- #1
FrogPad
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I'm not sure what the question is asking. Any help on getting started with this would be cool.
Q: Show that [itex] u = r\cos \psi +\frac{1}{2}r^{-2}\cos \psi [/itex] satisfies [itex] \nabla^2 u [/itex] and also [itex] \frac{\partial u}{\partial r}=0 [/itex] on the unit sphere. Find the velocity field [itex] \vec v = \nabla u [/itex] for flow past the sphere.
Now we have not done problems like this yet. We have only just done background work that will be leading up to these problems by definining the laplacian\div\grad\... in cylindrical\spherical coordinates.
The first part is straight forward, show that [itex] u [/itex] satisfies [itex] \nabla^2 u [/itex]. I did that. Now the next two parts are where I am confused.
[itex] \frac{\partial u}{\partial r} = 0 [/itex] on the unit sphere? I have no idea what to do here. Here are my thoughts on it:
* It is asking for me to show that the partial derivative of [itex] u [/itex] with respect to [itex] r [/itex] equals 0, ON the unit sphere.
* But what does ON the unit sphere mean? That if we are in a space that is within the unit sphere the rate of change with respect to r is 0? Well how do I express that mathematically.
And the last part [itex] \vec v = \nabla u [/itex] for flow past the sphere. I have no idea. Any help... please.
Q: Show that [itex] u = r\cos \psi +\frac{1}{2}r^{-2}\cos \psi [/itex] satisfies [itex] \nabla^2 u [/itex] and also [itex] \frac{\partial u}{\partial r}=0 [/itex] on the unit sphere. Find the velocity field [itex] \vec v = \nabla u [/itex] for flow past the sphere.
Now we have not done problems like this yet. We have only just done background work that will be leading up to these problems by definining the laplacian\div\grad\... in cylindrical\spherical coordinates.
The first part is straight forward, show that [itex] u [/itex] satisfies [itex] \nabla^2 u [/itex]. I did that. Now the next two parts are where I am confused.
[itex] \frac{\partial u}{\partial r} = 0 [/itex] on the unit sphere? I have no idea what to do here. Here are my thoughts on it:
* It is asking for me to show that the partial derivative of [itex] u [/itex] with respect to [itex] r [/itex] equals 0, ON the unit sphere.
* But what does ON the unit sphere mean? That if we are in a space that is within the unit sphere the rate of change with respect to r is 0? Well how do I express that mathematically.
And the last part [itex] \vec v = \nabla u [/itex] for flow past the sphere. I have no idea. Any help... please.