droplet splatting
Sorry for the delay in replying. I have been away for a while.
Oh no... I just found it funny (:#
In principle, both could do the job, but the finite element method can
capture moving interfaces far more easily than the finite difference
method. Capturing the free surface and contact line motion accuractely
is crucial in a drop spreading problem, where capillarity is a crucial
effect that has to be modelled.
With the finite difference method you start with a fixed grid
(Cartesian probably, but see my comments later), and discretise the
spatial derivative operators such as div and grad etc on the grid. If
your interfaces lie along grid lines and does not move then this works
very well. If your interface doesn't lie along the grid lines and is
not fixed like in our problem then what? a) The biasing of the
discretised versions of the derivative operators becomes far more
complicated. b) How do you handle the fact that the number of unknowns
in your solution will be changing at each time step etc. c) How do you
know where your interface actually is? For example, if the contact
line position that you are solving for lies between two grid points,
then where exactly is it, and how do you accurately apply boundary
conditions there?
There are some ways of trying to overcome these problems, like using
body fitted coordinates so that the mesh remains fixed with respect to
the drop. However, in practise this method only works for very small
deformations and is not suitable for drop impact problems. If you are
going to go down the finite difference route, then I suggest you do
something like the following: Use a coordinate system which is centred
on and moves with the contact line, which is polar local to the
contact line, and Cartesian far away from it. Such coordinate systems
exist (I know this because I base my FE grid on one), and this will at
least mean that you capture the interface accurately where it matters
most: Near the contact line. Be warned though, that deriving the
discretised form of the operators will be so complicated as to be
virtually impossible to get correct. Remember that in order to
calculate the curvature in the normal stress condition you will need
to accurately calculate second derivates of free surface shape.
With the finite element method on the other hand, you split the domain
up into regions with shape of your own choosing, and approximate the
functional form of the solution over each of those regions rather than
discretising derivative operators. Typically the solution is
approximated by quadratic and linear forms. You can then choose the
shape and position of the elements such that the interface is
accurately captured. The mesh will have to move with time as the
interface moves however, and become quite deformed during impact, so
you would have to think carefully about how to handle this. One way is
to use a Lagrangian formulation from the start, another to use an
Eularian formulation but to map the positions of the nodes of the
elements with the flow. I would argue against using either of these
methods, since the fluid flow inside the drop is so complicated that
the mesh deteriorates very quickly. I suggest using director based
method or some other critereon. A hint/tip for you is that since you
are dealing with an integral form of the equations you can do
integration by parts on the curvature and only every have to calculate
first derivatives of free surface shape.
Hope this helps!
Very interesting. Prof Veldman obviously has a HUGE amount of
computing power to hand. No mention of the model used for spreading
though. I am always curious when people don't mention how they have
overcome the problems with the fact that there is no solution to the
classical fluid mechanical model. I'd be interested to know how he got
his contact line to move.
All the best,
Paul
