Flow of a second grade fluid through a cylindrical permeable tube

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Abstract

In the present work, a second grade fluid flow in a cylindrical tube is investigated. The perturbations were expanded in terms of the second order material coefficient, which means that either the second order parameter is sufficiently small or the characteristic diameter is very large. After obtaining the solutions for the impermeable tube, problem is examined for suction or injection through the surfaces of the tube. The general solutions are obtained for a tube of circular cross-section and also applied to a tube of arbitrary cross-section by using conformal transformation.

Introduction

There are several models that have been proposed to predict the non-Newtonian behavior of various type of materials. One class of fluids which has gained considerable attention in recent years is the fluids of grade n. A great deal of information for these types of fluids can be found in the work of Dunn and Rajagopal [1]. Among these fluids, one special subclass associated with second order truncations is called second-grade fluids. The constitutive equation of a second grade fluid is given by the following relation for incompressible fluids:t=-pI+μA1+α1A2+α2A12,where t is the stress tensor, p is the pressure, μ is the classical viscosity, α1 and α2 are the material coefficients. A1 and A2 are the first two Rivlin–Ericksen tensors defined byA1=L+LT=2d,A2=A˙1+A1L+LTA1,where L = grad v (Lij = vj,i), v is the velocity field, d is the deformation tensor and the overdot represents the material derivative with respect to timef˙=dfdt=ft+v·f.This type constitutive relation was first proposed by Coleman and Noll [2]. Dunn and Fosdick [3] have shown that the following conditions:α1+α2=0,μ0,α10must be satisfied for the second-grade fluid to be entirely consistent with classical thermodynamics and the free energy function achieves its minimum in equilibrium.

In recent years considerable efforts have been devoted to the study of fluid flows of non-Newtonian fluids because of their practical and fundamental importance associated with many applications. Several authors have studied the theory of non-Newtonian fluids in various geometrical configurations. For the useful models for second grade fluids we refer the reader the following works [4], [5], [6], [7], [8], [9]. As a boundary value problem, suction and injection have been considered by several authors. El Dabe et al. [10] considered the incompressible fluid flow slowly varying circular cross-section having a permeable wall. Hayat and Hutter [11] studied second order incompressible fluid by an infinite porous plate. Hayat et al. [12] examined the flow of a second order incompressible fluid flow bounded by a porous disk. Maxwell fluids through a tube with porous walls are considered by Menon et al. [13] and Larson [14]. Incompressible third order fluid due to non-coaxial rotations of a porous disk is studied by Hayat et al. [15]. Recently Fetecau [16] examined unsteady unidirectional transient flows of an Oldroyd-B fluid in pipe-like domains. He also considered second grade fluids. Hayat et al. [17] studied incompressible, unsteady second order fluid flows for Couette flow, flow between two parallel plates and Poiseuille flow. Existence and uniqueness of stationary non-Newtonian fluid flows in channel-like and pipe-like domains are examined by Fentelos [18] under the assumption of large kinematic viscosity and the results are applied to some non-Newtonian fluid flows including second grade fluids.

Section snippets

Equations of motion

Dunn and Fosdick [3] have derived the field equations for an incompressible second order fluid in the absence of body forces as follows:ρvt+w×v+grad|v|22=-gradp+μ2v+α2vt+2w×v+gradv·2v+14|A1|2,where ρ is the density, w is the usual vorticity vector defined by w =  × v, α = α1 = α2, ∣A12 = 4dijdij and vt=vt. We next define non-dimensional variables byv=vU,p=pρU2,t=tUL,x=xLand introduce the following non-dimensional parametersRe=ρULμ,ε=αρL2,where U is a characteristic velocity, L is a

Formulation of the problem

We consider the steady flow through a cylindrical tube and assume that the velocity vector is dependent only on two cross-sectional variables x and y. It is then possible to define the velocity field asv=(ψy,-ψx,ω),where ψ is the stream function ψ = ψ(x, y) and ω is the axial velocity component ω = ω(x, y). Subscripts represent partial derivatives with respect to indicated variables. Substituting the vector field (7) into the equation of motion (6) we have(ψyωx-ψxωy)y=1Re2ω+ε[ψy(2ω)x-ψx(2ω)y]y,(ψxω

Conformal mapping

In Section 3, the velocity field of the second-grade fluid flow for a circular cross-sectional cylindrical tube is obtained. These results can be generalized for a tube of arbitrary cross-section by using conformal mapping. The expression z = h(σ) establishes a correspondence between the points of complex planes σ and z. Since z = h(σ) is analytical in a R domain of σ plane, the values of z will be analytical in a R′ domain of the plane z. Consequently R is transformed to R′ by using the expression

Acknowledgement

One of the authors (E.S. Şuhubi) acknowledges the support provided by Turkish Academy of Sciences.

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