Pressure in MCSC

Dear friends,
I am looking for the right equation of pressure in multicomponenet SC model. In Sukop’s lectures, I found the following:
But in other references, I found an additional term. For example, for D2Q9:
Can any one let me know which one is correct?

The second one … the second term gives you the pressure at the interface. You see that if you are in the bulk of a bubble for example … the second term goes to zero.


Thank you Andrea. I have another question:
when we initialize bubble test with two different rho0,1 and rho0,2 (a circle region of fluid1 is surrounded by fluid2), are the following correct for calculating pressure at the equilibirium state:

I derived the above equation based on P-P0=1/3(rho-rho0).

I’m not sure I understand, I’m sorry. When you talk about equilibrium state I think you mean no significant change in radius of your bubble with time. Is this what you mean?

Well in this case why don’t you use the formula you just wrote before for a lattice node inside the bubble and a second one outside the bubble and you cal Pin the first one and Pout the second one?

Anyway, I’m not sure that what you wrote in the last message is alright because I don’t see the parameter G in it. But maybe we are talking about two different things.


Sorry Andrea. my writing is not clear. I want to simulate surface tension by bubble test. My problem is calculating pressure. Suppose we initialize the simulation with a circular region of a higher density fluid 1 (rho0=2) surrounded by fluid 2 (rho0=1) in a rectangular domain. At the end of simulation we get rho1=2.01 and rho2=.01 within the bubble and rho2=.98 and rho1=.001 out of bubble. Now please let me know how to calculate presseure? Is the following correct?

Pin= 1/3*(2.01+.01)+12G2.01*.01 and Pout=1/3*(.98+.001)+12G.98*.001 and delta P= Pin-Pout

well, I think you are right yes …

Pin-Pout=Delta P we agree no?


yes. I agree on delta P= Pin- pout. But if what I wrote is correct then when we initialize with higher initial densities (for example rho01=3 and rho01=1) we get higher delta P. Is it correct?

However, I think it is not true. Because the definition of sound speed is:
Cs^2=d p/d rho; where in lattice units Cs^2=1/3. So after integration, we have P=P0+1/3*(rho-rho0).
Thus for previous example, we should write Pin=P0+1/3*((2.01-2)+(.01-0))+…; and Pout=P0+1/3*((.98-1)+(.001-0))+…
Is the latter wrong??

Many thanks for your attention.

Can someone explain where we get this from cs^2 =1/3 and RT = 1/3?
I need this for my report and wouldn’t mind being able to cite this along with some good explanation.

Hi ghassemi,

are you sure about 12*G ? Should not be 1/cs2 (which is 3 in my case for D2Q9 ) instead of 12?


Hi Andrea, I am not sure about 12G. I found the above equation in an unsure reference. So you use the following:
Grho1rho2 ? May I ask you to introduce your reference for this?
Finally, I am looking for your reply to my previous question about the pressure:
P=1/3*(rho1+rho2)+… or P=P0+1/3*(rho1-rho01+rho2-rho02)+…?


Hi ghassemi.

I took my equation for the pressure from:
Huang, H., D.T. Thorne, M.G. Schaap, and M.C. Sukop, 2007. Proposed approximation for
contact angles in Shan-and-Chen-type multicomponent multiphase lattice Boltzmann
models, Phys. Rev. E 76, 066701. doi: 10.1103/PhysRevE.76.066701

Anyway … there are the original references to the Shan Papers inside.

About your second question, I’m sorry but I don’t know what to answer.


Thank you Andrea.

Dear Ponja,

Actually c_s^2=1/3 comes from the discretization of velocity space. When you specify velocity set and specify weights to ensure density and momentum conservations and after performing Chapman-Enskog expansion you come up with the term p=rho/3 (because of weights for Gauss-Hermite quadrature). The sound of speed is defined as c_s^2=dp/drho, therefore p=c_s^2 rho. Actually you can specify other weights and ensure density and momentum conservations. But usual choice is 1/3.


thank you very much alex