Unsteady Stagnation-point Flow of a Second-grade Fluid

The unsteady two-dimensional stagnation point flow of second-grade fluid impinging on an infinite plate is examined and solutions are obtained. It is assumed that the infinite plate at y = 0 is oscillating with velocity U cosΩt, the fluid occupies the entire upper half plane y > 0 and it impinges obliquely on the plate. The governing partial differential equations are reduced to a system of ordinary differential equations by assuming a form of the streamfunction a priori. The resulting equations are, then, solved numerically using a shooting method for various values of the Weissenberg number, We. It is observed that the effect of the Weissenberg number is to decrease the velocity near the wall as it increases. Furthermore, analytical solutions are obtained for small and large values of frequency.


Introduction
In the past, the fluid flow near a stagnation point has been investigated extensively.Hiemenz [1] derived an exact solution of the steady flow of a Newtonian fluid impinging orthogonally on an infinite flat plate.Stuart [2], Tamada [3] and Dorrepaal [4] independently investigated the solutions of a stagnation point flow when the fluid impinges obliquely on the plate.Beard and Walters [5] used boundary-layer equations to study two-dimensional flow near a stagnation point of a viscoelastic fluid.Dorrepaal et al [6] investigated the behaviour of a viscoelastic fluid impinging on a flat rigid wall at an arbitrary angle of incidence.Labropulu et al [7] studied the oblique flow of a viscoelastic fluid impinging on a porous wall with suction or blowing.
Unsteady stagnation point flow of a Newtonian fluid has also been studied extensively.Rott [8] and Glauert [9] studied the stagnation point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane.Srivastava [10] investigated the same problem for a non-Newtonian second grade fluid using the Karman-Pohlhausen method [11] to solve the resulting equations.Labropulu et al [12] used series methods to solve the unsteady stagnation point flow of a Walters' B' fluid impinging on an oscillating flat plate.Matunobu [13,14] and Kawaguti and Hamano [15] examined the fundamental character of the unsteady flow near a stagnation point for a Newtonian fluid.Takemitsu and Matunobu [16] studied the oblique stagnation point flow for a Newtonian fluid and obtained the general features of a periodic stagnation point flow.The case when the stagnation point fluctuates along a solid boundary is especially interesting from the biomechanical point of view.This is because the wall shear stress experienced by blood vessels may be thought to be increased by pulsating blood flow near the mean position of fluctuating stagnation point [15,17] and lead to vascular diseases [18].
In this work, the unsteady stagnation point flow of a viscoelastic second-grade fluid is examined and solutions are obtained.We assume that the infinite plate at  = 0 is oscillating with velocity  cos Ω , the fluid occupies the entire upper half plane  > 0 and the fluid impinges obliquely on the plate.The governing partial differential equations are reduced to a system of ordinary differential equations by assuming a form of the streamfunction a priori.The resulting equations are, then, solved numerically using a shooting method for various values of the Weissenberg number, .It is observed that the effect of the Weissenberg number is to decrease the velocity near the wall as it increases.Furthermore, analytical solutions are obtained for small and large values of frequency.

Flow Equations
The flow of a viscous incompressible non-Newtonian second-grade fluid, neglecting thermal effects and body forces, is governed by when the constitutive equation for the Cauchy stress tensor  which describes second-grade fluid given by Rivlin and Ericksen [19] is Here  is the velocity vector field,  the fluid pressure function,  the constant fluid density,  the constant coefficient of viscosity and ₁, ₂ the normal stress moduli.Dunn and Fosdick [20] and Dunn and Rajagopal [21] have shown that if the second-grade fluid described by (3) is to undergo motions which are compatible with Clausius-Duhem inequality [22] and the assumption that the free energy density of the fluid be locally at rest, then the material constants must satisfy the following restrictions: Considering the flow to be plane, we take  = ((, , ), (, , )) and  = (, , ) so that the flow equations ( 1) to (3) take the form where     is the kinematic viscosity.
Continuity equation ( 5) implies the existence of a streamfunction ( , , ) x y t Substitution of ( 8) in equations ( 6) and ( 7) and elimination of pressure from the resulting equations using Having obtained a solution of equation ( 9), the velocity components are given by ( 8) and the pressure can be found by integrating equations ( 6) and (7).

Solutions in the Fixed Frame of Reference
Following Takemitsu and Matunobu [16], we assume that We assume that the infinite plate at  = 0 is oscillating with velocity  Ω and that the fluid occupies the entire upper half plane  > 0. Furthermore, we assume the streamfunction far from the wall is given by  = 1 2  2 +  (see Stewart [2]).Thus, the boundary conditions are given by (0) (0) 0, (0, ) 0, (0, ) where  is a non-dimensional constant characterizing the obliqueness of oncoming flow.It is assumed that only the real part of a complex quantity has its physical meaning.
Substitution of equation ( 11) in ( 9) yields and   4   4 −  3   2 +  1   5   4 +  (  3 Integrating equations ( 13) and ( 14) once with respect to  using the conditions at infinity, we have and   3 Using the non-dimensional variables , in equations ( 15) and ( 16), and boundary conditions (12a) and (12b), we obtain System (18 a-b) has been solved numerically by Garg and Rajagopal [23] and Ariel [24,25].Following Bellman and Kalaba [26] and Garg and Rajagopal [23], the quasi-linearized form of equation ( 18a) is where the subscript  and ( + 1) represents the  ℎ and ( + 1) ℎ approximation to the solution.Since the above equation is non-homogeneous, the solution at any approximation level can be written as =  homogeneous +  particular .Further, the homogeneous solution,  homogeneous , is a linear combination of two linearly independent solutions -namely  ℎ 1 and  ℎ 2 .The details of this technique are well described by Garg and Rajagopal [23].
Letting () =  1 ′ (), then system (22 a-b) becomes The only parameter in equation (24a) is the frequency .Two series solutions valid for small and large  respectively will be obtained.For small values of the frequency , we assume that where the numerical values for  0 ′ (0),  1 ′ (0) and  2 ′ (0) are given in Table 1 for different values of .
For large values of the frequency , we let and it was found that where  ≠ 1 and the numerical values of  ′′ (0) are given in Table 1 for different values of .

Solutions in the Moving Frame of Reference
We assume that the Cartesian coordinates (, ) are moving with the plate, the -axis is along the plate and the -axis is normal to the plate.In this case, following Takemitsu and Matunobu [16], we assume that the streamfunction is given by  and the boundary conditions are given by (0) (0) 0, (0, ) 0, (0, ) 0 We note that the flow is oscillating with velocity − cos Ω at infinity.Using equation (28) in (9), equating different powers of  to zero and integrating once with respect to  using the conditions at infinity, we obtain ) = 0,   (0, ) =   ,   (∞, ) =  (