I am a little confused by how you got square of d/dz ϕ(z,t,V,a) in the pde. Shouldn't it just be d/dz ϕ(z,t,V,a) as per the original equation. Kindly clarify.
The original equation given (by you) was NOT (something like):
Which means that when the
d/dphi of f(phi)
V * d/dz of phi(z,t)
the square of d/dz of phi(z,t) emerged.
If I look at the article that you took the formulae from, the situation doesn't get much clearer.
Considering that for the derivative of f(phi) the notation f'(phi) was used, one could even argue that it should have read:
d/dt of f(phi)
d/dphi of f(phi).
I think you are more an expert on the subject at hand, so you should guide as to what the PDE should be. After that, I can help you solve it.
To get the (eventually resulting) formulae into Prime, you'll just have to retype them, or use the Mathcad to Prime converter. Note that you might need to do some rework after conversion.
Also note that, because (I think) you are a licensed user of Prime, you are entitled (=licensed !) to install and use Mathcad 15 on your PC. After installing Mathcad 15, for licensing point it to the very same license file already on your computer that you used for licensing Prime. You'll need Mathcad 15 installed if you want to convert any Mathcad (15 and before) files to Prime. (There's no backwards conversion. That is, there's NO tool whatsoever to convert a Prime sheet to a format supported by any previous Prime version, or Mathcad version.)
Sorry, I am still a bit confused. The original equation is given by dϕ/dt +df(ϕ)/dz=0. In the equivalent form, we can write dϕ/dt +(df(ϕ)/dϕ)*dϕ/dz=0. Now f(ϕ)=ϕ*V. So shouldn't the final PDE be dϕ/dt +V*dϕ/dz=0 since df(ϕ)/dϕ=V.
I am relatively new to Mathcad. So far I have only used prime (version 2.0 and up). I do have Mathcad 15 and I find it to be bit different (slightly confusing) to prime. That is why I asked for a way to convert it to prime. Thanks for the help.
Sorry, I made a mistake: read one thing, interpreted as another.
You're right. The PDE, simplified by knowing that f(phi)=phi*V, should read:
d/dt of phi(z,t) + V * d/dz of phi(z,t).
I'll see if I can make this thing work.
I believe the boundary condition should be
ϕ(z,t0,V,a) = ϕ0 (initial condition) - uniform volume fraction throughout the mixture initially
ϕ(z0,t,V,a) = ϕinfinity for t >0 (BC1) - sedimentation starts and volume fraction at the base starts to increase to final value.
ϕ(L,t,V,a) = 0 for t>0 (BC2) - concentration at surface z=L is 0 as sedimentation starts (clarification).
I think you have it slightly mixed up.
The settling plot should like what you see in the attachment. Any ideas?
So if none of the phi values solve the PDE, then it means that it is probably not the solution. That's a bummer. That kind of leaves me a bit stuck.
Can you please share your mathcad file with me.