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The transient for Stokes's oscillating plate: a solution in terms of tabulated functions

  • Ronald Panton (a1)
Abstract

The motion of a semi-infinite incompressible fluid caused by the sinusoidal oscillation of a plane flat plate is termed Stokes's problem. When the plate starts from rest in a still fluid a transient solution must be added to Stokes's well-known steady-state result. This paper presents a closed-form expression for the transient solution. Previous answers have contained a non-standard integral which could not be evaluated. The answer presented herein contains exponentials and error functions of a complex argument. These functions are readily available in newer mathematical tables. Graphs of the transient solution are presented for both sin (T) and – cos (T) boundary conditions. Velocity distributions in the fluid are also plotted and it is found that the transient period is essentially complete in one-half cycle for the cosine oscillation and in a full cycle for the sine wave case.

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References
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Abramowitz, B. M. & Stegum, I. A. (eds.) 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: U.S. Gov. Printing Office.
Arpaci, V. S. 1966 Conduction Heat Transfer. Reading: Addison-Wesley.
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford University Press.
Salzer, H. E. 1951 Math. Tables Aids Comp. 5, 67.
Schlichting, H. 1960 Bounday Layer Theory. New York: McGraw-Hill.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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