# Boundary conditions for fluid flow

## Main Question or Discussion Point

What are the general boundary conditions for nonviscous, incompressible fluid flow? I am trying to find the velocity of fluid at the surface of a sphere with the incident fluid having uniform velocity. I am surprised to find in the solution that the radial velocity at the surface does not vanish. For magnetostatics, div B = 0 implies B-perp is continuous. Would not div v= 0 imply the same for v-perp?
What about the same problem but incident upon an infinite plane? Would the velocity not vanish at the surface?

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Andy Resnick

From Fluid Mechanics by Frank M. White

$$\psi = -\frac{1}{2} U r^2 sin^2 \theta + \frac{\lambda}{r} sin^2\theta$$

$$\psi = 0 => r = a = (\frac{2\lambda}{U})^{1/3}$$

$$v_r = -\frac{1}{r^2 sin\theta} \frac{\partial \psi}{\partial \theta}$$

$$v_r = U cos\theta (1 - \frac{a^3}{r^3})$$

so the radial component does appear to vanish at the surface $$r = a$$