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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… and… – 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

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and gets nice vertex flows:

enter image description here

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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:

enter image description here

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

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

enter image description here

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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

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