# The Rayleigh-Plateau instability

This is an ongoing work! Not yet validated!

This is the instability of an infinite cylinder of liquid due to surface tension. The system is unstable for wavelengthes longer than its perimeter.

The code is based on pipe_sym.m, but I have changed the names of the different velocity components. Here there is no base flow. I also use a way like in that code to enforce symmetry about the axis of the cylinder, so that I only need to enforce the boundary conditions at the free-surface.

clear all; clf; format compact

% parameters
N=80;      % number of grid points
Re=1000;     % Reynolds number
alpha=0.5;        % axial wavenumber
sig=1; % surface tension

% differentiation matrices
[d.y,d.yy,d.wy,y]=dif1D('cheb',-1,2,2*N);

% enforce symmetry about axis (even or odd)
sp=1:N; sn=2*N+1-sp; % selection and flipping vectors
d.ye=d.y(sp,sp)+d.y(sp,sn);
d.yye=d.yy(sp,sp)+d.yy(sp,sn);
d.yo=d.y(sp,sp)-d.y(sp,sn);
d.yyo=d.yy(sp,sp)-d.yy(sp,sn);
y=y(sp);

Z=zeros(N,N); I=eye(N);
d.x=i*alpha*I; d.xx=-alpha^2*I;


# The state vectorq

The state is the axial and radial velocity components U and v. And there is as well pressure p and interface displacement \eta (which is a scalar wheres u and v are defined as a function of y). \displaystyle q=\begin{pmatrix} u\\v\\p\\\eta\end{pmatrix}

% useful
l.u=(1:N)'; l.v=l.u+N; l.p=l.v+N; l.e=l.p(end)+1; % location vectors
II=eye(3*N+1);
Iu=II(l.u,:); Iv=II(l.v,:);Ip=II(l.p,:);Ie=II(l.e,:); % selection matrices

% Laplacian
lape=d.xx+d.yye+diag(1./y)*d.ye;
lapo=d.xx+d.yyo+diag(1./y)*d.yo;


# System matrices

The system equations are the Navier-Stokes equations in cylindrical coordinates linearized about a zero base flow, and the advection of the interface \eta by the flow (which once inearized about a zero ase flow reduces to the vertical advection of \eta as \eta_t=v) Eq_t=Aq \displaystyle \left( \begin{array}{rcl} u_t&=&-p_x+(u_{xx}+u_{yy}+u_y/y)/Re\\ v_t&=&-p_y+(v_{xx}+v_{yy}+v_y/y-v/y^2)/Re\\ 0&=&u_x+v_y+v/y\\ \eta_t&=&v|_1 \end{array} \right.

% System matrices
A=[lape/Re*Iu-d.x*Ip; ...
(lapo-diag(1./y.^2))/Re*Iv-d.ye*Ip; ...
d.x*Iu+(d.yo+diag(1./y))*Iv; ...
Iv(N,:); ...
];
E=[Iu; Iv; 0*Ip; Ie];


# Boundary conditions

They are: zero tangential stress at the free surface, and also at the free surface: normal stress equal to the pressure jump due to the curvature of the interface \displaystyle \left(\begin{array} 0&=&u_y|_1+v_x|_1 \\ 0&=&p-\sigma \left[\frac{1}{\eta(1+\eta_x^2)^{1/2}}-\frac{\eta_xx}{1+\eta_x^2}\right] \end{array}\right. The first one is already linear, and we need to linearize the second one about an interface at y=1+\eta, which gives for the perturbations \displaystyle \left(\begin{array} 0&=&u_y|_1+v_x|_1 \\ 0&=&p+\sigma (1-\alpha^2)\eta \end{array}\right.

% Boundary conditions
loc=[l.u(N);l.v(N)];
C=[d.ye(N,:)*Iu+d.x(N,:)*Iv; ... % uy+vx=0 at free surface
Ip(N,:)+sig*(1-alpha^2)*Ie]; % jump in normal stress

A(loc,:)=C;
E(loc,:)=0;

% computing eigenmodes
[U,S]=eig(A,E);
s=diag(S);  [t,o]=sort(-real(s)); s=s(o); U=U(:,o);
rem=abs(s)>1000; s(rem)=[]; U(:,rem)=[];

s(1:10)

% x=linspace(0,1,ss(2)); y=linspace(0,-1,ss(1));
% image(x,y,a); axis xy;
% hold on
%
% plot(-imag(s)/k,real(s),'ro')
% xlabel('wave speed'); ylabel('exponential growth rate')
% grid on; hold on

set(gcf,'paperpositionmode','auto')
print('-dpng','-r100','rayleigh_plateau.png')