EXISTENCE, UNIQUENESS, AND QUASILINEARIZATION RESULTS FOR NONLINEAR DIFFERENTIAL EQUATIONS ARISING IN VISCOELASTIC FLUID FLOW

Because of their various applications during the past several years, generalizations of the Navier-Stokes model to highly nonlinear constitutive laws have been proposed and studied (see [4, 5, 7]). Several different models have been introduced to explain such nonstandard features, as normal stress effect, rod climbing, shear thinning, and shear thickening. Among the differential-type models, Oldroyd models received special attention [2]. These models are rather complex from the point of view of partial differential equations theory. Nevertheless, several authors in fluid mechanics are now engaged with the equations of motion of non-Newtonian fluids of Oldroyd two-, three-, six-, and eight-constant models. Several authors [2, 6] considered an Oldroyd three-constant model which is a special case of the Oldroyd six-constant model. This has been used recently by Baris [1] for dealing with the steady and slow flow in the wedge between intersecting planes, one fixed and the other one moving. The Cauchy stress T in an incompressible Oldroyd six-constant-type fluid is related to the fluid motion by


Introduction
Because of their various applications during the past several years, generalizations of the Navier-Stokes model to highly nonlinear constitutive laws have been proposed and studied (see [4,5,7]).Several different models have been introduced to explain such nonstandard features, as normal stress effect, rod climbing, shear thinning, and shear thickening.Among the differential-type models, Oldroyd models received special attention [2].These models are rather complex from the point of view of partial differential equations theory.Nevertheless, several authors in fluid mechanics are now engaged with the equations of motion of non-Newtonian fluids of Oldroyd two-, three-, six-, and eight-constant models.Several authors [2,6] considered an Oldroyd three-constant model which is a special case of the Oldroyd six-constant model.This has been used recently by Baris [1] for dealing with the steady and slow flow in the wedge between intersecting planes, one fixed and the other one moving.
The Cauchy stress T in an incompressible Oldroyd six-constant-type fluid is related to the fluid motion by 2 Existence, uniqueness, and quasilinearization results (for details see [2]), where −pI is the indeterminate part of the stress due to the constraint of incompressibility.The extra stress tensor S is defined by where μ, λ 1 , λ 2 , λ 3 , λ 4 , λ 5 are six material constants.A 1 is the first Rivlin-Ericksen tensor defined by where DS/Dt is the upper-convected derivative of S and is defined as Recently, Wang et al. [8] studied magnetohydrodynamic steady Poiseuille channel flow of an Oldroyd six-constant fluid and obtained the numerical solution using the predictor corrector method.However, they did not show existence and uniqueness results.
In this paper, we study the existence, uniqueness, and behavior of exact solutions of second-order nonlinear differential equations arising in Oldroyd six-constant fluid flows in a channel.Furthermore, we obtain numerical solutions by using the quasilinearization technique.

Formulation of the problem
In this paper, steady plane shearing flows are considered for which the equation for the fluid flow (for details see Wang et al. [8]) is d dy μ(du/dy) + μα 1 (du/dy) 3  1 + α 2 (du/dy) 2 where We leave the issue of boundary conditions for later.Defining nondimensional variables F. T. Akyildiz and K. Vajravelu 3 and substituting (2.3) in (2.1), we obtain (after dropping the stars) d dy The appropriate no-slip boundary conditions are First, we define L = ((du/dy) + α 1 (du/dy) 3 )/(1 + α 2 (du/dy) 2 ) so that Now, (2.6) can be solved for du/dy in terms of L. In order to do this we assume the transformation This transformation effectively gets rid of the quadratic first derivative term yielding where The solution of this is where We note that (2.8) always has one real solution irrespective of the value of B 2 − 4R 3 .Also, if (B 2 − 4R 3 ) ≤ 0, then it is easy to see that (2.8) has three real solutions, hence there is no unique solution, so, throughout this paper, we assume that (B 2 − 4R 3 ) > 0.

Existence and uniqueness results
Theorem 3.1.There exists a classical solution of (2.4) which can be written as with Proof.We employ the Schauder fixed point theorem.First, from (2.6), we see that the solution can be written as (2.14).Let B be the Banach space of continuous functions u(y) on the interval 0 ≤ y ≤ 1 which vanish at 0 and 1 with the norm Define F : B → B, where (Fu)(y) is equal to the right-hand side of (2.14).

A priori bounds.
The Schauder fixed point theorem requires us to show that F is a continuous mapping of a convex compact subset of B into itself.To do this we need to derive estimates on (Fu)(y) and (Fu) (y).Since dp/dx = k (constant), k is known, and y ∈ [0,1], we have from (2.13) that L = ky + c.This gives us an estimate of (Fu)(y) and (Fu) (y).From the triangle inequality, we get where Since C 4 is independent of the function w, we see F : Z → V , where is a subset of B and hence Similarly, it is easy to show that (Fw) (y) ≤ C 5 . (3.9) Since C 5 is independent of w, we have F : Z → V c , where which is convex and compact via the Ascoli-Arzela theorem.Consequently, we have F : Z → Z.The continuity of F is an elementary calculation based on the estimates, and it is easy to see from (3.5) that Proof.The proof is by contradiction.We assume that (3.1) has two solutions u and v satisfying the conditions (3.2).Set z = u − v.We get where a 1 = 3α 1 − α 2 and a 2 = α 1 α 2 .We can write this equation in the form with boundary conditions Using the boundary condition (3.14), we find that z = 0.This proves the theorem.

Results and discussion
We use the quasilinearization method which has been explained in detail in [3].The quasilinear process equations for our differential equation are where By means of the finite difference method a linear algebraic equation system is derived and solved for each iterative step.A sequence of functions u 0 (y),u 1 (y),... is determined in the following manner: if an initial estimate u 0 (y) is given, then u 1 (y),u 2 (y),... are calculated successively as the solution of the boundary-value problem (4.1).The solution is assumed to converge when the difference between two successive iterations is less than the infinitesimal number ε = 1 × 10 −10 .In Figures 4.1 and 4.2, we show the effects of the parameters (α 1 , α 2 ), and the pressure gradient on the velocity field.In these figures, we also compared our results with the results of Wang et al. [8].For small values of α 2 , there is no appreciable difference between the two solutions.However, if α 2 is large enough, these two solutions are different; this mathematical problem is of interest and will be the subject of our future investigation.If α 1 is not large, these two solutions are identical as shown in Figures 4.3

2 y 0 e
.14) 6 Existence, uniqueness, and quasilinearization results Equation (3.13) can be solved easily to get z = e 1 + e − kFdt dt.(3.15) and 4.4.Here, the parameters α 1 , α 2 represent material constants; when they are zero, the model reduces to the linear Oldroyd-B model.Hence, we can regard the effects of the parameters α 1 , α 2 on the velocity field as due to nonlinearity.F. T. Akyildiz and K.