*faraday-dec27.tex
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**Irrotational Faraday Waves on
a Viscous Fluid**

T.Funada^{*}, J.Wang^{†},
D.D.Joseph^{†‡} & N.Tashiro^{*}

^{*} Department of Digital Engineering, Numazu
National College of Technology, 3600 Ooka, Numazu, Shizuoka, 410-8501, Japan

^{†} Department of Aerospace Engineering and Mechanics, University of
Minnesota, 110 Union St. SE,

Abstract

An analysis
of irrotational Faraday waves on an inviscid fluid was given by Benjamin and Ursell 1954. Here we extend the analysis of the same
problem to purely irrotational waves on a viscous fluid. Following our earlier
work on free surface problems, two irrotational theories are presented. In the
first theory (VPF) the effects of viscosity enter only through the viscous
normal stress term evaluated on the potential. In the second irrotational
theory (VCVPF), a viscous contribution is added to the Bernoulli pressure;
otherwise the second theory is the same as the first. The second theory VCVPF
gives rise to the same damped Mathieu equation as the dissipation method.
Pressure fields are not required and not used in the dissipation method. The dissipation
method is a purely irrotational theory, though it depends on viscosity, in
which only irrotational velocity fields _{} are needed. The two
purely irrotational theories VPF and VCVPF are not restricted to small
viscosities; they are restricted to small vorticity
and do not apply near no-slip wall where Vorticity is generated.

Our VCVPF
and dissipation theories give the same damped Mathieu equation as the
phenomenological approximation of Kumar and Tuckerman 1994. The damping term in
VCVPF is twice the damping rate of VPF. The growth rates of unstable
disturbances computed by VPF are uniformly larger than those computed by VCVPF
(or equivalently by Kumar and Tuckerman). Comparisons with the exact solution
and the Rayleigh-Taylor instability show that thresholds and growth rates for
viscously damped waves are better described by VPF than VCVPF.

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^{‡ }Author to whom correspondence should be
addressed. Telephone: (612) 625-0309; fax (612) 626-1558; electronic mail:
joseph@aem.umn.edu