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\title{Faraday crispations}
\author{Predrag Cvitanovi\'c}
\date{Nov 26 2002}

\section{Faraday instability}

The Faraday instability, produced by vertical oscillation of an incompressible liquid with a free upper surface, makes it possible to explore a variety of issues related to nonlinear pattern formation. 

We consider the linear stability of the interface between two incompressible Newtonian fluids. The lower fluid is a liquid layer of large depth $h$. 
The upper layer is air whose density and hydrodynamics we neglect.
We summarize the main points of the theoretical analysis, following ref.~\cite{BET96}.

Elimination of the pressure in the linearized Navier- Stokes equations leads to 
\begin{equation}
 (\partial_t-\nu\nabla^2)\nabla^2 w = 0
\label{2.1}
\end{equation}
where 
$w(x,y,z,t)$ and $\nu$ are the vertical velocity and kinematic viscosity, respectively.
 In a reference frame that oscillates vertically with the container and whose origin $z=0$ coincides with the flat interface, the boundary conditions are 
\begin{equation}
 w= \partial_z w = 0 \qquad\mathrm{at} \qquad z=-h,\,\infty
\label{2.2}
\end{equation} 
The interface position is $z= \zeta(x,y,t)$ and is advected by the fluid 
\begin{equation}
\partial_t \zeta =w
\,.
\label{2.3}
\end{equation} 
Several conditions must be imposed at the interface. Continuity of velocity and of tangential stress imply that 
\begin{eqnarray}
w = \partial_{z} w &=& 0 
\label{2.4a}\\
\rho \nu (\nabla_H^2 - \partial_{zz}) w &=& 0
\,, 
\label{2.4b}
\end{eqnarray}  
where $\nabla_H^2$ is the horizontal Laplacian 
evaluated at the interface, and $ \rho $ is the fluid density.

Curvature of the interface yields a normal stress discontinuity of 
$- \sigma\nabla_H^2 \zeta$,
 where $\sigma$ is the surface tension. The total balance of forces normal to the interface is
\begin{equation}
  \rho [ \partial_t  - \nu (3\nabla_H^2 + \partial_{zz})]\partial_{z}w = 
   \rho[g+f(t)]\nabla_H^2 \zeta - \sigma \nabla_H^4 \zeta, 
\label{2.4c} 
\end{equation}
where $g$ is the usual gravitational acceleration and $f (t)$
 is its effective modulation by the imposed vertical oscillation.
 
The full hydrodynamic problem would require the interface conditions (\ref{2.4a})-(\ref{2.4c}) to be applied at $z=  z(x,y,t)$, but linearization justifies the more tractable use of $z=0$.
The linear system (\ref{2.1})-(\ref{2.4b}) is homogeneous in the horizontal directions and periodic in time. With the assumption of boundedness
(as $x,y \to \infty$),
the solutions are of the Floquet-Fourier form 
\begin{eqnarray}
w(x,y,z,t) &=& \sin(k_x x + k_y y) e^{(\mu + i \alpha)t} 
		\sum_n w_{n}(z) e^{in \omega t} + c.c. 
\label{2.5a}\\
\zeta(x,y,t) &=& \sin(k_x x + k_y y) e^{(\mu + i \alpha)t} 
		\sum_n \zeta_{n}(z) e^{in \omega t} + c.c. 
\,, 
\label{2.5b}
\end{eqnarray}  
where $2\pi/\omega$ is the period of the vertical acceleration $f (t)$,
 $k = \sqrt{k_x^2 +k_y^2}$
 is the horizontal wave number, and $ \mu + i \alpha$ the Floquet exponent, with  
$0 \leq \alpha\leq \omega/2 $.
 The $z$ dependence of $w$ is determined by substituting 
 (\ref{2.5a}) into (\ref{2.1}):
\begin{equation}
w_n(z) = a_n e^{kz} + b_n e^{-kz} + c_n e^{q_n z} + d_n e^{-q_n z}
\,
\label{2.6a} 
\end{equation}
where 
\begin{equation}
q_n^2 = k^2 +  {\mu + (\alpha + n \omega)t \over \nu}
\,.
\label{2.6b}
\end{equation}  
Equations (\ref{2.2}) and (\ref{2.3}), which do not couple the different temporal Fourier components, allow the four coefficients in (\ref{2.6a}) to be expressed in terms of $ \zeta_n $. 
The remaining equation (\ref{2.4c}) can then be written as 
\begin{equation}
A_n \zeta_n = 2 [f \zeta]_n , 
\label{2.7} 
\end{equation} 
where $ A_n $ is an algebraic function of the physical parameters and 
$ \mu$, $\alpha$, $k  $ and $ [ f z]_n $ is the $n$th Fourier component of the product of 
$f (t)$ and $\sum_{n'} \zeta_{n'} e^{in' \omega t}$. 
With the vertical acceleration given by 
\begin{equation}
f(t) = a \cos(\chi) \cos(m \omega t)
\,,
\label{2.8} 
\end{equation} 
we have 
\begin{equation}
[f \zeta]_n = {a \over 2} \cos(\chi)(\zeta_{n-m}+ \zeta_{n+m})
\,, 
\label{2.9}
\end{equation}
 with appropriate modifications to ensure that $ \zeta $ is real and is represented by a truncated Fourier series in 
(\ref{2.5b}). Equation 
(\ref{2.7}) can therefore be written as 
\begin{equation}
{\cal A}\hat{\zeta} = a{\cal B}\hat{\zeta}
\, , 
\label{2.10}
\end{equation}
where $\hat{\zeta}$
is the vector of temporal Fourier coefficients $ \zeta_n $, $\mathcal{A}$ is the diagonal matrix of $A_n$ , and $\mathcal{B}$ is a banded matrix containing two super- and subdiagonals.

With $\mathcal{A}$ or $\mathcal{B}$ invertible, Eq. (\ref{2.10}) constitutes
an eigenvalue problem for the forcing amplitude a and can be solved by calling a standard subroutine, in this case RG from EISPACK. At most 20 temporal Fourier coefficients suffice to resolve $\hat{\zeta}$ for typical calculations such as $(m,l)=(4,5)$ and 
$a<30g$. With fixed physical parameters, the solution of (\ref{2.10}) yields a set of amplitudes a that depend on $ \mu$, $\alpha$ and  $k  $. To obtain marginal stability curves, we fix $k$ and $\alpha$, set the growth rate $ \mu$ to 0, and select the smallest real positive $\alpha$ obtained. We find that setting the imaginary part a of the Floquet exponent to any value other than 0 and $\omega/2$ yields only physically unrealizable complex values of $\alpha$, indicating that the time dependence of the marginal modes is always either harmonic $(\alpha= 0)$
or subharmonic $(\alpha =\omega/2)$. 

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