[Sashankh]

Hi,

I am trying to solve a liquid-liquid two phase laminar stratified flow in a channel (2D problem) for the velocity distribution. The liquids are immiscible and Newtonian. The interface is stationary. I am solving the Poisson equation in both domains. I impose the Neumann (stress balance) and Dirichlet (velocity balance) condition at the interface. I solve the problem as follows:

1. Initialize the velocity in both domains.
2. Solve domain A using the velocity condition (dirichlet) at the interface.
3. Determine the gradient of velocity at the interface.
4. Share the gradient with domain B.
5. Solve domain B using the gradient condition (Neumann) at the interface as the boundary condition.
6. Determine the velocity at the interface in domain B.
7. Share the velocity with domain A.
8. Solve domain A with the new velocity condition (dirichlet) at the interface.
9. Iterate until convergence.

However, I see that my velocity solution blows up if I increase the viscosity ratio between the two liquids. Is there a numerical stability criteria in solving such problems? I am using a finite-difference scheme.

Any help would be appreciated. Thanks.

[kbaker]

Did you solve Navier Stokes Equations for steady or unsteady state case? what software you tried to solve the governing equations?

kbaker

Originally Posted by Sashankh

Hi,

I am trying to solve a liquid-liquid two phase laminar stratified flow in a channel (2D problem) for the velocity distribution. The liquids are immiscible and Newtonian. The interface is stationary. I am solving the Poisson equation in both domains. I impose the Neumann (stress balance) and Dirichlet (velocity balance) condition at the interface. I solve the problem as follows:

1. Initialize the velocity in both domains.
2. Solve domain A using the velocity condition (dirichlet) at the interface.
3. Determine the gradient of velocity at the interface.
4. Share the gradient with domain B.
5. Solve domain B using the gradient condition (Neumann) at the interface as the boundary condition.
6. Determine the velocity at the interface in domain B.
7. Share the velocity with domain A.
8. Solve domain A with the new velocity condition (dirichlet) at the interface.
9. Iterate until convergence.

However, I see that my velocity solution blows up if I increase the viscosity ratio between the two liquids. Is there a numerical stability criteria in solving such problems? I am using a finite-difference scheme.

Any help would be appreciated. Thanks.

[rmh26]

I'm a little confused on the domain of your problem. Are you looking at flow down a channel (z direction) and solving for the z and y (height) components of the velocity and assuming there is no variation in the x direction?. You are also forcing the interface to be stationary. Is the normal component of the flow at the interface zero? This is the kinematic boundary condition. Also how are you initilializing the velocities?

I am currently working with a two phase flow solver for transient simulation and I do see a decrease in stability when using different viscosities. This comes from the fact that the "cross" terms in the divergence of the stress have to be handled explicitly.

Define your problem a little better and maybe we can find a solution.

[Sashankh]

I am solving the Poisson equation

grad.(grad(u)) = 1,

where u is the velocity. The interface conditions are equal velocity and equal shear stress.

I am currently using the Matlab pde toolbox to solve it. This toolbox uses FEM. I have also tried solving this problem using finite-difference in Matlab. Both times I get the same problem and the solution blows up when I use the algorithm listed in my earlier message.

Also, this is a 1D problem. My earlier post says it is 2D which is a typing error. Once I figure out the algorithm that will work for this simple problem I will extend it to my real problem in 2D.

I really appreciate your help. Thanks.

[Sashankh]

Hi,

I am sorry but there was an error in my previous post.

I am solving the 1D problem of channel flow (not 2D). The z axis points in the direction of the flow while the y axis is normal to the flow direction (height). Since the flow is fully developed the velocity w (z-component of velocity) does not vary in the z-direction but only in the y direction. Hence I am solving for the velocity w(z-component of velocity) as a function of only y. Hence it is a 1D problem.

There are two fluids flowing parallel to each other. I am solving the Poisson equation in both fluids as discussed in my above post. The interface is rigid and does not change shape. The normal component of velocity is zero. The fluids are immiscible.

Since we have two conditions at the interface (Dirichlet and Neumann) I use the algorithm mentioned in my previous post to solve each domain separately and then solve them iteratively.

The above problem has an analytical solution. I am trying to validate my numerical program with this simple problem before extending it to a more complicated domain.

Thanks for your help and please let me know if there is any other information you need with regards to this problem.

[kbaker]

Hello Sashankh you mentioned that you solve multiphase flow problem? Usually there are three models to solve multiphase flow problems homogeneous model, in-homogeneous model and drift flux model? if you said that your fluids are immisible then you need to follow the homogeneous model but you said that you solve poisson eqn? what the relation of poison equation with the flow? Second thing did you consider the interface as flat?

[rmh26]

I'm not sure of what exactly MATLAB is doing but assuming that the equation is being discretized and solved correctly then we can assume the issue is with your algorithm. Have you tried a under relaxing method yet. Everytime you solve for velocity you would update your solution as the some combination of your old solution and your new solution.

U(k-1) = solution from previous boundary condition
U* = solution from the current boundary conditions

U(k) = accept new solution

U(k) = (1-w)*U* + w*U(k-1)

if stability is an issue sometimes picking a w close to 1 will help.

[Sashankh]

Hi Khalid,

I am solving a Stokes flow problem. Hence my inertia terms drop out of the Navier Stokes equation leaving me with the Poisson equation to solve. I also consider the interface to be flat. Please let me know what you think. Thanks.

[Sashankh]

Hi,

I think you are correct about the stability problem with my algorithm. I am trying the under-relaxation method. I will keep you posted on the result.
Is there any other iterative algorithm (one in which I can solve each domain separately and then share node values or gradients at the interface) that you are aware of to solve such problems?
Thanks a lot for your help and I will let you know how the under-relaxation methods works out.