5.4.2. Spreading of a droplet
We can use the same equation class to calculate the spreading of a droplet. For that, we have to switch to an axisymmetric coordinate system and also make sure that the droplet can spread at all. For the latter, we must make sure that the droplet does not end at its radius \(R\) with \(h(R,t)=0\), since this would exclude any change in the height according to (5.6). A conventional way to resolve this is the addition of a precursor film with a thin thickness compared to the droplet. The thickness of this film will control the spreading velocity. Additionally, one may add a disjoining pressure. Thereby, one can e.g. enforce the spreading to stop at a finite contact angle:
from lubrication import *
class DropletSpreading(Problem):
def __init__(self):
super(DropletSpreading,self).__init__()
self.hp=0.0075 # precursor height
self.sigma=1 # surface tension
self.R,self.h_center=1,0.5 # initial radius and height of the droplet
self.theta_eq=pi/8 # equilibrium contact angle
def define_problem(self):
self.set_coordinate_system("axisymmetric")
self.add_mesh(LineMesh(N=500,size=5)) # simple line mesh
h=var("h") # Building disjoining pressure
disjoining_pressure=5*self.sigma*self.hp**2*self.theta_eq**2*(h**3 - self.hp**3)/(3*h**6)
eqs=LubricationEquations(sigma=self.sigma,disjoining_pressure=disjoining_pressure) # equations
eqs+=TextFileOutput() # output
h_init=maximum(self.h_center*(1-(var("coordinate_x")/self.R)**2),self.hp) # Initial height
eqs+=InitialCondition(h=h_init)
self.add_equations(eqs@"domain") # adding the equation
if __name__=="__main__":
with DropletSpreading() as problem:
problem.run(1000,outstep=True,startstep=0.01,maxstep=10,temporal_error=1)
Fig. 5.11 Spreading of a droplet until the equilibrium contact angle is reached, which is enforced by the disjoining pressure.