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 Seamus O'Shaughnessy November 27, 2007 08:58

I am trying to solve a problem involving Marangoni convection in which an air bubble is surrounded by a liquid oil layer. The governing equations are Navier-Stokes and The energy Eqn. I am trying to benchmark against data in the literature but my data points are all out by a constant factor, so I am assuming I have made some error in the non-dimensionalisation of the problem, or rather, in the implementation of the non-dimensional problem when using a commercial CFD code.

The original eqns were non-dimensionalised using the same reference scales as the author to which I am comparing results, and I get the same non-dimensional form of the eqns.

Original NS: (&rho;_&alpha;)*(&part;u/&part;t + (u.V)u) = V[-pI + (&mu;_&alpha;)*(Vu + (Vu)^t)] - (&rho;_op)*(&beta;_&alpha;)*g*(T-T_op)

Non_Dimensional NS: (&rho;_&alpha;/&rho;_oil)*(&part;u'/&part;t' + (u'.V')u') = V'[-p'I +(&mu;_&alpha;/&mu;_oil)* (V'u' + (V'u')^T)] - (&rho;_&alpha;/&rho;_oil)*(&beta;_&alpha;/&beta;_oil)*(Ra/Pr)*T'z

Original Energy: (&rho;_&alpha;)*(Cp_&alpha;)*(&part;T/&part;t + u.VT) = (k_&alpha;)*(V^2)T

Non_Dimensional Energy: (&rho;_&alpha;/&rho;_oil)*(Cp_&alpha;/Cp_oil)*Pr*(&part;T'/&part;t' + u'.V'T') = (k_&alpha;/k_oil)*(V'^2)T'

The important boundary condition (BC) is at the liquid-vapor interface, where the shear stress is related to surface tension effects.

Dimensional BC: -&mu;(&part;u_tang/&part;n) = (&part;&sigma;/&part;T)*(&part;T/&part;s)

Non-Dimensional BC: (&part;u_tang'/&part;n') = (Ma/Pr)*(&part;T'/&part;s')

Apologies for the appearance of the eqns. The prim symbol denotes dimensionless quantities. &rho;,&mu;,Cp,k,&beta; refer to density, dynamic viscosity, specific heat, thermal conductivity and volume expansion coefficient respectively. u is the velocity, and V is the differential operator. g is gravitational acceleration, T is temperature, and Ra, Pr, Ma are the Rayleigh, Prandtl and Marangoni numbers. &part;&sigma;/&part;T is the tempeture derivative of surface tension. The subscript alpha refers to the reference fluid, so if the reference fluid is chosen to be the oil (as properties of liquid generally much greater in magnitude to properties of vapor/air), should all those ratios in the non-dimensional form should go to unity?

I have tried to construct this problem using both FLUENT and COMSOL codes, but with no success so far. In both cases, I have non-dimensionlised the original geometry (0.5mm radius bubble, 10mm liquid layer depth) using the reference length scale, the radius. Hence I have a dimensionless geometry with bubble radius of 1 and liquid layer depth 20. I must set the following parameters: density, dynamic viscosity, specific heat, thermal conductivity, volume expansion coefficient, gravitational acceleration, and temperature derivative of surface tension.

My problem is what values do I give these properties??? I am also confused by the appearance of the Pr term in my non-dimensional energy eqn. Any help will be greatly appreciated. Thanks you.

 pankaj saha November 30, 2007 16:30

First, be sure whethere the commercial software solver is based on dimension or non-dimensional....i hope Fluent is a dimensional software..it cannot take values non-dimensonally....just check....i am also not sure ...

 Seamus O'Shaughnessy November 30, 2007 19:00