I am trying to simulate liquid film evaporation with free boundary conditions (in cartesian coordinates) and my boundary conditions are thus:
$$
\frac{\partial h}{\partial x} = 0, \qquad (1)
$$
$$
\frac{\partial^2 h}{\partial x^2} = 0, \qquad (2)
$$
$$
\frac{\partial^3 h}{\partial x^3}=0. \qquad (3)
$$
However, I need only two of the above three conditions to satisfy my 4th order non-linear partial differential equation for film thickness, which looks something like.
$$
\frac{\partial h}{\partial t} + h^3\frac{\partial^3 h}{\partial x^3} + … = 0
$$
My question is: what does a combination of 1st and 2nd derivative conditions mean and what does a combination of 2nd and 3rd derivatives mean?
If I apply (1) and (2), does it mean that slope and curvature are zero and if I apply (1) and (3), does it mean that slope and shear stress are zero (from analogies of bending beams etc.)
Best Answer
Here is the answer that I gathered from months of looking at these boundary conditions:
(1) and (2) would mean that the slope is zero and the bending moment / curvature at the ends is zero.
(1) and (3) mean that the slope is zero and the shear stress at the end is zero.