1 Introduction

The second grade fluid is a non-Newtonian fluid where the equations of motion are more complicated and more non-linear with higher order than the Navier-Stokes equations for New- tonian fluids. Thus,these make the job acquiring the exact solutions very difficult. The extension from the theory of Newtonian fluid mechanics to non-Newtonian fluids is shown to be not straightforward. Despite of the fact that the exact solutions are restricted to specific blends of easy geometry and boundary conditions,they provide a great insight into more difficult flow situations and can also serve as a measurement for assessing the accuracy of numerical estimate and experimental practices.

Some important and fundamental studies have been conducted for the flow of a viscous fluid which is described by the Navier-Stokes equation. The first closed form of the steady velocity for an incompressible fluid past an infinite oscillating plate was presented by Stokes^[1]. The closed form expression for the transient velocity was presented by Panton^[2],while the closed form solution for the starting velocity was given by Erdogan^[3]. Another new exact solution to the same problem was presented by Fetecau et al.^[4]. Recently,Cruz and Lins^[5] derived a more general solution when the fluid transpiration at the wall was considered.

Among the class of problems,the problems with known exact solutions are related on the motion imposed on the fluid by an oscillating wall boundary,which is referred to as the Stokes’ problems. In line with this,the exact solution for the second grade fluid past an infinite oscillating plate was first presented by Rajagopal^[6],where the steady-state velocities for the fluid were presented in several states of motion. This solution has a serious drawback because there is no expression for the starting velocity field. Asghar et al.^[7] used the Laplace transform and perturbation techniques to treat this problem and obtained the closed form expression for the starting velocity due to oscillation of the wall boundary. Also,Nazar et al.^[8] solved the same problem and obtained the starting velocity field due to oscillation of the wall boundary by means of the Laplace transform method. Recently,Ali et al.^[9] extended this problem to the electrically conducting second grade fluid passing through a porous space and established new exact solutions to Stokes’ second problem. Yao and Liu^[10] studied the effects of the side walls on the unsteady flow of a second grade fluid over a plane wall.

All the above mentioned studies of Stokes’ problems for a second grade fluid are based on the hypothesis that the bounding plate has no transpiration velocity. A far more general solution to Stokes’ problems for the second grade fluid can be derived when the fluid suction or injection is considered at the bounding plate. The model formulations must then be modified with the help of a brand new term representing the momentum introduced into the flow through the transpiration of fluid. Because of the important transpiration of fluid in the boundary layer control with good examples in aeronautical systems,manufacturing techniques,and mechanical and chemical engineering processes,the study is highly important.

In this paper,the exact solution to Stokes’ problems for the second grade fluid with addition of the wall fluid suction or injection is presented. To the very best of the authors’ knowledge, this is the very first time that such a closed solution will be presented. The solution shown here in the case of the Newtonian fluid reduces to the solution of Cruz and Lins^[5] and in the case of the zero-transpiration rate reduces to the solution of the second grade fluid presented by Asghar et al.^[7].

The organization of the rest of this paper is as follows. In Section 2,the statement of the problem is given. Section 3 deals with the analytical solution followed by Section 4 where the limiting solution for the Newtonian fluid is derived. Section 5 contains a periodic solution to the problem,whereas in Section 6,we provide detailed discussion on the results. Finally,the conclusions of this paper are drawn in Section 7.

2 Statement of problem

Consider a second grade fluid occupying a half-space z > 0 and bounded by an infinite plane wall situated in the xy-plane system of the Cartesian coordinate. Initially,both the plate and the fluid are at rest. At time t = 0+,the wall moves in the x-direction with the velocity V_w(t). The fluid velocity v(z,t) is given by the governing equation,

where ν is the kinematic viscosity,V_w is the transpiration velocity,and α = α₁/ρ ,in which α₁ is the viscoelastic parameter for a second grade fluid,and ρ is the constant density of the fluid.

The initial and boundary conditions are

where V₀ is the amplitude of wall oscillations,ω > 0 is the frequency of the wall velocity,and i is the imaginary unit. Using the wall velocity υ_w(t) given by Eq. (3),the cosine and sine oscillations can be obtained by taking the real and imaginary parts of the velocity field υ(z,t).

Considering the following dimensionless quantities:

we obtain the non-dimensional initial-boundary value problem (dropping the ∗ notation) 3 Solution to problem

The solution to the problem (6)-(8) will be obtained by means of the Laplace transform and the regular perturbation method. By applying the Laplace transform with respect to the time t,we get the following problem:

where v(z,s) = R∞ 0 v(z,t)e−stdt is the Laplace transform of the function v(z,t). In order to solve Eq. (9) with condition (10),we use the regular perturbation technique and expand the function v(z,s) in terms of the parameter β (β ≪ 1).

Let us consider the following perturbed solution:

Substituting Eq. (11) into Eqs. (9) and (10) and collecting the coefficients of equal powers of β yield

The solutions to the above mentioned problems are given by

Inversing the Laplace transform of Eq. (16) gives where It is noted that the solution (17) can be written as the sum between the permanent solution, also called the steady-state solution and the transient solution which tends to zero as t → ∞. The permanent solution (21) can be written in the simpler form as where It is important to point out that,if the transpiration velocity is zero ( = 0),then Eqs. (17)- (20) represent the closed form solution to Stokes’ problem for a second grade fluid,which was given by Asghar et al.^[7].

4 Case of Newtonian fluid

Letting β = 0 in Eqs. (17)-(20),we obtain the closed form solution to Stokes’ second problem for a Newtonian fluid with transpiration. This solution is given by

Moreover,for !ν = V 2 0 ,our solution (25) becomes identical with the solution given by Cruz and Lins^[5].

5 Periodic solution

In this section,we determine a periodic solution corresponding to the problem (6)-(8). This solution is useful after the start up phase effect disappears,and the fluid motion is harmonic with the same frequency of the flat plate. A similar solution for the flow of Newtonian fluids with transpiration was determined by Cruz and Lins^[5].

Consider that the velocity of the fluid is in the form of

where A and B are two complex constants.

By substitution of Eq. (26) into Eq. (6),we obtain that the function u(') is the solution to the following problem:

where u′ =du/d'. The general solution to Eq. (27) is given by where C1,C2,and C3 are arbitrary constants,and Using (29),we get C1 = C3 = 0,and Eq. (30) becomes with k = B A . Now,by using (28),we obtain C2 = 1,and k is a root of the following cubic align: Finally,we obtain the solution where k is a root of the align (33) with Im(k) > 0.

On basis of the Routh-Hurwitz criteria^[11],it shows that Eq. (33) has only one root whose imaginary part is positive.

We note that for β = 0,in the case of Newtonian fluid,Eq. (33) becomes

whose roots are and the periodic solution for the velocity is where m and n are given by Eq. (24). This solution is identical with that obtained by Cruz and Lins^[5].

6 Results and discussion

In the present analysis,exact solutions for the impulsive and oscillating motion of a second grade fluid with wall transpiration are derived using the Laplace transform,the perturbation techniques,and an extension of the variable separation technique together with similarity argu- ments. These solutions are written as the sum between the permanent and transient solutions. The variations of fluid behaviors with various physical parameters are shown graphically in Figs. 1-4. These dimensionless parameters are the transpiration velocity ,the second grade parameter β,the frequency of the wall velocity δ,and the time t.

In all these figures,velocity profiles are shown for both cosine and sine oscillations of the plate. The sine and cosine oscillations can be treated by taking the real and imaginary parts of the velocity field. The velocity related to the cosine oscillation is computed from Eq. (17) by taking the real part of v(z,t),whereas the velocity related to the sine oscillation is computed from Eq. (17) by taking the imaginary part of v(z,t). Note that the cosine oscillation presents a discontinuity at t = 0,when the wall velocity jumps from zero to u0 differently from the sine function,which represents a more realistic situation.

The velocity profiles shown in Fig. 1 are plotted for two different values of the dimensionless time t = 2 and t = 4 in order to see the variation in . In this figure,three different flow cases are compared such that they correspond to the case of injection (γ < 0),the case of suction (γ > 0), and the case of no transpiration velocity (γ = 0) for both the cosine and sine oscillations of the plate. Thus,for small values of the time t,the second grade fluid with injection flows much faster than the fluid without transpiration as well as the fluid with suction. This property is preserved for increasing t,and the sinusoidal oscillations of the plate are observed. Physically,it is justified because the net effect of the suction reduces the overshooting tendency and slows down the flow.

The velocity profiles for both the cosine and sine oscillations of the plate when the transpi- ration parameter shows injection = −0.9 as well as suction = 0.3 are shown in Fig. 2 and Fig. 3 for two different time t = 0.2,0.6 and for different values of the second grade param- eter β. It is noted that the influence of the second grade parameter β on the fluid motion is significant and clearer only for small values of the time t. We can see from Fig. 2 that as we increase the second grade parameter β,the cosine velocity of the fluid decreases continuously near the plate and approaches a constant value and then increases again. Physically,it is true because higher values of β have greater stability than the smaller values. However,for large values of the independent variable z,the starting velocity approaches the steady-state velocity. A similar situation is observed in Fig. 3 when the wall transpiration admits suction. However, it is observed that in the suction case,the starting velocity approaches the steady-state quite early in comparison with the injection cases in Fig. 2. This is because that the momentum transmitted to the fluid by the wall is sucked away. From both of these figures,we can also compare the velocity of the second grade fluid with the velocity corresponding to the Newtonian fluid (β = 0). For the sine oscillation of the plate and small values of time t,the second grade fluid with wall transpiration flows faster than the Newtonian fluid with wall transpiration. For increasing t,the Newtonian fluid is faster near the plate and slower when z increases.

The starting velocity v(z,t) is written as the sum of the permanent solution vp(z,t) given by Eq. (21) and the transient solution vt(z,t) given by Eq. (22). Because lim t→∞ vt(z,t) = 0, the transient solution can be neglected for the large values of the time t. In this case,the flow accords with the permanent solution. These aspects are exhibited in Fig. 4. The starting solutions v(z,t) corresponding to a second grade fluid with and without transpiration together with the permanent solution vp(z,t) are plotted versus t and z,respectively,for two different values of the second grade parameter β = 0.1,0.2 and several values of the non-dimensional transpiration parameter . The diagrams from Fig. 4 indicate that for small values of the time t, the difference between the starting solutions and the permanent solutions is significant. For large values of t,the curves corresponding to the starting solutions become identical with the curves corresponding to the permanent solutions. In the considered case,t = 7 is the moment beyond which the motion of the fluid can be approximated with the permanent motion,described by the steady state solution. It is clear that after this value of the time t,the transient solution can be neglected. From several numerical cases,we find that the required time to achieve the steady state is less for cosine oscillations. Also,the values of this time are decreasing for increasing values of the oscillation frequency.

7 Conclusions

In this work,the exact solution of the moving boundary transient flow of a second grade fluid with wall transpiration is derived with the help of the Laplace transform method,the perturbation techniquess,and the extended version of variable separation techniques. The present solutions are more general,and the existing solutions in the literature can be regarded as special cases. Furthermore,in the present analysis,the results for the horizontal velocity are plotted and discussed for both the cosine and sine oscillations of the plate. The starting and permanent velocity profiles are also analyzed. The results show that the effects of the wall transpiration parameter,the second grade parameter,and the time exert great influence on the general flow pattern,by enhancing or decelerating the fluid flow. The significant findings are summarized as below.

(i) The second grade fluid with injection flows faster than the second grade fluid with suction and no transpiration as the time increases in the small interval of the time for both types of oscillations.

(ii) The speed of the second grade fluid with transpiration is larger than the Newtonian fluid with transpiration.

(iii) The transient solution can be neglected as the time increases. The amplitude of velocity becomes large in the case of injection and becomes smaller in the case of suction. (iv) The influence of the second grade parameter on the fluid motion is significant only for the small values of the time.

(v) The variation of the starting and permanent solutions mainly depends on small values of the time. For the large values of the time,the two solutions are identical. The period of time can be determined before the transient solution vanishes. For the cosine case,t = 4,whereas for the sine case,t = 7,after which the transient solution can be neglected.

Acknowledgements The authors would like to acknowledge the Ministry of Education (MOE) and Research Management Centre of Universiti Teknologi Malaysia for the financial support through vote numbers 04H27 and 4F255 for this research.