Example of the use of the principle of linear superposition.

$\phi _i (\vec{x})$ is a unit-source flow located at $\vec{x}_{i}$.

A more general potential function can be written as the linear superposition of the unit-sources, as follows:

$$\phi(\vec{x}) = \sum_{i} m_{i}\phi_{i}(\vec{x}) \notag$$

where the value of the constants $m_{i}$ are found such that:

$$\phi = \sum_{i} m_i \phi_i (\vec{x}) \mbox{ \ satisfies the KBC on the boundaries } \notag$$

Caution: $\phi(\vec{x})$ must be regular for $x \in V$ (fluid domain), so it is required that $\vec{x} \notin V$.

 

Figure: Note: $\vec{x}_{j}, j=1,\ldots,4$ are not in the fluid domain $V$.

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