- #1

happyparticle

- 421

- 20

- Homework Statement
- Consider the steady flow pattern produced when an impenetrable rigid spherical obstacle is placed in a uniformly flowing, incompressible, inviscid fluid.

- Relevant Equations
- ##r(r,\theta, \phi) = V \hat{z} , r \to\infty##

##\vec{v} = - \nabla \Phi##

I'm trying to find how the author finds the boundary condition at ##r\to\infty## is ## \Phi(r,\theta, \phi) = - V r cos \theta##.

Using the spherical coordinates.

##- V \hat{z} = \nabla \Phi##

##- V ( cos \theta \hat{r} - sin \theta \hat{\theta}) = \frac{d \Phi}{dr}\hat{r} + 1/r \frac{d \Phi}{d \theta} \hat{\theta} + \frac{1}{r sin \theta} \frac{d \Phi}{ d \phi} \hat{\phi} ##

I'm not sure to understand why most of the terms vanishes.

Using the spherical coordinates.

##- V \hat{z} = \nabla \Phi##

##- V ( cos \theta \hat{r} - sin \theta \hat{\theta}) = \frac{d \Phi}{dr}\hat{r} + 1/r \frac{d \Phi}{d \theta} \hat{\theta} + \frac{1}{r sin \theta} \frac{d \Phi}{ d \phi} \hat{\phi} ##

I'm not sure to understand why most of the terms vanishes.