# How to solve fluid flow problem based on Navier-Stokes equations?

Does anyone know or can provide any examples how fluid flow problem can be formulated and solved in Wolfram Language? Simplest cases of 1D or 2D flows based on Navier-Stokes equations or even their linearized version would be great to see.

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What have you tried? There is a simple example in the What's New in Mathematica 10, just plug into the Navier-Stokes instead of pure Stokes flow. – m0nhawk Aug 15 '14 at 7:25
Change the equation to Indicial notation and follow mathematica.stackexchange.com/questions/15307/… and mathematica.stackexchange.com/questions/14654/… – Algohi Aug 15 '14 at 7:40

This blog is a good start:

Using Mathematica to Simulate and Visualize Fluid Flow in a Box

It fully solves 2D problem of one moving boundary

and gets nice vertex flows:

There are detailed descriptions of proper equations and numerical discretization. You can generalize to 3D. I would look also in latest V10 functionality to see if anything can be used there to upgrade the methods of the blog - it was written before V10 came out. For instance, take a look at this Stokes Flow example which is a simplification of NS:

The same example is discussed in Solving Partial Differential Equations with Finite Elements tutorial

that goes a bit further and solves for pressure and other quantities:

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In routine maintenance I was preparing to delete this old closed question but I noticed your answer of general interest. Since this question is otherwise abandoned would you please consider editing it to specifically fit this answer, or reposting this to a self-Q&A? – Mr.Wizard Jul 25 '15 at 13:48
@Mr.Wizard I edited the question, also added a new link and images in the answer at the end. Hope it'll do. – Vitaliy Kaurov Jul 26 '15 at 4:47
Did you vote to reopen? – Mr.Wizard Jul 26 '15 at 10:59
@Mr.Wizard I do not see an option for this. – Vitaliy Kaurov Jul 26 '15 at 11:15
I have trouble remembering the details of site mechanics; perhaps you cannot vote to reopen since you voted to close originally. I'll just reopen directly. – Mr.Wizard Jul 26 '15 at 11:25