﻿ Run-Up Flow of a Maxwell Fluid through a Parallel Plate Channel

American Journal of Computational Mathematics
Vol.3 No.4(2013), Article ID:39769,7 pages DOI:10.4236/ajcm.2013.34039

Run-Up Flow of a Maxwell Fluid through a Parallel Plate Channel

Syed Yedulla Qadri1, M. Veera Krishna2

1Department of Mathematics, St. Josephs PG College, Kurnool, India

2Department of Mathematics, Rayalaseema University, Kurnool, India

Copyright © 2013 Syed Yedulla Qadri, M. Veera Krishna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received August 17, 2013; revised September 11, 2013; accepted September 18, 2013

Keywords: Run-Up Flow; Maxwell’s Fluid; Laplace Transforms; Reynolds Number and Parallel Plate Channels

ABSTRACT

We consider the flow of an incompressible viscous Maxwell fluid between two parallel plates, initially induced by a constant pressure gradient. The pressure gradient is withdrawn and the upper plate moves with a uniform velocity while the lower plate continues to be at rest. The arising flow is referred to as run-up flow. The unsteady governing equations are solved as initial value problem using Laplace transform technique. The expression for velocity, shear stresses on both plates and discharge are obtained. The behavior of the velocity, shear stresses and mass flux has been discussed in detail with respect to variations in different governing flow parameters and is presented through graphs.

1. Introduction

2. Formulation and Solution of the Problem

Consider the flow of an incompressible Maxwell fluid between two infinite rigid parallel plates y = 0 and y = h along the direction of x-axis (Figure 1). Since the flow is along the x-direction, we take the velocity , which satisfies the continuity equation.

We consider a Cartesian system so that the fluid flow takes place within boundary plates y = 0 and y = h. The linear momentum equation governing the flow u(y, t) is given by (1)

where the density, p is the pressure, is the coefficient of viscosity and is the relaxation time. The boundary conditions are (2)

We consider the run-up flow of the Maxwell’s fluid through the parallel plate channel. Initially the flow due to a prescribed pressure gradient with boundaries at rest and at the time t > 0, the pressure gradient is withdrawn and the upper plate moves with a uniform velocity while the lower plate continues to be at rest. The equation gov- Figure 1. Configuration of the problem.

erning the initial flow is (3)

The corresponding boundary conditions are (4)

We introduce the non-dimensional variables Using non-dimensional variables, the governing equation are (dropping the asterisk) (5) (6)

where, is the Reynolds’s number, is the Maxwell’s fluid parameter, is the constant of pressure gradient. Using the boundary conditions (4), the Equation (6) reduces to (7)

Applying the Laplace transform to the Equation (5) and using the Equation (7), we get (8)

Now the transformed boundary conditions are (9)

On solving Equations (8) using (9); we get (10)

On taking the inverse Laplace transform  for the Equation (10); we get (11)

The shear stresses on the upper plate and the lower plate are given by (12)

and (13)

The mass flux Q is given by (14)

3. Results and Discussion

The flow governed by the non-dimensional parameters R the Reynolds number, the Maxwell fluid parameter. The velocity, the shear stresses on the plates and discharge between the plates are evaluated analytically and computationally discussed for different variations in the governing parameters R and . It is interesting to note that the behavior of the flow very much depends on the pressure gradient, in accordance with the run-up flow, the initial steady flow is due to the prescribed pressure gradient, while the perturbed flow is due to the sudden movement of the boundary in absence of the pressure gradient. The flow in direction of the movement of the boundary may be considered as actual flow, where as the flow caused by the pressure gradient assumed to be against the direction of the boundary movement as the reversal flow. Figures 2-9 represent the behaviour of the velocity component u for variations in low and high Reynolds number with Maxwell fluid parameter and for various values of time t. Tables 1-3 show stresses on both boundaries and mass flux.

Figure 2 and 3 depict the variation of the velocity u slightly increases for increase in low Reynolds number R while it experience enhancement with increase in high Figure 2. The velocity profile for u on low Reynolds number R with . Figure 3. The velocity profile for u on high Reynolds number R with . Figure 4. The velocity profile for u on the Maxwell fluid parameter with low Reynolds number R = 20, P = 1,  Figure 5. The velocity profile for u on the Maxwell fluid parameter with high Reynolds number R = 20, P = 1,  Figure 6. Time development of the velocity component u for with low Reynolds number R = 20, P = 1. Figure 7. Time development of the velocity component u for with high Reynolds number .

Reynolds number being the other parameter fixed. The velocity profiles (4 and 5) for u on the Maxwell fluid parameter with low and high Reynolds number R = 20 and R = 1000 respectively. The similar behaviour is observed as earlier mentioned. We notice that the Maxwell fluid parameter with the Reynolds number affects the flow in the entire region. An interesting phenomenon is observed in Figures 6 and 7, the magnitude of the velocity enhances with increase in time while fixing R and . Figures 8 and 9 indicates that for a fixed t, R and Figure 8. The velocity profile for u with pressure gradient P on low Reynolds number with . Figure 9. The velocity profile for u with pressure gradient P on low Reynolds number with . Table 1. The shear stresses on the upper plate. Table 2. The shear stresses on the lower plate. , as P increases the velocity decreases for any y. This is in accordance with the fact that an increase in P implies a decrease in pressure which naturally results in a decrease of velocity.

The magnitude of the stresses on upper and lower plates enhance with increasing in , and reduces in Table 3. The Discharge between the plates.

lower plate increases in upper plate with increasing R (Tables 1 and 2). The discharge between the plates enhances with increase in both R and (Table 3).

4. Conclusion

The run-up flow of an incompressible Maxwell fluid between two infinite parallel plates is studied using Laplace transform technique. Analytical expressions for the fluid velocity field are obtained in Laplace transform domain. The magnitude of the velocity enhances with the increasing in both R and and reduces with the increasing in P. The stresses on upper and lower plates enhance with the increasing in , reduce in lower plate and increase in upper plate with increasing R. The discharge between the plates enhances with the increasing in both R and .

5. Acknowledgements

The authors are thankful to Prof. R. Siva Prasad, Department of Mathematics, Sri Krishnadevaraya University, Anantapur, Andhra pradesh, India, and AJCM Journal for the support to develop this document.

REFERENCES

1. N. B. Naduvinamani, P. S. Hiremath and G. Gurubasavaraj, “Squeeze Film Lubrication of a Short Porous Journal Bearing with Couple Stress Fluids,” Tribology International, Vol. 34, No. 11, 2001, pp. 739-747.
2. N. B. Naduvinamani, P. S. Hiremath and G. Gurubasavaraj, “Surface Roughness Effects in A Short Porous Journal Bearing with a Couple Stress Fluid,” Fluid Dynamics Residuals, Vol. 31, No. 5-6, 2002, pp. 333-354.
3. N. B. Naduvinamani, P. S. Hiremath and G. Gurubasavaraj, “Effects of Surface Roughness on the Couple Stress Squeeze Film between a Sphere and a Flat Plate,” Tribology International, Vol. 38, No. 5, 2005, pp. 451-458.
4. N. B. Naduvinamani, S. T. Fathima and P. S. Hiremath, “Hydro Dynamic Lubrication of Rough Slider Bearings with Couple Stress Fluids,” Tribology International, Vol. 36, No. 12, 2003, pp. 949-959.
5. N. B. Naduvinamani, S. T. Fathima and P. S. Hiremath, “Effect of Surface Roughness on Characteristics of Couple Stress Squeeze Film between Anisotropic Porous Rectangular Plates,” Fluid Dynamics Residuals, Vol. 32, No. 5, 2003, pp. 217-231.
6. J. R. Lin and C. R. Hung, “Combined Effects of NonNewtonian Couple Stresses and Fluid Inertia on the Squeeze Film Characteristics between a Long Cylinder and an Infinite Plate,” Fluid Dynamics Residuals, Vol. 39, No. 8, 2007, pp. 616-639.
7. J. Y. Kazakia and R. S. Rivlin, “Run-Up and Spin-Up in a Visco-Elastic Fluid I,” Rheologica Acta, Vol. 20, No. 2, 1981, pp. 111-127.
8. R. S. Rivlin, “Run-Up and Spin-Up in a Visco-Elastic Fluid II,” Rheologica Acta, Vol. 21, No. 2, 1982, pp. 107- 111.
9. R. S. Rivlin, “Run-Up and Spin-Up in a Visco-Elastic Fluid III,” Rheologica Acta, Vol. 21, No. 3, 1982, pp. 213-222.
10. R. S. Rivlin, “Run-Up and Spin-Up in a Visco-Elastic Fluid IV,” Rheologica Acta, Vol. 22, No. 3, 1983, pp. 275-283.
11. N. Ch. Pattabhi Ramacharyulu and K. A. Raju, “Run-Up in a Generalized Porous Medium,” Indian Journal of Pure Applied Mathematics, Vol. 15, No. 6, 1984, pp. 665- 670.
12. D. Ramakrishna, “Some Problems in the Dynamics of Fluids with Particle Suspensions,” Ph.D. Thesis, Kakatiya University, Warangal, 1986.
13. G. Honig and U. Hirdes, “A Method for the Numerical In-version of Laplace Transforms,” Journal of Computational Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132.
14. M. Diwakar and T. K. V. Iyengar, “Run-Up Flow of a Couple Stress Fluid between Parallel Plates,” Nonlinear Analysis: Modelling and Control, Vol. 15, No. 1, 2010, pp. 29-37.
15. V. Sugunamma, M. S. Latha and N. Sandeep, “Run-Up Flow of a Rivlin-Ericksen Fluid through a Porous Medium in a Channel,” International Journal of Mathematical Archive, Vol. 2, No. 12, 2011, pp. 2625-2639.
16. M. Veera Krishna, S. V. Suneetha and R. Siva Prasad, “Hall Current Effects on Unsteady MHD Flow of Rotating Maxwell Fluid through a Porous Medium,” Journal of Ultra Scientist of Physical Sciences, Vol. 22, No. 1, 2010, pp. 133-144.
17. R. Reddy Sheelam, “Computational Techniques in Transient Magneto Hydro Dynamics Dusty Viscous and RunUp Flows,” Ph.D. Thesis, Osmania University, Hyderabad, 1992, pp. 47-65.
18. M. Basha, “Visco-Elastic Fluid Flow and the Heat Transfer through Porous Medium,” Ph.D. Thesis, SriKrishnadevaraya University, Anantapur, 1994.
19. D. Malleswari, “Unsteady Flow of a Rivlin-Ericksen Fluid through Plannar Channels with Porous Lining,” Ph.D. Thesis, Sri Padmavathi Mahila Viswavidyalayam, Tirupati, 2010.

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