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Nereus December 5, 2011 15:08

Navier-Stokes solution and tracers

I have an embarrassingly basic question to ask. As far as I understand it, solving the Navier-Stokes equations furnishes a velocity field for the considered domain subject to the imposed assumptions. If this is indeed the case, my question is, say I wanted to describe the spread of an inert tracer (for example) in that domain, do I only need to know this velocity field? Alternatively, do I also need to consider mixing coefficients or are they already incorporated in the velocity field during the solution of the Navier-Stokes equations?

Thanks in advance! :)

SergeAS December 5, 2011 18:07

For calculation of concentration field of an inert tracer, the system of Navier-Stokes equations complemented the transport equation for the inert tracer, which has the same form with the equation of continuity (+diffusion member if diffusion significant)

\frac{\partial \rho  Y }{\partial t} +
\frac{\partial}{\partial x_j}\left[ \rho Y u_j \right] - \rho D \frac{\partial^2 Y}{\partial x^2_j} = 0

where Y, - percentage of inert tracer
D - binary diffusion coefficient

The number of additional equations is equal to N-1
where N - the number of mixture components

\sum\limits_{i=1...N} {Y_i }  = 1

Nereus December 6, 2011 10:52

Thanks for the reply, SergeAS! In that equation, does the diffusion term reflect molecular diffusion or diffusive mixing? That is to say, should D be the mixing coefficient that features in the Navier-Stokes equations or the molecular diffusion coefficient? (Also, I'm a bit confused by the term "binary diffusion coefficient". Why binary?)

SergeAS December 6, 2011 11:41

Molecular diffusion and diffusion term in the Navier - Stokes equations are of the same nature.

Diffusion is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas.

In your case (single inert tracer) we have binary diffusion (In the process involves only two components)

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