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.     

Heat transfer in laminar flow of Bingham material through a circular pipe

Summary The fully developed heat transfer in laminar flow of Bingham material through a straight circular pipe has been analysed when the effect of dissipation is taken into consideration. The temperature distribution, the mixed-mean temperature and the Nusselt number are calculated and found to depend on the Modified Reynolds number, the Brinkman number and the rate of heat transfer at the wall. The effect of dissipation is to increase the temperature and the mixed-mean temperature of the material while the Nusselt number is found to decrease with increasing dissipation.Nomenclature r, , z space coordinates - u, v, w velocity components - density of the fluid - modulus of rigidity (constant) - bulk modulus - e _kk, the dilation - ₁ coefficient of viscosity (constant) - e _ik strain tensor - d _ik rate of strain tensor - p _ik stress tensor primes denote deviatoric components of tensors, e.g. - p _ik p _ik p _ik, p=–1/3p _kk - yield value (constant) - D/Dt material derivative with regard to time following the particle - R radius of the pipe - r ₀ radius of the yield surface (cylindrical) - r=r ₀/R non-dimensional radius of the yield surface - T ₀ temperature of the pipe - K conductivity - Modified Reynolds number - Br Brinkman number     

Ion slip effect on a steady flow through a circular pipe of a dusty conducting Oldroyd 8-constant fluid

In this paper, a steady magnetohydrodynamic (MHD) flow of a dusty incompressible electrically conducting Oldroyd 8-constant fluid through a circular pipe is examined with considering the ion slip effect. A constant pressure gradient in the axial direction and an external uniform magnetic field in the perpendicular direction are applied. A numerical solution is obtained for the governing nonlinear momentum equations by using finite differences. The effect of the ion slip, the non-Newtonian fluid characteristics, and the particle-phase viscosity on the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is reported.     

Unsteady hydromagnetic Couette flow through a porous medium in a rotating system

This paper investigates the unsteady hydromagnetic Couette fluid flow through a porous medium between two infinite horizontal plates induced by the non-torsional oscillations of one of the plates in a rotating system using boundary layer approximation. The fluid is assumed to be Newtonian and incompressible. Laplace transform technique is adopted to obtain a unified solution of the velocity fields. Such a flow model is of great interest, not only for its theoretical significance, but also for its wide applications to geophysics and engineering. Analytical expressions for the steady state velocity and shear stress on the plates are obtained, and the case of single oscillating plate is also discussed. The influence of pertinent parameters on the flow is delineated, and appropriate conclusions are drawn.     

Unsteady flow through a porous medium in the presence of chemical reactions

The underground leaching of uranium ores and nonferrous and precious metals under natural conditions is one of the latest methods of mineral extraction [1]. It consists of pumping into isolated formations through reaction wells an acid solution which upon reacting with the rock yields a readily soluble salt that can be brought to the surface with water through extraction wells. Together with the acid solution, it is also possible to pump in other reactants to participate in the chemical reaction, for example, gases such as oxygen. Moreover, secondary gases may be formed as a result of the chemical reaction. Thus, the chemical reaction proceeds in the presence of a one or two-phase flow in the porous medium. The mathematical modeling of these processes is usually based on the approximation of one-phase flow without allowance for the changes in the porosity and permeability of the medium as a result of the reaction [2]. In this paper we solve the problem of unsteady flow in the presence of a chemical reaction for a two-phase system taking into account the changes in the flow parameters of the porous medium. The condition of stability of the plane reaction front is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 82–87, January–February, 1987.The author is grateful to R. I. Nigmatulin for his useful comments and interest in the work.     

Steady flow and heat transfer of a magnetohydrodynamic Sisko fluid through porous medium in annular pipe

In this paper, the steady flow and heat transfer of a magnetohydrodynamic fluid is studied. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field and occupies the porous space in annular pipe. The governing nonlinear equations are modeled by introducing the modified Darcy's law obeying the Sisko model. The system is solved using the homotopy analysis method (HAM), which yields analytical solutions in the form of a rapidly convergent infinite series. Also, HAM is used to obtain analytical solutions of the problem for noninteger values of the power index. The resulting problem for velocity field is then numerically solved using an iterative method to show the accuracy of the analytic solutions. The obtained solutions for the velocity and temperature fields are graphically sketched and the salient features of these solutions are discussed for various values of the power index parameter. We also present a comparison between Sisko and Newtonian fluids. Copyright © 2011 John Wiley & Sons, Ltd.     

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 circular Couette flow of a Bingham plastic with the Augmented Lagrangian Method

Numerical simulations are undertaken for unsteady flows of an ideal Bingham fluid in a circular Couette viscometer. The main difficulties in such simulations are caused by the non-differentiability of the constitutive equation and the need to determine the position and shape of the yield surface separating the yielded zones from the unyielded ones. In this work, these difficulties are overcome by using a numerical method based on variational inequalities, i.e. the augmented Lagrangian/Uzawa method. The start-up and cessation of circular Couette flows of a Bingham fluid are solved numerically assuming that only one of the cylinders is rotating. An improved theoretical upper bound for the stopping time in the case of cessation is derived. The numerical estimates for the stopping time compare well with the theoretical bounds. Moreover, with the adopted method the evolution of the velocity profiles and the locations of yielded/unyielded surfaces are accurately calculated. In flow cessation, we observe an interesting effect, namely the appearance of a small unyielded region adjoined to the outer cylinder shortly before cessation.     

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.     

Very slow flow of Bingham viscoplastic fluid around a circular cylinder

Numerical simulations have been used to study the flow of a Bingham viscoplastic fluid around a circular cylinder in an infinite medium with negligible inertia effects. Papanastasiou's regularisation technique has been adopted to approximate the model. The case corresponding to preponderant plasticity effects has been particularly studied and convergence of the solutions examined in detail. The flow kinematics and stresses have been determined. The rigid zones have been identified and characterised. At large Oldroyd numbers, when plasticity effects become preponderant, a viscoplastic boundary layer appears around the cylinder. The characteristics of this viscoplastic boundary layer are quantified. The results are compared with existing theoretical results, concerning particularly the predictions of the viscoplastic boundary layer theory and the plasticity theory.     

Entropy generation in developing laminar fluid flow through a circular pipe with variable properties

Entropy generation in a circular pipe is analyzed numerically. A two-dimensional solution for the velocity ant temperature profiles is obtained considering temperature dependent thenmophysical properties. Uniform wall heat flux case is considered as the thermal boundary condition. The distribution of the entropy generation rate is investigated throughout the volume of the fluid as it flows through the pipe. Engine oil is selected as the working fluid. In addition, ethylene glycol and air are used in a parametric study. The total entropy generation rate is calculated by integration over the various cross-sections as well as over the entire volume. The results are compared with those obtained for the constant viscosity case. A considerable discrepancy is found between the two cases since the viscosity of these fluids is highly sensitive to the temperature variation.     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Axial flow of gelling materials through a circular pipe

Summary A set of constitutive equations for a group of incompressible materials of technological importance previously proposed by the authors is used to analyze the axial flow of a gelling Binghamlike material. Through five material parameters these constitutive equations account for the phenomena of breakdown in rigidity after deformation has occurred and of recovery in rigidity in the state of continued deformation or in the state of rest. Due to the complexity of predicting the flow of such nonlinear materials only the steady state behaviour in general seems to be tractable.The solution presented here describes the steady state axial flow of this type material through a circular pipe. It is shown that, depending on the choice of material parameters, two separate solutions may occur. As in the case of axial flow of a Bingham material or a retarded Bingham material a concentric core with rigid body motion is found. Analytically the radius of this core enters into the formulation of the requirements that must be fulfilled to establish a continuous flow field.It is further shown that volume flow rate dependence on the current pressure gradient may be a function of the loading history. For a specific case this dependence is shown in graphical form.     

Unsteady rotative flow of non-Newtonian fluid in an annular pipe

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Hall effects on the oscillatory MHD flow through porous medium in a rotating fluid

Taking Hall currents into account the unsteady magnetohydrodynamical flow through a porous medium bounded by an infinite vertical limiting surface in a rotating frame of reference is theoretically investigated when a strong magnetic field is imposed in a plane which makes an angle a with the normal to the plate. The influence of Hall currents on the velocity and temperature distribution are shown graphically for various values of .

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Hall-Effekte in oszillierender magnetohydrodynamischer Strömung durch ein poröses Medium in einer rotierenden Flüssigkeit

Zusammenfassung Bei der Betrachtung von Hallströmen wird die unstetige magnetohydrodynamische Strömung durch ein poröses Medium, begrenzt durch eine unendlich lange senkrechte Fläche in einem rotierenden Rahmen, für den Fall theoretisch untersucht, daß ein starkes magnetisches Feld in einer Ebene, die den Winkel a mit der normalen auf die Fläche einschließt, angelegt wird. Der Einfluß der Hallströme auf die Geschwindigkeit und die Temperaturverteilung wird für verschiedene Werte von graphisch dargestellt. Wärme- und Stofflibertragung 23 (1988)

     

Surface deposition from fluid flow in a porous medium

The changes to porosity and permeability resulting from surface deposition and early dissolution in an initial rhombohedral array of uniform spheres are studied. Very rapid decreases in permeability result from early deposition, with 48 percent reduction predicted in permeability from 8 percent reduction in porosity. After deposition has caused about a 1 percent increase in the radii of the spherical array, relative permeability reductions vary approximately as the square of relative changes in porosity. These theoretical results are matched with experimental data of Itoi et al. and Moore et al. on deposition of silica. Satisfactory results are obtained in some cases, but for other cases a more complex model of the porous medium is needed.