Is it possible to use simple potential function to construct more complex potential functions ?

If $\phi_{1}, \phi_{2}, \ldots$ are harmonic functions (they satisfy Laplace's equation, i.e. $\nabla^{2}\phi_{i} = 0$), then $\phi = \sum \alpha_{i}\phi_{i}$, where $\alpha_{i}$ are constants, are also harmonic, and is the solution for the boundary value problem provided the boundary conditions (kinematic boundary condition) are satisfied, i.e.

$$\frac{\partial \phi}{\partial n} = \frac{\partial}{\partial n... ...ha_{2}\phi_{2}+\ldots \right) = U_{n} \mbox{ \ on \ B}. \notag$$

The key is to combine known solution of the Laplace equation in such a way as to satisfy the K.B.C. (kinematic boundary condition).

The same is true for the stream function $\psi$. K.B.C.s specify the value of $\psi$ on the boundaries.

• Potential function $\phi$.

  
  

  
  
  

• Stream function $\psi$.

  
  

  
  
  

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