Diffusion in a sphere of radius a on the basis of spherical coordinates. Develop the solution for concentration as a function of radial distance and time in terms of the spherical Bessel functions.
The correct answer and explanation is :
The diffusion equation in spherical coordinates for a spherically symmetric concentration ( C(r,t) ), where ( r ) is the radial distance and ( t ) is time, can be written as:
[
\frac{\partial C(r,t)}{\partial t} = D \left( \frac{\partial^2 C(r,t)}{\partial r^2} + \frac{2}{r} \frac{\partial C(r,t)}{\partial r} \right)
]
where ( D ) is the diffusion coefficient.
To solve this, we use separation of variables by assuming a solution of the form:
[
C(r,t) = R(r)T(t)
]
Substituting this into the diffusion equation, we get:
[
T'(t)R(r) = D \left( R”(r)T(t) + \frac{2}{r} R'(r)T(t) \right)
]
Dividing both sides by ( D R(r) T(t) ), we obtain:
[
\frac{T'(t)}{D T(t)} = \frac{R”(r)}{R(r)} + \frac{2}{r} \frac{R'(r)}{R(r)}
]
Since the left side depends only on ( t ) and the right side only on ( r ), both must equal a constant, say ( -\lambda^2 ).
Now, solving for ( R(r) ), we get the ordinary differential equation:
[
r^2 R”(r) + 2r R'(r) + \lambda^2 r^2 R(r) = 0
]
This is a standard form of the spherical Bessel differential equation, with general solutions for ( R(r) ) being the spherical Bessel functions of the first and second kinds, ( J_\lambda(r) ) and ( Y_\lambda(r) ), respectively. For a physical problem with boundary conditions (e.g., ( C(a,t) = 0 ) for a sphere of radius ( a )), we choose the solution that satisfies the boundary conditions, which generally leads to using only the spherical Bessel function of the first kind, ( J_\lambda(r) ).
The time dependence ( T(t) ) leads to the solution:
[
T(t) = e^{-D\lambda^2 t}
]
Thus, the general solution for the concentration is:
[
C(r,t) = \sum_{n=1}^{\infty} A_n J_{\lambda_n}(r) e^{-D \lambda_n^2 t}
]
where ( \lambda_n ) are the roots of the equation ( J_\lambda(a) = 0 ), which ensures the boundary condition at ( r = a ) is satisfied.
The spherical Bessel functions of the first kind are crucial in solving diffusion problems in spherical geometries, especially with specific boundary conditions.
Here is the plot showing the spherical Bessel functions ( J_n(r) ) for different modes ( n = 1, 2, 3 ). The radial distance ( r ) is on the x-axis, and the corresponding function values are on the y-axis. This visual representation helps in understanding how these functions behave, which is important when solving diffusion problems in spherical geometries.