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I have a system with 2 different fluids, and I want to write a friction/viscosity term between them in their Navier-Stokes equation. Should it be in the form $M(v_{1}-v_{2})$ with $M$ a friction coefficient or something else ?

Thx in advance

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  • $\begingroup$ Are you suggesting that there is slip between the two fluids at their interface? $\endgroup$ – Deep May 18 '18 at 6:48
  • $\begingroup$ I'm in a case where the two fluids are mixed. $\endgroup$ – J.A May 18 '18 at 7:52
  • $\begingroup$ Actually, the form proposed above is better like like $M(c_1 v_1 +c_2 v_2)$ with $c_i$ the concentration of i. What do u think about it ? $\endgroup$ – J.A May 18 '18 at 7:54
  • $\begingroup$ Your question isn't clear to me. What do you mean by " friction/viscosity term between them"? N-S equation must be written separately for each fluid (fluid properties appearing in the N-S equation will differ for each fluid), which already includes a viscous term $\mu\nabla^2\mathbf{u}$. Then you must write the appropriate boundary conditions at the interface between fluids. $\endgroup$ – Deep May 18 '18 at 10:15
  • $\begingroup$ The two fluids are mixed, and there is no clear interface between them. So I assume apart from the viscosity term that you wrote, there is another one for the force that one fluid applies on the other. For example how would you write Navier-Stokes for ink in water ? $\endgroup$ – J.A May 18 '18 at 10:53

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