## Complete solutions to extended Stokes’ problems.(English)Zbl 1149.76016

Summary: We theoretically solve the viscous flow of either a finite or infinite depth, which is driven by moving plane(s). Such a viscous flow is usually named as Stokes first or second problem, which indicates the fluid motion driven by impulsive or oscillating motion of the boundary, respectively. Traditional Stokes problems are firstly revisited, and three extended problems are subsequently examined. Using some mathematical techniques and integral transforms, we derive complete solutions which can exactly capture the flow characteristics at any time. The corresponding steady-state and transient solutions are readily determined on the basis of complete solutions. Current results have wide applications in academic researches and are of significance for future studies taking more boundary conditions and non-Newtonian fluids into account.

### MSC:

 76D07 Stokes and related (Oseen, etc.) flows

### Keywords:

boundary motion; integral transforms
Full Text:

### References:

 [1] G. G. Stokes, “On the effect of the internal friction of fluids on the motion of pendulums,” Transaction of the Cambridge Philosophical Society, vol. 9, pp. 8-106, 1851. [2] M. E. Erdogan, “A note on an unsteady flow of a viscous fluid due to an oscillating plane wall,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 1-6, 2000. · Zbl 1006.76028 · doi:10.1016/S0020-7462(99)00019-0 [3] C.-M. Liu and I.-C. Liu, “A note on the transient solution of Stokes’ second problem with arbitrary initial phase,” Journal of Mechanics, vol. 22, no. 4, pp. 349-354, 2006. [4] R. Panton, “The transient for Stokes’s oscillating plate: a solution in terms of tabulated functions,” Journal of Fluid Mechanics, vol. 31, no. 4, pp. 819-825, 1968. · Zbl 0193.56302 · doi:10.1017/S0022112068000509 [5] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, NY, USA, 1979. · Zbl 0434.76027 [6] Y. Zeng and S. Weinbaum, “Stokes problems for moving half-planes,” Journal of Fluid Mechanics, vol. 287, pp. 59-74, 1995. · Zbl 0843.76015 · doi:10.1017/S0022112095000851 [7] C. Y. Wang, “Exact solutions of the unsteady Navier-Stokes equations,” Applied Mechanics Reviews, vol. 42, no. 11, part 2, pp. S269-S282, 1989. · Zbl 0753.76046 · doi:10.1115/1.3152400 [8] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer, New York, NY, USA, 1973. · Zbl 0285.65079
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