Journal of Fluid Flow, Heat and Mass Transfer (JFFHMT)

ISSN: 2368-6111

1. 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 is oscillating with velocity  , the fluid occupies the entire upper half plane 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.

2. 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 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  such that

Substitution of (8) in equations (6) and (7) and elimination of pressure from the resulting equations using  yields 

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).

The shear stress component   of the Cauchy stress  is given by

3. Solutions in the Fixed Frame of Reference

Following Takemitsu and Matunobu [16], we assume that

We assume that the infinite plate at  is oscillating with velocity  and that the fluid occupies the entire upper half plane Furthermore, we assume the streamfunction far from the wall is given by  (see Stewart [2]). Thus, the boundary conditions are given by

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

Integrating equations (13) and (14) once with respect to using the conditions at infinity, we have

and

Using the non-dimensional variables

in equations (15) and (16), and boundary conditions (12a) and (12b), we obtain

and

where  is the Weissenberg number.

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  represents the  and  approximation to the solution. Since the above equation is non-homogeneous, the solution at any approximation level can be written as . Further, the homogeneous solution, , is a linear combination of two linearly independent solutions – namely  and . The details of this technique are well described by Garg and Rajagopal [23].

Using the quasi-linearization technique described by Garg and Rajagopal [23], we find that  when . This value is in good agreement with the value obtained by Takemitsu and Matunobu [16]. Numerical values of  for different values of  are shown in Table 1. These values are in good agreement with the values obtained by Garg and Rajagopal [23] and Ariel [24]. Figure 1 shows the profiles of  for various values of . We observed that as the elasticity of the fluid increases, the velocity near the wall decreases.

Letting , then system (19) gives

and

Letting , then system (21 a-b) gives

System (23 a-b) is solved numerically using a shooting method and it is found that for ,  Since , then for ,  which is in good agreement with the value obtained by Takemitsu and Matunobu [16]. Numerical values of  for different values of  are shown in Table 1. Figure 2 depicts the profiles of  for various values of

Letting , 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  and  are given in Table 1 for different values of .

 For large values of the frequency , we let

 and

and it was found that

where  and the numerical values of  are given in Table 1 for different values of . Figures 3-5 depict the variations of ,  and  for various values of .

4. 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

 We note that the flow is oscillating with velocity  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

and

 Non-dimensionalizing using

we obtain

and

System (33 a-b) has been solved numerically in section 3. Letting , system (34 a-b) gives

and

Numerical solutions of system (35 a-b) have been obtained in section 3. It can easily be shown that the function

is a solution of system (36 a-b) since it satisfies both the equation and the boundary conditions. In equation (37), the functions  and  have been found in section 3.

4. Discussion and Conclusions

The unsteady second grade stagnation-point flow impinging obliquely on an oscillatory flat plate is studied. Numerical results for this flow are found for various values of the Weissenberg number . Figure 1 shows the variations of  for various values of . The effect of the Weissenberg number, , is to decrease the velocity  near the wall as it increases. Figure 2 depicts the variations of  for various values of  and shows that  decreases near the wall as  is increasing. The variations of  with various values of  are shown in Figure 3. From this figure we observed that   is decreasing as  is incresing. Figure 4 shows the variations of  for various values of  and Figure 5 depicts the variations of  for various values of . From Table 1,  is decreasing as  is increasing.

References