# Partial differential equation (multidimensional, Navier–Stokes)

What is generic approach to solving this system?

$$\rho\left({\partial v_i \over\partial t}+v_k{\partial v_i \over \partial x_k} \right) = - {\partial p \over\partial x_i} + {\partial \over\partial x_k}\left\{ \mu \left({\partial v_i \over\partial x_k}+{\partial v_k \over\partial x_i} - {2\over 3}\delta_{i,k}{\partial v_l \over\partial x_l}\right) \right\}+{\partial \over\partial x_k}\left( \zeta {\partial v_l \over\partial x_l}\delta_{i,k}\right)$$

$$\vec{v}|_{\partial \Omega} = 0, \quad \vec{v}|_{t=0} = \vec{v}_{0}$$

I finished a course in differential math and PDE, but I can't even say what is the kind of the task.

The task is to be solved numerically.

PS: or can anybody say something about last two formulas? What is "\partial \Omega" ($\partial \Omega$) usually means?

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I don't suppose "run away screaming" qualifies as a generic approach? :) –  Zarrax Oct 13 '11 at 17:28
$\partial \Omega$ is the boundary of $\Omega$. –  Beni Bogosel Oct 16 '11 at 8:25

This is the Navier–Stokes for compressible fluid exactly as was recorded here http://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F_%D0%9D%D0%B0%D0%B2%D1%8C%D0%B5_%E2%80%94_%D0%A1%D1%82%D0%BE%D0%BA%D1%81%D0%B0

and the second formula seems to be "No Slip Condition", e.g. here http://scienceworld.wolfram.com/physics/NoSlipCondition.html for non-movable container. and the '\partial \omega' sounds as ... walls of the container. So, fluid at the points, just next to the walls is stopped and notmovable.

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