Abstract

This investigation concerns the analytical solutions for the Riabouchinsky time-dependent flows of an incompressible second-grade fluid. Semi-inverse method has been used for the solutions of highly non-linear differential equations.

1. Introduction

The inadequacy of the classical Navier-Stokes theory to describe rheological complex fluids such as polymer solutions, blood, paints, certain oils and greases, has led to the development of several theories of non-Newtonian fluids. Amongst these the differential type have received special attention. One particular subclass of differential type fluids for which one can reasonably hope to get the analytic solution is the second-grade fluid.

Motivated with the above facts, the analytical solutions for unsteady Riabouchinsky flows of second-grade fluids are constructed in this note. The analytical solutions of the governing non-linear equations are obtained using inverse methods.

In studying Riabouchinsky flows [4] the streamfunction is taken to be linear in one of the space dimensions. In this way the fifth order compatibility partial differential equation in terms of stream functions can readily be written by two coupled fifth order ordinary differential equations. Solutions are then obtained by a suitable assumption of one equation. Actually, Riabouchinsky considered [psi](x, y) = yf (x); the plane steady flow which represents the flow where flow is separated in the two symmetrical regions by a vertical or horizontal plane. Recently, Hamdan [2] gave an alternate approach to find exact solutions of Riabouchinsky flows. He after choosing [psi] (x, y) = yf (x) + g (x) obtained two coupled fourth order equations in f and g; and choose a solution of g and then find the function f. The advantage of this approach is that; the constant appearing in Riabouchinsky flows can easily be obtained. However, the solutions assumed by Hamdan [2] cannot be all applied in second grade fluid and accordingly the alternate approach will not be feasible. In the present work we consider unsteady flows which have the stream function, [psi] (x, y, t) = x[xi] (z, t), z = y + Kx. The Riabouchinsky's [4] and Hamdan's [2] solutions can be recovered from the present investigations.

In terms of the stream function, the governing equation for unsteady flow is [3]

[rho] [[[partial derivative]/[partial derivative]t][[nabla].sup.2][psi] - {[psi], [[nabla].sup.2][psi]}] = ([mu] + [[alpha].sub.1][[partial derivative]/[partial derivative]t])[[nabla].sup.4][psi] - [[alpha].sub.1] {[psi], [[nabla].sup.4], [psi]}, (1)

in which [[alpha].sub.1] is the viscoelasticity of the second-grade fluid, [rho] is the density, [mu] is the dynamic viscosity, [psi] is the stream function and

{[psi], [[nabla].sup.2]} = [[partial derivative][psi]/[partial derivative]x][[[partial derivative]([[nabla].sup.2][psi])]/[partial derivative]y] - [[partial derivative][psi]/[partial derivative]y] [[[partial derivative]([[nabla].sup.2][psi])]/[partial derivative]x].

It is worthmentioning to note that Eq. (1) is highly non-linear. The exact solutions in closed form is impossible. Even, the closed form solution in case of Newtonian fluid ([[alpha].sub.1] = 0) is not possible. Similar to several previous studies, our interest lies in finding some analytical solutions of Eq. (1) with appropriate form of the stream function. Thus we consider the stream function as [2]

[psi] (x, y, t) = x[xi] (z, t), z = y + Kx, (2)

[psi] (x, y, t) = x[xi] (z, t) + [eta] (z, t). (3)

2. Solutions of Specific Flows

2.1 Flow where [psi] (x, y, t) = x[xi] (z, t), z = y + Kx

Substitution of Eq. (2) into Eq. (1) gives

[[xi]".sub.t] + [xi]'[xi]" - [xi][xi]'" = [nu] (1 + [K.sup.2])[[xi].sup.IV] + [alpha] (1 + [K.sup.2]) [[[xi].sub.t.sup.IV] + [xi]'[[xi].sup.IV] - [xi][[xi].sup.V]], (4)

where [xi] (z, t) is an arbitrary function of z, t, subscript t indicate the derivative with respect to time, primes (', IV) denote the derivatives with respect to z and

[nu] = [mu]/[rho], [alpha] = [[alpha].sub.1]/[rho], (5)

where [nu] is the kinematic viscosity. For the solution of Eq. (4) we write

[xi] (z, t) = V + [lambda] (z + Vt) = V + [lambda](s), s = z + Vt, (6)

where V is an arbitrary constant and [lambda] satisfies the following non-linear ordinary differential equation

[d[lambda]/ds][[[d.sup.2][lambda]]/[d[s.sup.2]]] - [lambda][[[d.sup.3][lambda]]/[d[s.sup.3]]] = [nu] (1 + [K.sup.2])[[[d.sup.4][lambda]]/[d[s.sup.4]]] + [alpha](1 + [K.sup.2])[d[lambda]/ds][[[d.sup.4][lambda]]/[d[s.sup.4]]] - [lambda][[[d.sup.5][lambda]]/[d[s.sup.5]]]. (7)

The first integral of Eq. (7) for zero constant of integration is

(d[lambda]/ds)[.sup.2] - [lambda][[[d.sup.2][lambda]]/[d[s.sup.2]]] = [nu] (1 + [K.sup.2])[[[d.sup.3][lambda]]/[d[s.sup.3]]] + [alpha](1 + [K.sup.2])[2[d[lambda]/ds][[d.sup.3][lambda]/d[s.sup.3]] - ([d.sup.2][lambda]/[d[s.sup.2]])[.sup.2] - [lambda][[[d.sup.4][lambda]]/d[s.sup.4]]]. (8)

Taking

[lambda](s) = A(1 + [ce.sup.as]) (9)

into Eq. (8) we can write

A = [[nu] (1 + [K.sup.2])a]/[[alpha] (1 + [K.sup.2]) [a.sup.2] - 1], (10)

where A, c and a are arbitrary real constants. From Eq. (2) we have

[psi](x, y, t) = Vx + [[[nu] (1 + [K.sup.2])a]/[[alpha] (1 + [K.sup.2]) [a.sup.2] - 1]] (1 + c[e.sup.a(y + Kx + Vt)]) (11)

and the corresponding velocity fields are

u = [[[a.sup.2]c[nu] (1 + [K.sup.2])]/[[alpha](1 + [K.sup.2]) [a.sup.2] - 1]] [e.sup.a(y + Kx + Vt)], (12)

v = -V - [[cK[a.sup.2][nu] (1 + [K.sup.2])]/[[alpha] (1 + [K.sup.2]) [a.sup.2] - 1]] [e.sup.a(y+Kx+Vt)]. (13)

2.2 Flow where [psi] (x, y, t) = x[xi] (z, t) + [eta] (z, t), z = y + Kx

Here, we use Eq. (3) into Eq. (1) and obtain the following non-linear partial differential equations

[[xi]".sub.t] + [xi]'[xi]" - [xi][xi]'" = [nu] (1 + [K.sup.2])[[xi].sup.IV] + [alpha](1 + [K.sup.2])[[[xi].sub.t.sup.IV] + [xi]'[[xi].sup.IV] - [xi][[xi].sup.V]], (14)

[[eta]".sub.t] + [eta]'[xi]" - [xi][eta]'" = [2K/[1 + [K.sup.2]]] ([xi][xi]" - [[xi]'.sub.t]) + [nu] [4K[xi]'" + (1 + [K.sup.2])[[eta].sup.IV]] + [alpha][4K ([[xi]'".sub.t] - [xi][[xi].sup.IV]) + (1 + [K.sup.2]){[[eta].sub.t.sup.IV] + [eta]'[[xi].sup.IV] - [xi][[eta].sup.V]}]. (15)

We note that the partial differential equation for [xi] is the same as in section 2.1. When [xi] is known, the second equation is a linear partial differential equation for the determination of [eta] (z, t). In order to get the time independent equations, we introduce the following transformations

[xi] (z, t) = V + [lambda] (z + Vt) = V + [lambda] (s), s = z + Vt, (16)

eta (z, t) = V + [theta] (z + Vt) = V + [theta] (s), (17)

and get the non-linear ordinary differential equations in terms of [lambda] and [theta]. The differential equation for [lambda] is same as in subsection 2.1 and the differential equation satisfied by [theta] is

[d[theta]/ds] [[[d.sup.2][lambda]]/[d[s.sup.2]]] - [lambda] [[[d.sup.3][theta]]/[d[s.sub.3]]] = [2K/[1 + [K.sup.2]]][lambda][[[d.sup.2][lambda]]/[d[s.sup.2]]] + [nu] [4K [[[d.sup.3][lambda]]/[d[s.sub.3]]] + (1 + [K.sup.2])[[[d.sup.4][theta]]/[d[s.sup.4]]]] + [alpha] [(1 + [K.sup.2]){[d[theta]/ds][[[d.sup.4][lambda]]/[d[s.sup.4]]] - [lambda][[[d.sup.5][theta]]/[d[s.sup.5]]]} - 4K[lambda] [[[d.sup.4][theta]]/[d[s.sup.4]]]]. (18)

The first integral of Eq. (18) is

[d[theta]/ds] [d[lambda]/ds] - [lambda][[[d.sup.2][theta]]/[d[s.sup.2]]] = [2K/[1 + [K.sup.2]]] ([lambda][d[theta]/ds] - [integral] ([d[lambda]/ds])[.sup.2] ds)+ [nu] [4K [[[d.sup.2][lambda]]/[d[s.sup.2]]] + (1 + [K.sup.2])[[[d.sup.3][theta]]/[d[s.sup.3]]]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

where the constant of integration is equal to zero. Using Eq. (9) into Eq. (19) and making lengthy algebraic calculations we get

-[alpha] (1 + [K.sup.2]) A(1 + c[e.sup.as])[PHI]'" + (1 + [K.sup.2])([nu] + [alpha]a Ac[e.sup.as])[PHI]" (20)

+A[1 + c[e.sup.as] - (1 + [K.sup.2])[alpha][a.sup.2]c[e.sup.as]][PHI]' - a Ac[e.sup.as][PHI] = [OMEGA](s)

where [PHI] = d[theta]/ds and

[OMEGA](s) = 2Ka[A.sup.2]c[e.sup.as] (1 + [1/2]c[e.sup.as]) ([1/[1 + [K.sup.2]]] - 2[alpha][a.sup.2]) + 4K[nu][a.sup.2]Ac[e.sup.as]. (21)

Consequently, we may reduce the order of the equation by means of the consecutive substitutions [PHI](s) = P (s) [e.sup.as] and P' (s) = R(s) to obtain

-[alpha] (1 + [K.sup.2]) A(1 + c[e.sup.as])R" + (1 + [K.sup.2])[[nu] - [alpha]a A (3 + 2c[e.sup.as])]R' + [A(1 + c[e.sup.as]) + (1 + [K.sup.2])(2a[nu] - [alpha][a.sup.2]A(3 + 2c[e.sup.as]))]R = -[OMEGA](s[e.sup.-as]. (22)

In order to find the solution of Eq. (22) we discuss few of its special cases:

Case I. The solution of Eq. (22) for c = 0 is given by

R(s) = [C.sub.1][e.sup.-[X.sub.7]s] + [C.sub.2][e.sup.[X.sub.8]s], (23)

where

[X.sub.7] = [[lambda].sub.1] + [square root of ([[lambda].sub.1.sup.2] - 4[[lambda].sub.2])], [X.sub.8] = -[[lambda].sub.1] + [square root of ([[lambda].sub.1.sup.2] - 4[[lambda].sub.2])],

[[lambda].sub.1] = [3[alpha]aA - [nu]]/[alpha]A, [[lambda].sub.2] = [3[alpha][a.sup.2]A - A - 2a[nu]]/[alpha]A.

Now, the expressions for [eta] (z, t) can be written as

[eta] (z, t) = V + [C.sub.4] + [[[C.sub.3][e.sup.as]]/a] - [[C.sub.1]/[[X.sub.7] (a - [X.sub.7])]][e.sup.(a-[X.sub.7])s] + [[C.sub.2]/[[X.sub.8] (a + [X.sub.8])]][e.sup.(a+[X.sub.8])s], (24)

in which [C.sub.r] (r = 1 to 4) are arbitrary constants. The stream function and the velocity components are respectively of the following form

[psi] (x, y, t) = x (V + a) + V + [C.sub.4] + [[[C.sub.3][e.sup.a](y+Kx+Vt)]/a] -[[C.sub.1]/[[X.sub.7] (a - [X.sub.7])]] [e.sup.(a-[X.sub.7])(y+Kx+Vt)] + [[C.sub.2]/[[X.sub.8] (a + [X.sub.8])]] [e.sup.(a+[X.sub.8])(y+Kx+Vt)], (25)

u = [C.sub.3][e.sup.a(y+Kx+Vt)] - [[C.sub.1]/[X.sub.7]][e.sup.(a-[X.sub.7])(y+Kx+Vt)] + [[C.sub.2]/[X.sub.8]] [e.sup.(a+[X.sub.8])(y+Kx+Vt)], (26)

v = -(V + a) - [C.sub.3]K[e.sup.a(y+Kx+Vt)] (27)

+ [K[C.sub.1]/[X.sub.7]] [e.sup.(a-[X.sub.7])(y+Kx+Vt)] - [[K[C.sub.2]]/[X.sub.8]][e.sup.(a+[X.sub.8])(y+Kx+Vt)].

Case II. The Eq. (22) for K = 0 ([OMEGA](s) = 0) becomes

-[alpha]A(1 + c[e.sup.as])R" + [[nu] - [alpha]aA (3 + 2c[e.sup.as])]R'

+[A(1 + c[e.sup.as]) + 2a[nu] - [a.sup.2]A(3 + 2c[e.sup.as])]R = 0. (28)

In order to solve Eq. (28) we put [sigma] = [e.sup.as] and get

-[alpha] A[a.sup.2] (1 + c[sigma]) [[sigma].sup.2] [[[d.sup.2]R]/[d[[sigma].sup.2]]] + a [[nu] - [alpha]aA (4 + 3c[sigma])] [sigma][dR/d[sigma]]

+[Ac (1 - 2[alpha][a.sup.2])[sigma] + 2a[nu] + A(1 - 3[alpha][a.sup.2]A)]R = 0. (29)

The solution of Eq. (29) for a = c = 1, is found through MATHEMATICA 4.1 and is given as

R([sigma]) = [C.sub.5][[sigma].sup.-[[delta].sub.1]][.sub.2.F.sub.1] ([X.sub.1], [X.sub.2]; [X.sub.3]; -[sigma]) + [C.sub.6][[sigma].sup.[[delta].sub.2]][.sub.2.F.sub.1] [[X.sub.4], [X.sub.5]; [X.sub.6]; -[sigma]], (30)

where [C.sub.5] and [C.sub.6] are arbitrary constants and

[[delta].sub.1] = (3[alpha][A.sub.1] - [nu] + [[delta].sub.3])/2[A.sub.1][alpha], [[delta].sub.2] = (-3[alpha][A.sub.1] + [nu] + [[delta].sub.3])/2[A.sub.1][alpha],

[[delta].sub.3] = ([square root of ([A.sub.1.sup.2][alpha] (4 - 3[alpha]) + [nu] (2[A.sub.1][alpha] + [nu]))]), (31)

[X.sub.1] = -[1/2] - [square root of ([1/[alpha]] - 1)] + [[[nu] - [[delta].sub.3]]/[2[A.sub.1][alpha]]], [X.sub.2] = -[1/2] + [square root of ([1/[alpha]] - 1)] + [[[nu] - [[delta].sub.3]]/[2[A.sub.1][alpha]]],

[X.sub.3] = 1 - [[[delta].sub.3]/[[A.sub.1][alpha]]], [X.sub.4] = 1 + [[[delta].sub.3]/[[A.sub.1][alpha]]],

[X.sub.5] = -[1/2] - [square root of ([1/[alpha]] - 1)] + [[[nu] + [[delta].sub.3]]/[2[A.sub.1][alpha]]], [X.sub.6] = -[1/2] + [square root of ([1/[alpha]] - 1)] + [[[nu] + [[delta].sub.3]]/[2[A.sub.1][alpha]]].

In Eq. (30), [.sub.2.F.sub.1] is the hypergeometric function denoted as [.sub.2.F.sub.1](a, b; c; z), which has the following properties.

** The hypergeometric function [.sub.2.F.sub.1] has the following series expansion:

[.sub.2.F.sub.1](a, b; c; z) = [[infinity].summation over (k=0)] [[(a)[.sub.k] (b)[.sub.k]]/[(c)[.sub.k]]] [[z.sup.k]/k!]. (32)

** The hypergeometric [.sub.2.F.sub.1](a, b; c; z) has a branch cut discontinuity in the complex z-plane running from 0 to [infinity].

The stream function and the velocity fields are

[psi] (x, y, t) = V + x (V + [A.sub.1] (1 + [e.sup.s])) + [integral] [[integral] R(s) ds][e.sup.s]ds, (33)

u (x, y, t) = x[A.sub.1][e.sup.s] + [[partial derivative]/[partial derivative]y] [integral] [[integral] R(s) ds][e.sup.s]ds, (34)

v (x, y, t) = -(V + [A.sub.1]) - [A.sub.1][e.sup.s] - [[partial derivative]/[partial derivative]x] [integral] [[integral] R(s) ds][e.sup.s]ds, (35)

where [A.sub.1] = [[nu] - [phi]]/[[alpha] -1], ([alpha] [not equal to] 1).

Case III. The solution of Eq. (29) for [alpha] = 0 is also found through MATHEMATICA 4.1 as

R([[theta].sub.1]) = -[[X.sub.9]/[1 + [K.sup.2]]][e.sup.c[[theta].sub.1][X.sub.9]][[theta].sub.1.sup.-1+[X.sub.9]] (36)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[theta].sub.1] = [e.sup.as],

[A.sub.2] = [nu]a (1 + [K.sup.2]), (37)

and ExpIntegral E[n, z] gives the exponential integral function [E.sub.n] (z), which is defined by

[E.sub.n] (z) = [[integral].sub.1.sup.[infinity]][e.sup.-zt]/[t.sup.n]dt. (38)

ExpIntegral E[n, z] has a branch cut discontinuity in the complex z-plane running from 0 to [infinity].

The stream function and the velocity components in this case are respectively given by

[psi] (x, y, t) = V + x (V + [A.sub.2] (1 + c[e.sup.as])) + [integral] [[integral] [R.sub.1] (s) ds][e.sup.as]ds, (39)

u (x, y, t) = xca[A.sub.2][e.sup.as] + [[partial derivative]/[partial derivative]y] [integral] [[integral] [R.sub.1] (s) ds][e.sup.as]ds, (40)

v (x, y, t) = -(V + [A.sub.2]) - [A.sub.2]c[e.sup.as] - [[partial derivative]/[partial derivative]x] [integral] [[integral] [R.sub.1] (s) ds][e.sup.as]ds. (41)

3. Concluding Remarks

The four analytic solutions of the involved non-linear equation for unsteady flow of second-grade fluid are constructed. Almost all such studies in the literature are restricted to the analysis of steady flows. Expressions for stream function and velocity components are obtained in each case. It is noted that Eq. (1) for [[alpha].sub.1] = 0 reduces to the Newtonian case [1]. Moreover, the results in all the cases for K = 0 recovers the results of reference [3]. This tends confidence in the mathematical calculations of the present analysis. Thus, the analytical solutions of the non-linear equations for the unsteadiness of the second-grade fluid is a step forward and may add a significant contribution to the literature.

References

[1] R. Berker, Integration des equations du mouvemont d'un fluide visqueux incompressible, Handbuch der Physik VII, Springer-Verlag, Berlin, 1963.

[2] M. H. Hamdan, An alternative approach to exact solutions of a special class of Navier-Stokes flows, Appl. Math. Comput. 93 (1998), 83-90.

[3] T. Hayat, M. R. Mohyuddin, S. Asghar, Some inverse solutions for unsteanian fluid, Tamsui Oxford J. Math. Sci. 21(1) (2005), 1-20.

[4] D. Riabouchinsky, Some considerations regarding plane irrotational motion of a liquid, C. R. Acad. Sci. Paris 179 (1936), 1133-1136.

S. Asghar*

Department of Mathematics, COMSATS Institute of Information Technology H-8 Islamabad 44000, Pakistan

M. R. Mohyuddin ([dagger]), T. Hayat ([double dagger])

Department of Mathematics, Quaid-i-Azam University 45320 Islamabad 44000, Pakistan

A. M. Siddiqui ([section])

Pennsylvania State University, York Campus York, Pennsylvania 17403, U.S.A.

Received August 25, 2005, Accepted November 9, 2005.

* E-mail:s_asgharpk@yahoo.com

([dagger]) E-mail:m_raheel@yahoo.com (Corresponding Author)

([double dagger]) E-mail:t_pensy@hotmail.com

([section]) E-mail:E-Mail: ams5@psu.edu