User:Egm6321.f12.team4.mishra/Homework7 R7.5

Problem R*7.5 – Heat conduction on a cylinder
sec40-5

Statement
Given,

$$ x= rcos\theta = \xi_1cos\xi_2 $$

$$ y= rsin\theta = \xi_1sin\xi_2$$

$$ z= \xi_3 $$

1) Express, the infinitesimal change in length, in Cartesian coordinates, in terms of Cylindrical coordinates

2) Find, $$ ds^2 = \sum_i (dx)^2 = \sum_k (h_k)^2(d\xi_k)^2) $$, the length in cylindrical coordinates

3) Find $$ \Delta u $$ in cylindrical coordinates

4) Use separation of variable to find the separated equations and compare it to the Bessel Equation.

Solution
1.) From the problem statement,


 * {| style="width:100%" border="0"

$$ x= \xi_1 cos \xi_2 $$ (7.5.1)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

Differentiating the above,


 * {| style="width:100%" border="0"

$$ dx= d\xi_1 cos \xi_2 - \xi_1 sin \xi_2 d\xi_2$$ (7.5.2)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

Similarly for $$ dy $$ & $$ dz $$ ,


 * {| style="width:100%" border="0"

$$ dy= d\xi_1 sin \xi_2 + \xi_1 cos \xi_2 d\xi_2$$ (7.5.3)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$ dz = d\xi_3 $$ (7.5.4)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

2.) Squaring Eq. (7.5.2), (7.5.3) & (7.5.4) and adding ,


 * {| style="width:100%" border="0"

$$ ds^2 = dx^2 + dy^2 + dz^2 $$ (7.5.5)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$ ds^2 = (d\xi_1 cos \xi_2 - \xi_1 sin \xi_2 d\xi_2)^2 + (d\xi_1 sin \xi_2 + \xi_1 cos \xi_2 d\xi_2)^2 + d\xi_3^2 $$ (7.5.6)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$ ds^2 = ((d\xi_1 cos \xi_2)^2 + (\xi_1 sin \xi_2 d\xi_2)^2 -2( d\xi_1 cos \xi_2\xi_1 sin \xi_2 d\xi_2)+(d\xi_1 sin \xi_2)^2 + (\xi_1 cos \xi_2 d\xi_2)^2 + 2(d\xi_1 sin \xi_2\xi_1 cos \xi_2 d\xi_2)+ d\xi_3^2 $$     (7.5.7)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

Cancelling terms we get,


 * {| style="width:100%" border="0"

$$ ds^2 = (d\xi_1)^2 + (\xi_1d\xi_2)^2 + (d\xi_3)^2 $$ (7.5.8)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

Comparing Eq. (7.5.8) with,


 * {| style="width:100%" border="0"

$$ ds^2 = (h_1)^2(d\xi_1)^2 + (h_2)^2(d\xi_2)^2 + (h_3)^2(d\xi_3)^2 $$ (7.5.9)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$ h_1 = 1 $$ (7.5.10)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$ h_2 = \xi_1 $$ (7.5.11)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$ h_3 = 1 $$ (7.5.12)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

3.) We know that the Laplace equation in general curvilinear coordinates is,


 * {| style="width:100%" border="0"

$$ \Delta u = \frac{1}{h_1 h_2 h_3} \sum^3_{i=1} \frac{\partial}{\partial \xi_i} \left [\frac{h_1 h_2 h_3}{(h_i)^2}\frac{\partial u}{\partial \xi_i} \right] $$ (7.5.13)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

Using the Eq. (7.5.10), (7.5.11), (7.5.12),


 * {| style="width:100%" border="0"

$$ \Delta u = \frac{1}{\xi_1}[\frac {\partial}{\partial \xi_1}(\xi_1 \frac{\partial u}{\partial \xi_1})+ \frac {\partial}{\partial \xi_2}(\frac{1}{\xi_1}\frac{\partial u}{\partial \xi_2})+ \frac {\partial}{\partial \xi_3}(\xi_1 \frac{\partial u}{\partial \xi_3})] $$ (7.5.13)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

which can be rewritten as,


 * {| style="width:100%" border="0"

$$ \Delta u = \frac{1}{\xi_1}\frac {\partial}{\partial \xi_1}(\xi_1 \frac{\partial u}{\partial \xi_1})+ \frac{1}{\xi_1^2}\frac {\partial}{\partial \xi_2}(\frac{\partial u}{\partial \xi_2})+ \frac{\partial^2 u}{\partial \xi_3^2}$$ (7.5.14)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

which is the Laplace equation in cylindrical coordinates.

Putting $$ \xi_1 = r $$, $$ \xi_2 = \theta $$ , $$ \xi_3 = z $$


 * {| style="width:100%" border="0"

$$ \Delta u = \frac{1}{r}\frac {\partial}{\partial r}(r \frac{\partial u}{\partial r})+ \frac{1}{r^2}(\frac{\partial^2 u}{\partial \theta^2})+ \frac{\partial^2 u}{\partial z^2}$$ (7.5.15)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

4)
 * {| style="width:100%" border="0"

$$ \Delta u = \frac{1}{r}\frac {\partial}{\partial r}(r \frac{\partial u}{\partial r})+ \frac{1}{r^2}(\frac{\partial^2 u}{\partial \theta^2})+ \frac{\partial^2 u}{\partial z^2} =0$$ (7.5.16)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

We start by assuming,


 * {| style="width:100%" border="0"

$$ u (R, \Theta,Z) = R(r) \, \Theta (\theta) \, Z(z)$$ (7.5.17)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Putting this in Eq. (7.5.16),


 * {| style="width:100%" border="0"

$$ \frac{\Theta Z}{r} \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r})+ \frac{R Z}{r^2}(\frac{\partial^2 \Theta}{\partial \theta^2}) + R \Theta \frac{\partial^2 Z}{\partial z^2}=0$$ (7.5.18)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Dividing by $$ R \Theta Z $$ and multiplying by $$ r^2 $$,


 * {| style="width:100%" border="0"

$$ \frac{r}{R} \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r})+ \frac{1}{\Theta}(\frac{\partial^2 \Theta}{\partial \theta^2}) + \frac{r^2}{Z} \frac{\partial^2 Z}{\partial z^2}=0 $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

which can be rewritten as ,


 * {| style="width:100%" border="0"

$$ \frac{r}{R} \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r}) + \frac{r^2}{Z} \frac{\partial^2 Z}{\partial z^2}= - \frac{1}{\Theta}(\frac{\partial^2 \Theta}{\partial \theta^2}) $$ (7.5.19)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Since L.H.S is a function of r and z, and RHS is a function of \Theta only, for them to be equal, both should be equal to a constant k.


 * {| style="width:100%" border="0"

$$ \frac{\partial^2 \Theta}{\partial \theta^2} = - k \Theta $$ (7.5.20)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$ \frac{r}{R} \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r}) + \frac{r^2}{Z} \frac{\partial^2 Z}{\partial z^2} = k $$ (7.5.21)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Dividing Eq. (7.5.21) by $$ r^2 $$ and rearranging,


 * {| style="width:100%" border="0"

$$ \frac{1}{rR} \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r})- \frac{k}{r^2} = - \frac{1}{Z} \frac{\partial^2 Z}{\partial z^2} = c $$ (7.5.22)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$ - \frac{1}{Z} \frac{\partial^2 Z}{\partial z^2} = c $$ (7.5.23)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$ \frac{1}{rR} \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r})- \frac{k}{r^2} = c $$ (7.5.24)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Eq. (7.5.24) can be written as,


 * {| style="width:100%" border="0"

$$ r \frac {\partial}{\partial r}(r \frac{\partial R}{\partial r})+ (cr^2 - k)R =0 $$ (7.5.25)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

Differentiating by parts,


 * {| style="width:100%" border="0"

$$ r^2 \frac{\partial^2 R}{\partial r^2} + r \frac{\partial R}{\partial r}+ (cr^2 - k)R =0 $$ (7.5.26)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

which is same as the Bessel Function. The other separated equations are, Eq. (7.5.20) and Eq. (7.5.23)

Author and References

 * Solved and Typed by -- Pushkar Mishra
 * Reviewed by --