volume of the discharged liquid, enters the system in an oscillatory manner, much like the “gulping” seen in an upended beer bottle. A mathematical model, based closly on that derived by Dougall & Kathiresan [Chem. Engng Commun. 8, 289–304 (1981)], has been applied to predict the pressure fluctuations in the closed tank. The rate of water discharge from the tank has been predicted and gives a much closer agreement with experimental results than a prediction based on a steady counter-current flooding limitation approach. A drift flux model has been used to describe the two-phase flow effect in the tube and the Wallis flooding criterion has been modified for use in the slug flow regime to describe the boundary conditions at the bottom of the tube. The pressure fluctuations in the sealed tank have been measured and compared with results obtained from the mathematical prediction for a variety of tube diameters.     

Flow of a dipolar fluid between parallel plates

The flow of a dipolar fluid between two parallel plates with and without heat transfer is studied. The following cases are discussed:

1. Isothermal flow due to the relative motion of the plates,
2. Isothermal flow due to a constant pressure gradient with the plates at rest,
3. Nonisothermal flow with linearly varying plate temperatures.

Case (ii) is of particular interest to the experimentalists as it shows the effect of the material constants even when there are no externally applied dipolar tractions on the plates.     

Squeezing flow between parallel plates

Summary The flow between two parallel plates (rectangular or circular) approaching or receding from each other symmetrically is analysed. The Xavier-Stokes equations have been transformed into an ordinary differential equation using a similarity transformation and the resulting equations are solved numerically. Results for the velocity components, pressure distribution and shearing stress on the wall are presented. In the case of squeezing flow between two circular plates the load supporting capacity of the upper plate has been calculated.

---

Quetschströmung zwischen parallelen Platten

Übersicht Untersucht wird die Strömung zwischen zwei parallelen Rechteck- bzw. Kreisplatten, die sich einander nähern oder entfernen. Die Navier-Stokes-Gleichungen werden durch eine Ähnlichkeitstransformation in eine gewöhnliche Differentialgleichung überführt. Die Lösung erfolgt numerisch. Ergebnisse für die Geschwindigkeitskomponenten, die Druckverteilung und die Wandschubspannung werden vorgestellt. Für die Quetschströmung zwischen zwei Kreisplatten wird die Tragkraft bestimmt.

     

Nonstationary model of immiscible fluid flow through porous media

Unsteady two-phase flow through a microinhomogeneous porous medium is considered. A forest growth model — a percolation model that enables nonequilibrium effects to be taken into account — is proposed for describing the dynamics of the process. In the context of the plane problem expressions are obtained for determining the saturation and the characteristic dimensions of the stagnation zones of trapped phase behind the displacement front.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 73–80, November–December, 1993.     

A numerical study of unsteady gas-solid flow between parallel porous plates submitted to a magnetic field

This paper studies unsteady laminar flow of dusty conducting fluid between parallel porous plates with temperature dependent viscosity and the Network Simulation Method (NSM) is used to solve the governing nonlinear partial differential equations. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates that are assumed to be porous. The NSM is applied to solve the steady-state and transient problems of flow and heat transfer for both the fluid and dust particles. With this method, only discretization of the spatial co-ordinates is necessary, while time remains as a real continuous variable. The velocity and temperature are studied for different values of the viscosity and magnetic field parameters.