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Nonlinear problem with almost no implicit terms

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Old   June 13, 2006, 07:20
Default dear Friends, I have highly
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rafal zietara
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dear Friends,

I have highly nonlinear problem that I want to solve in OF. please help me with implementation.

This problem is in steady state derived from system where three forces act on fluid (capillary, gravity and viscosity forces).
I simplified problem to be tackable numerically (the equation is dimmensionless form).
The continuity equation is expressed as:
div(A*u)=0 (1)
and u as:
u= - K/sqrt(A) grad A - g/mag(g) A + v (2)

where: (variables given in the order name - meaning, unit; type of variable(field) in OF)
u - liquid velocity, dimless; volVectorField defined on mesh
v - gas velocity, dimless; volVectorField defined on mesh
K - constant , m^-1 ; dimensionedScalar
A - area , dimless; volScalarField defined on mesh
g - gravity const. , ms-1 ; dimensionedVector

From those two equations i would like to calculate A.

when i plug (2) to (1), i get:

div(-K*sqrt(A) grad A - g/mag(g) A^2 + A v )=0 (3)

if we try to expand div we get:

- 0.5K/sqrt(A) grad A grad A - K*sqrt(A) laplacian(A) - 2A g/mag(g) grad(A) + A div(v) + v grad(A)=0 (4)

as you see in equation (4) we have almost no implicit terms.

How to tackle the problem?
How to implement this in OpenFoam?
Where to look for a solutions of a similar problems?

Any suggestions appreciated. Thank you in advance.

/Rafal

one thing more
Significance of terms in equation (3).
(- K*sqrt(A) grad A) - plays key role near 2 out of 5 of the boundaries in the rest of the domain neglegible.
(- g/mag(g) A^2) - significant in most of the domain
A v - an order less than previous
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Old   June 13, 2006, 17:32
Default Hi Rafal! At first the equa
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Bernhard Gschaider
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Hi Rafal!

At first the equations looked to me like something I have seen before (two phase flow in porous media), but then again they didn't.

Questions I have at a first glance:
- dimensionless velocities? (I know physicicsts do it all the time, usually by normalizing with c, but engineers?)
- the relative densities of liquid and gas go into K?
- which leads to the next question: both (gas and liquid) are incompressible?
- in (2) you wanted to write K/(sqrt(A) grad(A))? Otherwise I can't match the dimensions.
- You are looking for a stationary solution?

I think it can be done in OF, but I'm not aware of a solver in the distribution that is similar to your problem.
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Old   June 14, 2006, 07:02
Default Hi Bernhard, Thanks for ans
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rafal zietara
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Hi Bernhard,

Thanks for answer. I didn't want to provide to much details about the system not to complicate the problem to much.

Good intuition. Those equation describe motion of liquid in a soap foam (drainage of water from a foam due to gravity), which is similar to flow in porous media in a sens that we have also channels in which our fluid flows through static material. In this case material consist of close packed air bubbles moving upwards (velocity of gas v in eq.(2)). Channels are located in the junction of three bubbles.


First of all word of apology. I mistook dimension of K which is m (and NOT m-1 as i wrote above), sorry for confusion.

To give you more details about the equation and provide more comprehensive description of the problem i have prepared it in attachment with derivation of equations.

derivation.dvi

Velocity of gas (v) i assumed is influenced by water drainage. v is calculated separately (done this before). in this discussion we can assume is known.

I assume also that both fluids (gas,liquid) are incompressible.
I am looking for steady state solution.


Why normalisation to get dimensionless velocities?
A (cross-section area through channel) in case of foam is in order of 10-5 with velocities in the order of 10-1 may give numerical problems. I do not stick to this normalization it is only idea i read somewhere about. If it is a problem i can remove it.

Do not hasitate to ask for more details.
All help is appreciated. Thank you in advance for all suggestions and i invite everyone to discussion.
rafal
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Old   June 14, 2006, 07:09
Default IS: Velocity of gas (v) i assu
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IS: Velocity of gas (v) i assumed is influenced by water drainage
SHOULD BE: Velocity of gas (v) i assumed is NOT influenced by water drainage
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Old   April 24, 2008, 09:11
Default hi all, i am supposed to modi
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davey david
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hi all,
i am supposed to modify a code into a dimensionless one.how do i go about it?

thanks in advance
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