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Seamus November 21, 2007 10:37

Dimensionless Analysis
I am trying to solve a problem involving Marangoni convection. 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.

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_sub_alpha)*(dV/dt + (V.grad)V) = grad(-pI + mu_sub_alpha(gradV+ (gradV)^t)) - (rho_op_sub_alpha)*(beta_sub_alpha)*(T-T_op)*g

Non_Dimensional NS: (rho_sub_alpha)/(rho_sub_oil)*(dV'/dt' + (V'.grad')V') = grad'(-p'I + (mu_sub_alpha/mu_sub_oil)(grad'V'+ (grad'V')^t)) - (rho_op_sub_alpha/rho_sub_oil)*(beta_sub_alpha/beta_sub_oil)*(Ra/Pr)*grad'T'z

Original Energy: (rho_sub_alpha)*(Cp_sub_alpha)*(dT/dt + V.gradT) = (k_sub_alpha)*(grad^2)T

Non_Dimensional Energy: (rho_sub_alpha/rho_sub_oil)*(Cp_sub_alpha/Cp_sub_oil)*Pr*(dT'/dt' + V'.grad'T') = (k_sub_alpha/k_sub_oil)*(grad'^2)T'

Apologies for the appearance of the eqns. The subscript alpha refers to the reference fluid, so if the refence fluid is chosen to be the oil, all those ratios should go to unity, correct? Does that then mean that my density, viscosity, specific heat, thermal conductivity and volume expansion coefficient should all be set to 1 in my dimensionless model? Following on from that, it would appear that g = (Ra/Pr). However, I do not understand what the Pr term is doing to my solution in the dimensionless energy equation!!!

lyes43 February 28, 2011 13:18

contact me via i will tel u somethings about dimensionless equations

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