A Note on the Transient Solution of Stokes' Second Problem with Arbitrary Initial Phase

Abstract

The flow of a viscous fluid disturbed by an oscillating plate of arbitrary initial phase is studied in present note. The exact solutions of the velocity and the shear stress are solved using a Laplace transform method. The velocity is derived in terms of complementary error functions and the shear stress on the boundary is given in the form of Fresnel integrals. Since the steady-state solutions are well known, our discussions are focused on the transient solutions. The transient state will disappear faster for the wall stress than that for the velocity field. Comparing the results corresponding to different initial phases, the cosine case reaches to the steady state more rapidly than the sine case.

A Residual Theorem Approach Applied to Stokes’ Problems with Generally Periodic Boundary Conditions including a Pressure Gradient Term

“…The solutions of these problems in one-dimensional case and for unbounded domains are well established in literature, in particular for heat diffusion problems (see, for example, Watson [9], Batchelor [1], Erdogan [10], and Liu and Liu [11]). The strategy for the solution, as in the present paper, is to consider the Laplace transformation in time; however, the inverse transformation is not simple to achieve (see, for example, Devakar and Iyengar [12] where the inverse transform is obtained through a numerical procedure); therefore a clear and a more general strategy, which would allow the achievement of the analytical solution for different boundary conditions, has to be found.…”

Remarks on the solution of extended Stokes' problems

“…1-a their definitions (8) evaluated in t = 0. The functions H ± 2 (9) with H ± 2 (y, 0) ≡ 0 are then inserted into the formula (8) and the nondimensional quantities T = ωt and Y = y(ω/ν) 1/2 are used, according to [4]. In this way, the solution:…”

“…Solutions of the second problem in an infinite-depth flow have been discussed in [2], [3], [4] and in many other papers. They are usually written in terms of error functions of complex arguments, because in the corresponding real forms integrands containing oscillatory functions appear, the numerical integration of which can lead to severe errors [5].…”

Solution to the Stokes' First Problem for the Earthquake-Induced Flow

“…There are two kinds of Stokes' problems [1] describing the flows induced by an impulsively moving plate and the oscillating plate, respectively. The complete and steady-state solutions have been provided in many literatures (for details, see [3], [4] and [5]). For geophysicists interested in the flow induced by earthquakes, fracture of ice sheets and other boundary motions, the extension of the classical Stokes' problems is required to calculate practical geophysical flows.…”

“…Since (1) and (2) (4) 978-1-4244-2126-8/08/$25.00 ©2008 IEEE Accordingly, the total solution u can be determined by superpositioning the solutions of (3) and (4). As the former case stands for the classical Stokes' first problem, the exact solution to (3) is shown as (for details, see [4]) ul uo erfcj===j (5) As for the solution to (4), owing to the fluid motions for z > 0 and z < 0 being in the opposite direction, an additional condition of u2 has to be considered u2(y,z > O,t)= -u2(y,z < 0,t) I (6) which further leads to the following condition u2(y,z = O,t)= 0. (7) Based on the conditions shown in (6) Applying the inverse Fourier sine transform and inverse Laplace transform to (12), u2 can be obtained after some algebra.…”