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## Analytical solution to specific Stockes first problem PDE

June 12, 2017 compiled on — Monday June 12, 2017 at 01:18 AM

Solve Initial conditions Boundary conditions Let (2)

where is steady state solution that only needs to satisfy boundary conditions and satisfies the PDE itself but with homogenous B.C.  At steady state, the PDE becomes The solution is .  Hence (2) becomes Substituting the above in (1) gives With boundary conditions . This is now in standard form and separation of variables can be used to solve it. Now acts as a source term. The eigenfunctions are known to be where . Hence by eigenfunction expansion, the solution to (3) is (3A)

Substituting this into (3) gives (4)

Expanding using same basis (eigenfunctions) gives Applying orthogonality But since and the above simplifies to But , hence Therefore and (4) becomes This is an ODE in whose solution is From (3A) now becomes (5)

To find , from initial conditions, at the above becomes Hence Therefore (5) becomes And since then the solution is To simulate

Here is the animation from the above Here is the numerical solution to compare with

Here is the animation from the above 