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
    % enforce symmetry about axis (even or odd)
    sp=1:N; sn=2*N+1-sp; % selection and flipping vectors,sp)+d.y(sp,sn);
    Z=zeros(N,N); I=eye(N); 
    d.x=i*alpha*I; d.xx=-alpha^2*I;

    The state vector q

    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
    Iu=II(l.u,:); Iv=II(l.v,:);Ip=II(l.p,:);Ie=II(l.e,:); % selection matrices
    % Laplacian

    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**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
    C=[,:)*Iu+d.x(N,:)*Iv; ... % uy+vx=0 at free surface
        Ip(N,:)+sig*(1-alpha^2)*Ie]; % jump in normal stress 
    % computing eigenmodes
    s=diag(S);  [t,o]=sort(-real(s)); s=s(o); U=U(:,o);
    rem=abs(s)>1000; s(rem)=[]; U(:,rem)=[];
    % % loading the scanned figure for comparison
    % a=imread('pipe_spectra.png'); ss=size(a);
    % 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


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