- #1
member 428835
Hi PF!
I'm calculating the added mass of a sphere accelerating in a fluid, which I found here: http://web.mit.edu/2.016/www/handouts/Added_Mass_Derivation_050916.pdf
My thought process was slightly different from theirs, but I am not getting the same answer. My thoughts are to take the surface normal element of a sphere, in this case ##\vec{dS} = R^2 \sin \theta \, d\theta \, d\phi \hat{r}## and dot this with the direction of the sphere, say in the direction of the zenith angle, ##\hat{y}##. We know ##\hat{r} = \sin \theta \sin \phi \hat{x} + \sin \theta \sin \phi \hat{y} + r \cos \theta \hat{z}##, which implies ##\hat{r} \cdot \hat{y} = \sin \phi \sin \theta##, which means my surface element in the direction of motion would be ##R^2 \sin^2 \theta \sin \phi d\theta d\phi##. now if we integrate ##\phi## from ##[0,2\pi]## the ##\sin \phi## term takes this to zero. Even if it did give me ##2 \pi## like in the link, I still have an extra sine and lack a cosine.
Any ideas on how to amend my approach, and also why it's not working?
Thanks so much!
I'm calculating the added mass of a sphere accelerating in a fluid, which I found here: http://web.mit.edu/2.016/www/handouts/Added_Mass_Derivation_050916.pdf
My thought process was slightly different from theirs, but I am not getting the same answer. My thoughts are to take the surface normal element of a sphere, in this case ##\vec{dS} = R^2 \sin \theta \, d\theta \, d\phi \hat{r}## and dot this with the direction of the sphere, say in the direction of the zenith angle, ##\hat{y}##. We know ##\hat{r} = \sin \theta \sin \phi \hat{x} + \sin \theta \sin \phi \hat{y} + r \cos \theta \hat{z}##, which implies ##\hat{r} \cdot \hat{y} = \sin \phi \sin \theta##, which means my surface element in the direction of motion would be ##R^2 \sin^2 \theta \sin \phi d\theta d\phi##. now if we integrate ##\phi## from ##[0,2\pi]## the ##\sin \phi## term takes this to zero. Even if it did give me ##2 \pi## like in the link, I still have an extra sine and lack a cosine.
Any ideas on how to amend my approach, and also why it's not working?
Thanks so much!