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.)

  • 1
    $\begingroup$ @MaksimZholudev Thanks for making the edit. It looks much cleaner! $\endgroup$
    – dearN
    Jan 31, 2012 at 16:48

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.