# Viscous fluid flowing around obstacle: would it deflect earlier?

Consider a viscous fluid, flowing linearly (say, with velocity $\vec u = [1,0]$ everywhere). Then an obstacle is put in the flow. Would a highly viscous fluid start deflecting around the obstacle earlier (more up-stream) than one with low viscosity?

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The dominant cause behind the fluid deflecting before reaching an obstacle is sound wave propagation. One might guess then that there is little, if any, change with the viscosity of the fluid. In fact, for high Reynolds numbers that is indeed the case. For low Reynolds numbers I can't say that I know what will happen. The question really needs to be asked with regard to both Mach number and Reynolds number. – SimpleLikeAnEgg Feb 28 '14 at 18:45
For stokes-flow ($Re << 1$, $M << 1$) there is an analytic solution for flow over a sphere (see "Viscous Fluid Flow" by White). It turns out that the solution is entirely independent of the fluid's viscosity. I think you should rephrase your question to indicate fixed, moderate, Mach numbers and moderate Reynolds numbers. That is probably the range where what you are asking can occur. My intuition as to the how the deflection varies with Reynolds number isn't quite able to answer your question though. – SimpleLikeAnEgg Feb 28 '14 at 19:13

Your intuition is effectively correct. Basic corroboration can be found in the book Viscous fluid flow, second edition by Frank M. White. In section 3-9.2 he exams low Mach and Reynolds number flow, known as Stokes flow, over a sphere. Surprisingly, the analytic solution for the problem turns out to be independent of the fluid's viscosity. He then draws some comparison to the high Reynolds number, potential flow, solution to flow over a sphere:

When we compare streamlines past a fixed sphere, the two are superficially similar, except that the Stokes streamlines are displaced further by the body.

This statement essentially provides verification that as Reynolds number decreases, or viscosity increases, that it can be expected that streamlines are displaced further from an obstacle.

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If I read your answer correctly, the analytic solution for both high- and low-viscosity fluids is the same, but still the streamlines for high-viscosity fluids are displaced further? Isn't that a contradiction? – Supernormal Mar 2 '14 at 20:49
@Supernormal No, the solutions are not the same. The author describes them as "superficially similar" in that they share somewhat similar forms, but they are different. – SimpleLikeAnEgg Mar 4 '14 at 20:43