ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Exact Solutions and Analysis for a Class of Extended Stokes’ Problems

Discontinuity, Nonlinearity, and Complexity 2(1) (2012) 85--102 | DOI:10.5890/DNC.2012.12.003

L.Z. Zhang$^{1}$,$^{2}$, H.S.Tang$^{2}$, J.P. -Y.Maa$^{3}$, G.Q. Chen$^{4}$

$^{1}$ Department of Apply Mathematics, University of Shanghai Finance and Economics, Shanghai 200433, China

$^{2}$ Department of Civil Engineering, City College, City University of New York, New York, NY 10031, USA

$^{3}$ Virginia Institute of Marine Science, College of William and Mary, Gloucester Point, VA 23062, USA

$^{4}$ State Key Laboratory of Turbulence and Complex Systems, Department of Mechanics and Aerospace Technology, School of Engineering, Peking University, Beijing 100871, China

Abstract

This paper studies a class of unsteady flows as extensions of the classic Stokes’ problems to consider influence of solid walls, effect of pressure gradients, and situation of two-layer fluids. The flows are solved using the method of separating variables and the eigenfunction expansion method. With simplifications, the derived solutions will degenerate to solutions to the classic Stokes’ problems, the Couette flow, and the Poiseuille flow. The exact solutions of these flows clearly illustrate the complexity of the involved physics including evolution of flow velocity profiles and energy transferring at fluid boundaries. For a single-layer flow driven by a plate moving at a constant speed, energy transferred from the plate decreases with time and tends to a non-zero constant as a result of wall effect. In a single-layer flow with an oscillatory boundary, negative energy input may appear at the boundary. For an air-water flow with a finite depth, the interface velocity is proportional to the air velocity, which is a well-known observation in physical oceanography. In addition, there is no energy transferring at the interface between the two fluids in a purely pressure driven two-layer flow.

Acknowledgments

This work is supported by NOAA CREST. Support for LZZ also comes from China Scholarship Council.

References

1.  [1] Stokes, G.G. (1851), On the effect of the internal friction of fluids on the motion of pendulums, Transaction of the Cambridge Philosophical Society, 9, 8-106.
2.  [2] Kundu, P. K. (1990), Fluid Mechanics, Academic Press, New York.
3.  [3] Schlichting, H. (1968), Boundary Layer Theory,, Sixth edition, McGraw-Hill, New York.
4.  [4] Raffaele, F and Carl, W. (2009), Ocean circulation kinetic energy: reservoirs, sources, and sinks, Annual Review of Fluid Mechanics, 41,253-282.
5.  [5] Ozinsky, A.E. and Ekama, G.A. (1995), Secondary setting-tank modeling and design. Part 1: Review of theoretical and practical developments, Water SA, 21, 325-332.
6.  [6] Iorio, C.S., Goncharova, O. and Kabov, O. (2011), Influence of boundaries on shear-driven flow of liquids in open cavities, Microgravity Science and Technology, 23, 373-379.
7.  [7] Erdogan, M.E. (2000), A note on an unsteady flow of a viscous fluid due to an oscillating plane wall, International Journal of Non-Linear Mechanics 35, 1-6.
8.  [8] Liu, C.M. (2008), Complete solutions to extended Stokes' problems, Mathematical Problems in Engineering, 2008, 754262, doi:10.1155/2008/754262.
9.  [9] Tang, H.S., Zhang, L.Z., Maa, J.P.-Y., Li, H., Jiang, C.B. and Hussain, R. (2013), Fluid driven by tangential velocity and shear stress: Mathematical analysis, numerical experiment, and implication to surface flow. Mathematical Problems in Engineering, In press.
10.  [10] Muzychka, Y.S. and Yovanovich, M.M. (2010), Unsteady viscous flows and Stokes's first problem, International Journal of Thermal Sciences, 49, 820-828.
11.  [11] Tan, W. C. and Masuoka, T. (2005), Stokes first problem for an Oldroyd-B fluid in a porous half space, Physics of Fluids, 17, 023101.
12.  [12] Ali, F., Norzieha, M., Sharidan, S., Khan, I. and Hayat, T. (2012), New exacts olutions of Stokes' second problem for an MHD second grade fluid inaporousspace, International Journal of Non-Linear Mechanics, 47, 521-525.
13.  [13] Evans, L.C. (1998), Partial Differential Equations, American Mathematical Society, Provindence.