Abstract
The unsteady two-dimensional stagnation point flow of the Walters B' fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number and the Weissenberg number.
1. Introduction
During the past few decades, non-Newtonian viscoelastic fluids have become more and more important industrially. Among these fluids are the fluids of diffferential type such as the Walters B' fluid [1]. Behaviour of viscoelastic fluids cannot be accurately described by the Newtonian fluid model. The equations of motion of non-Newtonian fluids are highly non-linear and one order higher than the Navier-Stokes equations. For this reason, boundary conditions in addition to the non-slip condition are required to have a well-posed problem. Only in some special cases where the higher order nonlinear terms in these equations can be neglected thereby reducing their order, are the βno-slipβ condition sufficient to yield unique solutions. In general, Rajagopal [2], Rajagopal and Gupta [3], Rajagopal [4] and Rajagopal and Kaloni [5] have shown that the absence of this additional boundary condition leads to non-unique solutions for problems involving the flow of fluids of differential type such as Walters' B' fluid in a bounded domain.
The flow of an incompressible viscous fluid over a moving plate has its importance in many industrial applications. The extrusion of plastic sheets, fabrication of adhesive tapes and application of coating layers onto rigid substrates are some of the examples. If a magnetic field is present, viscous flows due to a moving plate in an electro-magnetic field, ie magnetohydrodymanic (MHD) flows, are relevant to many practical applications in the metallurgy industry, such as the cooling of continuous strips and filaments drawn through a quiescent fluid and the purification of molten metals from non-metallic inclusions.
In the history of fluid dynamics, considerable attention has been given to the study of 2-D stagnation point flow. Hiemenz [6] derived an exact solution of the steady flow of a Newtonian fluid impinging orthogonally on an infinite flat plate. Stuart [7], Tamada [8] and Dorrepaal [9] independently investigated the solutions of a stagnation point flow when the fluid impinges obliquely on the plate. Beard and Walters [10] used boundary-layer equations to study two-dimensional flow near a stagnation point of a viscoelastic fluid. Dorrepaal et al. [11] investigated the behaviour of a viscoelastic fluid impinging on a flat rigid wall at an arbitrary angle of incidence. Labropulu et al. [12] studied the oblique flow of a viscoelastic fluid impinging on a porous wall with suction or blowing. The Hiemenz flow of a Newtonian fluid in the presence of a magnetic field was first considered by Na [13] and later by Ariel [14]. The flow of the non-Newtonian Walters' B' fluid in the presence of a transverse magnetic field was studied by Ariel [15]. Attia [16] investigated the steady flow of a non-Newtonian second grade fluid at a stagnation point with heat transfer in an an external uniform magnetic field.
Furthermore, the unsteady or time dependent viscous flow near a stagnation-point has also been widely investigated. Glauert [17] and Rott [18] studied the stagnation-point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane. Srivastava [19] investigated the same problem for a non-Newtonian second grade fluid using the Karman-Pohlhausen method. Labropulu et al. [20] used series methods to solve the unsteady stagnation point flow of a viscoelastic fluid impinging on an oscillating flat plate.
In this work, the two-dimensional unsteady stagnation point flow of the Walters B' fluid impinging on an infinite plate in the presence of a magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at is making harmonic oscillations in its own plane and the magnetic field is transverse or perpendicular everywhere in the flow field. Solutions for small and large frequencies of the oscillations are obtained using a finite difference technique.
2. Flow Equations and Boundary Conditions
We consider the two-dimensional flow of an incompressible non-Newtonian Walters' B' fluid against an infinite plate normal to the flow in the presence of a magnetic field. The -axis is along the plate and the -axis is normal to the plate. We assume that the plate makes harmonic oscillations on its own plane with velocity in the -direction equal to where and are constants. The unsteady two-dimensional flow of a viscous incompressible non-Newtonian Walters B' fluid in the presence of a magnetic field is governed by (see Beard and Walters [10] and Na [13])where , are the velocity components, is the pressure, is the kinematic viscosity, is the fluid density, is the electrical conductivity, is the magnetic field and is a measure of the viscoelasticity of the fluid. It is assumed that , so that it is possible to neglect the effect of the induced magnetic field.
For this problem, the boundary conditions are given bywhere is a constant which has the units of inverse time. The quantity is the velocity of the fluid outside the boundary layer.
Continuity (2.1) implies the existence of a streamfunction such thatSubstitution of (2.5) in (2.2) and (2.3) and elimination of pressure from the resulting equations using yieldsHaving obtained a solution of (2.6), the velocity components are given by (2.5).
The boundary conditions in terms of the streamfunction take the form
The shear stress component is given by
3. Solutions
Following Glauert [17], we seek a solution in the formThe boundary conditions take the formUsing (3.1) in (2.6), we obtainNon-dimensionalizing usingwe getwhere , the Weissenberg number, is the ratio of the elastic effects over the viscous effects and is the Hartmann's number.
The asymptotic behaviour of far away from the plate (outside the boundary layer) is given bywhere is a constant that accounts for the boundary layer displacement.
Using this asymptotic behaviour, we obtain
Integrating (3.5) once with respect to and using the conditions at infinity, we have (see [10])
System (3.8) with has been solved numerically by many authors (Beard and Walter [10], Ariel [21]). Using the shooting method with the finite difference technique described by Ariel [21], we find that when and which is in good agreement with the values obtained by Hiemenz [6] and Glauert [17]. Numerical values of for different values of and are shown in Table 1. These values are in good agreement with the values obtained by Ariel [21] when , Ariel [15] for all and Labropulu et al. [20] when . These values are also in good agreement with the values obtained by Attia [16] when . Figure 1 shows the profiles of for various when . Figure 2 shows the profiles of for various when . Figure 3 depicts the profiles of for various when . We observed that as the elasticity of the fluid and the Hartman's number increase, the velocity near the wall increases.
Letting , then system (3.9) becomesThe only parameter in system (3.10) is the frequency ratio . Series solutions will be developed, valid for small and large values of , respectively.
3.1. Small Values of
Consider the case where , which implies that the plate velocity has the constant value . Letting , then system (3.10) gives
This system is solved numerically using a shooting method and it is found that for and , which is in good agreement with the value obtained by Glauert [17]. Numerical values of for different values of and are shown in Table 1. These values are in good agreement with the values obtained by Labropulu et al. [20] for . Figure 4 depicts the profiles of for various values of and . Figure 5 shows the profiles of for various when .
For small but non-zero values of , we let
Substituting (3.12) into (3.10), we get for
This system can be solved numerically either by using the perturbation technique or by a finite difference scheme. Numerical integration of system (3.13) for using a finite difference technique gives for and , which is in good agreement with Glauert's value [17]. Numerical values of for different values of and are shown in Table 1. These values are in good agreement with Labropulu et al. [20] for . Figure 6 shows the profiles of for various values of and and Figure 7 depicts the profiles of for various values of and
Numerical integration of system (3.13) for using a finite difference technique gives for and , which is in good agreement with Glauert's value [17]. Numerical values of for different values of and are shown in Table 1. These values are in good agreement with Labropulu et al. [20] when . Figure 8 depicts the profiles of for various values of and and Figure 9 shows the profiles of for various values of and .
The oscillating component of the shear stress on the wall is given bywhere and are given in Tables 1, 2, 3, and 4 for different values of and . When and , the value of the shear stress on the wall is in good agreement with the value obtained by Glauert [17].
3.2. Large Values of
When is large, we let
Letting , then and (3.10) takes the form
Since is small for most fluids which behave as second order fluids (see Markovitz and Coleman [22]), we follow Srivastava [19] and take to be of the order of . Thus, and (3.16) becomes
The expansion for near the wall iswhere .
Since for large values of the parameter is small, we let
The boundary conditions are
Substituting (3.20) in (3.17) and equating the coefficients of different powers of to zero, we find that the boundary-value problem for iswith solution provided .
The second equation gives that is zero. The next four equations for and are
Solving these equations and using the boundary conditions, we obtainprovided . If and , we recover the solutions for the Newtonian fluid obtained by Glauert [13] and if , we recover the solutions obtained by Labropulu et al. [16].
The oscillating component of the shear stress on the wall is given byIf and , the shear stress is in good agreement with the result obtained by Glauert [13].
4. Conclusions
The unsteady stagnation-point flow of a viscoelastic Walters' B' fluid in the presence of a magnetic field is examined. Results for this flow are obtained for various values of the Hartmann's number and the Weissenberg number . At the higher frequencies, the perturbation is a shear layer, exactly as on a plate oscillating in a fluid at rest. Figure 1 shows the variation of for various Weissenberg numbers when the Hartmann's number . The effect of the Weissenberg number is to increase the velocity near the wall as it increases. Figures 2 and 3 show the variation of for various Hartmann's number when and . The effect of the Hartmann's number is to increase the velocity near the wall as it increases. Figures 4 and 5 show that decreases near the wall as and are increasing. Also, from Tables 1 to 4, increases with Hartmann's number and Weissenberg number . The reason for this behaviour is that the magnetic field induces a force along the surface which supports the motion. As a result, the velocity along the surface increases everywhere and hence the shear stress on the wall increases with increasing Hartmann's number and Weissenberg number.
These conclusions are presented for the Walters' B' fluid. Future works will examine the unsteady stagnation-point flow in the presence of a magnetic field of other viscoelastic fluids.
Acknowledgment
The author would like to thank the reviewers for their valuable suggestions.