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Laplace equation

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Poirot

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Feb 15, 2012
250
A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature

\(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \)


Using the general solution for the Laplace equation \( u(r,\theta)=\sum A_n r^n P_n \) where the Pn are legendre polynomials, find the (axisymmetric) steady state temperature distribution \(u(r,\theta) \) within the shell.<<there was a long incomprehensible expression here which seems to have disappeared ?>>
You may assume the legrendre polynomials (first three) and that Legendre polynomials satisfy the orthogonality relation.
 
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CaptainBlack

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Jan 26, 2012
890
I'm afraid the some of your notation is incomprehensible. It is generally not possible to copy formatted mathematics into the question input box and have it render correctly.

I have changed some of your expressions to the LaTeX we use here, though some of the expressions look wrong, but there is still a large expression that makes no sense <<which seems to have disappeared not only from the post but from the edit history>>.

CB
 
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Poirot

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Feb 15, 2012
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Thanks CB. I did it in proper code and then my internet stopped working so I tried to copy and paste. I don't know what expression you are referring to but you have all you need.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,701
A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature


\(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \)


Using the general solution for the Laplace equation \( u(r,\theta)=\sum A_n r^n P_n \) where the Pn are legendre polynomials, find the (axisymmetric) steady state temperature distribution \(u(r,\theta) \) within the shell.<<there was a long incomprehensible expression here which seems to have disappeared ?>>
You may assume the legrendre polynomials (first three) and that Legendre polynomials satisfy the orthogonality relation.
The general solution for the Laplace equation should read $$u(r,\theta) = \sum A_nr^n P_n\color{red}{(\cos\theta)} = A_0P_0(\cos\theta) + A_1rP_1(\cos\theta) + A_2r^2P_2(\cos\theta) +\ldots.$$ If you substitute the formulas for the first three Legendre polynomials (which you are allowed to assume, so presumably you ought to know them), ignore the remaining terms, and put $r=a$, then you get the equation for the surface temperature in the form $A_0P_0(\cos\theta) + A_1aP_1(\cos\theta) + A_2a^2P_2(\cos\theta) = 1 + \cos\theta + 2\cos^2\theta.$ Now all you have to do is to compare coefficients of powers of $\cos\theta$ in order to determine the values of $A_0$, $A_1$ and $A_2.$
 
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Poirot

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Feb 15, 2012
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Ok I did that, substitued x=acosϕsin θ, and it didn't work out. Was I meant to do that?
 
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