%% A droplet sitting on a surface
% Ray Treinen and Nick Trefethen, October 2022
%%
% (Chebfun example ode-nonlin/Droplets.m)
% [Tags: #nonlinearODE, #fluids, #BVP]
%% 1. The equations
% Suppose we put a drop of liquid on a solid surface.
% Under idealized conditions
% it will form an axisymmetric droplet (a ``sessile drop'') whose shape is determined by the
% contact angle and by the capillary constant of the liquid,
% $$ \kappa = {\rho g\over \sigma}, $$
% where $\rho$ is the density, $g$ is the gravitational acceleration, and
% $\sigma$ is the surface tension.
% In engineering, the behavior of droplets is one of innumerable situations where
% surface tension plays a role, and we hope the reader has seen the movie [2]
% made by the second author's father at MIT in the
% 1960s, still available on YouTube after all these years. In mathematics,
% this is a prototypical problem in the area of capillary surfaces, on which a classic
% reference is the 1986 book [1] by Robert Finn at Stanford.
%%
% Suppose our goal is to calculate the generating curve that defines the shape
% of the droplet. As described in [3], we can do this by solving the equations
% $$ {dr\over ds} = \cos\psi, \quad
% {du\over ds} = \sin\psi, \quad
% {d\psi\over ds} = \kappa u - r^{-1}\sin\psi $$
% where $s$ is the arc length along the curve,
% $r(s)$ is the radius, $u(s)$ is the height, and
% $\psi(s)$ is the angle of inclination of the curve.
% In this formulation $s\in [-L,L]$, where $L$, the total arc length of
% the curve starting at the center, is unknown a priori. For convenience we can
% rescale by defining $\tau = s/L$, so that the computational domain
% becomes $\tau \in [-1,1]$ and the equations are
% $$ {dR\over d\tau} = L \cos\Psi, \quad
% {dU\over d\tau} = L \sin\Psi, \quad
% {d\Psi\over d\tau} = \kappa L U - L R^{-1}\sin\Psi. $$
%% 2. Example: borderline wetting
% Let's use Chebfun to compute a simple droplet. We set the contact radius to $3$ and
% we fix the contact angle as $\pi/2$, corresponding
% to a liquid at the borderline of wetting/not wetting the solid. We use
% the traditional scaling of the problem so that $\kappa = 1$.
% Here is the shape.
tic
b = 3; Psib = -pi/2;
N = chebop(-1,1);
N.op = @(t,R,U,Psi,L) [diff(R) - L*cos(Psi)
diff(U) - L*sin(Psi)
R*diff(Psi) + L*sin(Psi) - R*L*U
diff(L)];
N.bc = @(t,R,U,Psi,L) [R(-1)+b; R(1)-b; Psi(-1)+Psib; Psi(1)-Psib];
t = chebfun('t');
N.init = [b*t; 1+0*t; t*Psib; 2*b+0*t];
[R,U,Psi,L] = N\0;
U = U - min(U);
LW = 'linewidth'; blue = [.3 .4 1];
plot([-b-1;b+1],[0;0],'k',LW,1.5) , hold on
fill(R,U,blue), plot(R,U,'k',LW,1), hold off
axis equal, grid on, title('borderline wetting')
%% 3. Example: no wetting
% Let's change the contact angle to $\pi$, so the liquid minimally wets the
% surface. If a liquid has a contact angle greater than $\pi/2$, this is described
% as ``not wetting''.
Psib = -pi;
N.bc = @(t,R,U,Psi,L) [R(-1)+b; R(1)-b; Psi(-1)+Psib; Psi(1)-Psib];
N.init = [b*t; 1+0*t; t*Psib; 2*b+0*t];
[R,U,Psi,L] = N\0;
U = U - min(U);
plot([-b-1;b+1],[0;0],'k',LW,1.5) , hold on
fill(R,U,blue), plot(R,U,'k',LW,1), hold off
axis equal, grid on, title('no wetting')
%% 4. Computational remarks
% In Chebfun, our computations have proceeded in the simplest possible
% manner. We worked on $[-1,1]$, though $[0,1]$ would also be possible
% with Neumann boundary conditions. We multiplied through by $R(\tau)$
% to avoid the singularity at $R=0$. We included the unknown parameter $L$
% in the problem via a trivial additional differential equation $dL/dt = 0$,
% though one could also use an explicit parameter formulation as described
% in section 7.10 fo the Chebfun Guide.
%%
% Here is how long the computations took,
toc
%%
% and here are the lengths of the computed functions $R$, $U$, and $\Psi$:
[length(R) length(U) length(Psi)]
%%
% All this can be done more efficiently by bypassing Chebfun overhead, using
% spectral methods more carefully as described in [3].
%% 5. Prescribing the volume
% The prescription of the radius $b$ where the liquid meets the supporting plane
% is a mathematical construction to make for a nice boundary value problem.
% It might be more physically realistic to prescribe a volume of the liquid
% and treat that radius $b$ as a free parameter.
%%
% While it would be possible to compute the volume of the droplet numerically,
% a neat computation turns this into evaluating the solution in the following
% formula.
% To derive this, we note one form of the differential equations is
% $$
% (r\sin\psi)_r = ru
% $$
% and this holds for any portion of the curve except at the vertical point.
% We take the initial height $u = u_0 < 0$ when $r=0$.
% Then we use the method of washers to compute the volume
% $$ V = \int^{u_0}_u \pi r^2\, du
% = -\pi r^2u - 2\pi\int^0_r \rho u\, d\rho
% = -\pi r^2u - 2\pi\int^0_r (\rho\sin\psi)_\rho\, d\rho
% = -\pi r^2u + 2\pi r\sin\psi.$$
% This gives the volume formula
% $$
% V = 2\pi b\sin\psi_b - \pi b^2 U(1)
% $$
% for the computed curve before the vertical translation.
% We can include this formula to measure the volume of a drop we have
% computed with the modification.
[R,U,Psi,L] = N\0;
pi*b*(2*sin(Psib)-b*U(1))
%%
% We can also use this formula to build an equation that must be solved in
% the boundary conditions, while we add the radius $b$ to the list of
% parameters.
Psib = -pi; v0 = 10;
N = chebop(-1,1);
N.op = @(t,R,U,Psi,ell,b) [diff(R) - ell*cos(Psi)
diff(U) - ell*sin(Psi)
R*diff(Psi) + ell*sin(Psi) - R*ell*U
diff(ell)];
N.bc = @(t,R,U,Psi,ell,b) [R(-1)+b
R(1)-b
Psi(-1)+Psib
Psi(1)-Psib
pi*b*(2*sin(Psib)-b*U(1))-v0];
t = chebfun('t'); bguess = nthroot(v0,3)/2;
N.init = [bguess*t; bguess*(-2+cos(t*Psib)); t*Psib; 2*bguess+0*t; bguess+0*t];
[R,U,Psi,ell,b] = N\0;
U = U - min(U);
plot([floor(min(R));ceil(max(R))],[0;0],'k',LW,1.5) , hold on
fill(R,U,blue), plot(R,U,'k',LW,1), hold off
axis equal, grid on, title('prescribed volume')
%%
% Finally, we note that sometimes nonlinear problems can be quite sensitive
% to the supplied initial guess. In this volume constrained problem, we
% find that Chebfun gives the correct answer for the problem above, but if
% the contact angle is changed too much, then the solver does not find a
% solution. More robust initial guesses are discussed in [3].
%% References
%
% [1] R. Finn, _Equilibrium Capillary Surfaces_, Springer Science &
% Business Media, 2012 (originally published in 1986).
%
% [2] L. M. Trefethen, Surface Tension in Fluid Mechanics,
% https://www.youtube.com/watch?v=sfyAkFc66bE (movie created in 1963).
%
% [3] R. Treinen, Spectral methods for capillary surfaces described
% by bounded generating curves, arXiv:2205.02931v2, 2022.