ON SEMI-INVERSE SOLUTIONS FOR THE TIME-DEPENDENT FLOWS OF A SECOND-GRADE FLUID

This paper deals with analytical solutions for the time-dependent equations arising in a second-grade fluid. The solutions have been developed by assuming certain forms of the stream function. Expressions for velocity components are obtained for flows in plane polar, axisymmetric cylindrical, and axisymmetric spherical polar coordinates. The obtained solutions are compared with existing results.


Introduction
Under several assumptions, the Navier-Stokes equations can be linearized, and closedform solutions are available.It is known that in general, the governing equations of non-Newtonian fluids are nonlinear and much more complicated in comparison to Newtonain fluids.To obtain analytical solutions of such equations is not easy.In order to explain several nonstandard features, such as normal-stress effects, rod climbing, shear thinning, and shear thickening, Rivlin-Ericksen fluids [18] of differential type were introduced.The second-grade fluids form a subclass of the differential-type fluid.The Cauchy stress T in a second-grade fluid is [1,11,12,15,19] where in which V is the velocity vector, ( * ) is the matrix transpose, p is the indeterminate pressure constrained by the incompressibility, d/dt is the material derivative, μ, α 1 , and α 2 are material constants.They denote, respectively, the viscosity, elasticity, and crossviscosity.These material constants can be determined from viscometric flows for any real fluid.Second-grade fluids are dilute polymers and the differential-type models can be used to describe the response of dilute polymers.A detailed account of the characteristics of second-order fluids is well documented by Dunn and Rajagopal [4].Fosdick and Rajagopal [5] have studied that the relation (1.1) is compatible with thermodynamics (the Clausius-Duhem inequality and the assumption that the Helmholtz free energy is a minimum in equilibrium when the fluid is at rest) if the following restrictions are imposed on the material constants: By assuming a certain form of the stream function, solutions for second-grade fluids were obtained by Mohyuddin et al. [10], Kaloni and Huschilt [7], Siddiqui et al. [2,19,20], Hayat et al. [6], and Labropulu [8].
The main purpose of the present communication is twofold.Firstly, to present the equations for the unsteady plane and axisymmetric flows of a second-grade fluids in polar, cylindrical, and spherical coordinates.Secondly, to obtain some analytical solutions of the governing equations in each case.We mention here that no boundary value problem is considered.The governing equations of the second-grade fluids are highly nonlinear.Moreover, these equations are of higher order than the Navier-Stokes equations.Thus, in general, to solve a well-posed problem for such a fluid, one requires additional initial and/or boundary conditions.For a detailed discussion about this issue and for some interesting examples, we refer the readers to the works by Rajagopal and Gupta [16], Rajagopal and Kaloni [17], and Rajagopal [13,14].The semi-inverse approach is used to deduce solutions of the governing time-dependent equations.
The outline of the paper is as follows.In Section 2, the basic laws are given.Section 3 contains the modelling for unsteady flows for three cases.In Section 4, analytic solutions are calculated for each case.The analytical solutions obtained are new and comparisons are made with the solutions available in the literature.

Basic equations
The basic laws governing the motion of an incompressible, homogeneous, second-grade fluid are div V = 0, (2.1) where div is the divergence and grad is the gradient operator.Substituting (1.2) in (2.2), we obtain Muhammad R. Mohyuddin et al. 3 where t in the subscript denotes the partial derivative with respect to time and ∇ 2 is the Laplacian operator.

Compatibility equations
In this section, we give the compatibility equations in plane polar, axisymmetric cylindrical, and axisymmetric spherical polar coordinates.The equations are constructed by eliminating the pressure field from the resulting equations in component form of (2.3) along with the continuity equation (2.1).

Unsteady plane flows in polar coordinates.
Here, the velocity field is of the following form: Making use of (3.1) into (2.1) and (2.3), we obtain in the absence of body forces in which p is the modified pressure and ω is the vorticity vector.The expressions for p and ω are ) ) The existence of continuity equation (2.1) allows us to define a stream function ψ(r,θ,t) as 4 Unsteady semi-inverse solutions which satisfies (3.2) identically and (3.3) and (3.4) become where ω = −∇ 2 ψ.Differentiating (3.7) with respect to θ and (3.8) with respect to r and using the integrability condition p rθ = p θr , we obtain in which (3.10) It is to be noted that the material parameter α 2 does not contribute to the compatibility equation (3.9).However, it does appear in the pressure distribution given in (3.5a).

Unsteady axisymmetric flows in cylindrical coordinates.
The velocity field for this case is V = u(r,z,t),0,w(r,z,t) . (3.11) The continuity and momentum equations are ) where A , (3.14) Introducing the stream function ψ(r,z,t) through the continuity equation (3.12) is satisfied identically and (3.13) give where Elimination of p from the above equations yields 3.3.Unsteady axisymmetric flows in spherical coordinates.On using the velocity field 6 Unsteady semi-inverse solutions into (2.1) and (2.3), we obtain the following equations: where (3.23) Define the stream function in spherical polar coordinates as We see that (3.21) is satisfied identically and (3.22) lead to where (3.29)

Solutions
In this section, we apply the inverse methods to obtain the exact solutions of the nonlinear partial differential equations (appearing in Section 3) by considering specific forms of the stream function.
Equation (4.4a) implies that G = G(t) which shows that G is steady, and hence from (4.5) H is steady.From (4.4c), we assume ∂F/∂θ = 0 (since ∂F/∂θ = 0 ⇒ F = F(θ) which contradicts the assumption (4.1)) which implies that H = 0. Using this information in (4.4b), we get 2ρ ∂F ∂θ The solution of the above equation is where A(t) and B(t) are arbitrary functions.
Muhammad R. Mohyuddin et al. 9 Consequently, the expressions for stream function and velocity components are given through (3.2) and (4.1) as which gives rise to the following equations: ) where Equation (4.10a) indicates that G is steady, and hence through (4.11) H is steady and from (4.10c), we get whose solution is where A 1 (t) and B 1 (t) are arbitrary functions of t.Substitution of (4.13) into (4.10b)yields where C(t) is a function of integration.The stream function and velocity components are, respectively, given by where For viscous case (α 1 = 0), we get In order to solve (4.19), we introduce the travelling wave solution of the form which (for Q = 0) readily gives  For α 1 = 0, (4.17) gives in which C 1 (t) is a function of integration.Equation (4.28) together with (4.3) forms a nonlinear partial differential equation for the determination of F (except when n = 2), which is given as (4.29) The solution for n = 1 and C 1 (t) = 0 is given by The solution for n = 2 and C 1 (t) = 0 is as follows: where For steady case, the solution of (4.32) is ψ(r) = A 5 r 2 lnr + B 5 r 2 + C 3 lnr, (4.37) where A 5 , B 5 , and C 3 are arbitrary constants, and the corresponding velocity components are Here we remark that the solution given in (4.37) is in good agreement to that given by Mohyuddin et al. in [10].The solution of (4.35) for and the velocity components are For n = 2, (4.42) reduces to The first integral of (4.43) is where we have taken the function of integration in (4.44) equal to zero.In order to solve (4.44), we define and obtain the following equation: where N is an arbitrary constant.
Letting α 1 = α 2 = 0 in (4.46) and assuming that we get the following relation: On choosing λ = −1, we readily obtain A = 2ν (ν is the kinematic coefficient of viscosity).The expressions for the stream function (4.41) and velocity components (3.15) are It is noted that the solutions (4.49)-(4.51)reduce to that of Berker solutions [3] when For α 1 = 0 and α 2 = 0, we assume [9,21] that in (4.46) and get the following solution: where and C 0 is a constant.The stream function and velocity components now become The above expressions reduce to the results of Berker [3] for N = 0.For n = 0, (4.42) gives Since ∂F/∂z = 0, (4.58) implies that where a 1 (t) and a 2 (t) are arbitrary functions of time and the above expression leads to the following values of the stream function and velocity components: Writing F(z,t) = Φ(z)e λ1t (4.61) into (4.56) and then solving the resulting equation, we obtain in which a i (i = 3 to 6) are arbitrary constants of integration and where λ 1 is a constant.The stream function and velocity components are of the following form: 4.3.Flow where Ψ(R,σ,t) = R n F(σ,t).On specializing the solution of (3.28) to the form we obtain the following equation: where (4.67) For n = 0, we obtain Since ∂F/∂σ = 0, the above equation yields where C 0 and C 1 are arbitrary functions and the stream function and velocity components are Muhammad R. Mohyuddin et al. 17 Now (4.66) for n = 1 leads to the following: where Various steady cases of the above equation have been discussed by Squire [21].Integrating (4.74) and employing a similar procedure as used by Landau and Lifshitz [9] (after setting the functions of integrations equal to zero), we obtain a particular solution of the form where a is a constant.Substitution of (4.75) into (4.3) 3 yields

.76)
We note that the solution (4.75) cannot be obtained for all values of a. Siddiqui and Kaloni [19] obtained the solution for the steady case when a = −1, 0,1.For the sake of completeness, we include it here briefly for the convenience of the reader.Setting a = ±1, (4.76) is identically satisfied and (4.75) gives (4.77) 18 Unsteady semi-inverse solutions For a = 0 and 7α 1 + 2α 2 = 0, (4.75) becomes The stream function and velocity fields for the above three cases are (4.79) For n = 2, (4.66) yields the following form: in which Equation (4.80) 2 can also be written as

.82)
The solution of the above equation is and the expressions for the stream function and velocity components are For n = 3, (4.66) gives rise to where For the steady and viscous cases, (4.86) gives μH 1 = 0 which on using (4.87) 3 becomes in which k 1 and k 2 are integration constants.We see that the solution F obtained from (4.88) will only be satisfied by (4.85) when k 1 = k 2 = 0.The solution of (4.88) is given as  For k 1 = 0 and k 2 = 0, we have the following solutions: where C i (i = 3 to 6) are constants.

.96)
The first and third equations in (4.95) imply that G 1 is steady, and hence H 1 is steady.Since F = 0, (4.95) 2 gives ∂G 1 /∂σ = 0, which on using (4.94) have the following solution: The solution of (4.98) for steady case is given by Berker [3].In order to avoid repetition, we directly give the solution with the stream function and velocity components as where k 3 is a constant.

Concluding remarks
In this paper, the governing time-dependent equations for plane polar, axisymmetric cylindrical, and spherical coordinates are constructed.Moreover, the analytical solutions for eleven nonlinear equations involving three-coordinate systems are given.The solutions obtained are found to be in good agreement to that of the previous steady solutions for viscous and second-grade fluids.