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I am trying to calculate speed of viscous fluid in a rotating hollow sphere. The sphere is rotating round z axis. The fluid is incompressible. I used Navier-Stockes equations. The only boundry condition I have, is rotational speed of sphere. My code:

v = {vr[r, ϑ, ϕ], vt[r, ϑ, ϕ], vf[r, ϑ, ϕ]}

eq =  Curl[Cross[v, Curl[v, {r, ϑ, ϕ}, "Spherical"]], {r, ϑ, ϕ}, "Spherical"] + 
      Curl[Laplacian[v, {r, ϑ, ϕ}, "Spherical"], {r, ϑ, ϕ}, "Spherical"]

NDSolve[{eq[[1]] == 0, eq[[2]] == 0, eq[[3]] == 0, vf[1, ϑ, ϕ] == Sin[ϑ], 
vr[1, ϑ, ϕ] == 0, vt[1, ϑ, ϕ] == 0}, {vr, vt, vf}, {r, 0, 1}, {ϑ, 0, Pi}, {ϕ, 0, 2*Pi}]

Mathematica displays an error:

NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable. >>

I have no idea what is wrong, so if anyone can help me, I'll be very happy. Thanks.

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closed as unclear what you're asking by MarcoB, C. E., march, m_goldberg, halirutan Sep 30 '15 at 6:29

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

You should post actual Mathematica code, as it will help diagnose the problem. It looks like you have a syntax error with vf and sin, which should use brackets instead of parentheses (also mind capitalization). – bobthechemist Sep 28 '13 at 16:10
Your problem is not particularly well posed. First of all, you should set boundary conditions on vr and vt. Secondly, you need to be clear what type of problem you're dealing with, as NDSolve is only designed to work with initial-value types of problem - as the documentation on NDSolve and your NDSolve::ivone error makes clear. – Emilio Pisanty Sep 28 '13 at 18:21