I'm trying to simulate Stokes flow in 2D around an arbitrary polygon (representing a rigid body). I'd like to get both the effect of the body on the flow velocity and the forces on the body by the fluid. Basically, like how panel methods are used to examine inviscid flow around airfoils, I'm trying to examine "airfoils" in a Stokes flow.

If I'm reading it right, in The Method of Regularized Stokeslets, section 2.2, it suggests doing this by discretizing the surface of the body and solving for the "external forcing" given the velocities of the fluid at the surface of the body.

That is, the velocity of a Stokes flow at $\vec{x}$ given point forces $\vec{f_k}$ acting at $\vec{x_k}$ is:

$$u(x) = \sum_{k=1}^N \frac{-\vec{f_k}}{4 \pi \mu} \ln|r| + \vec{f_k} \cdot (\vec{x} - \vec{x_k}) \frac{\vec{x} - \vec{x_k}}{4 \pi \mu |r|^2}$$

With $r = |\vec{x} - \vec{x_k}|$. The paper suggests inverting that to find the forces given the velocities. However, I don't entirely understand what is meant by "external forcing" in the case of flow around a body. Is it the same as the fluid's force on the body (equal but opposite)? If so, it doesn't seem to match the pressure forces, given that the paper says the pressure is given by

$$p(x) = \sum_{k=1}^N \frac{\vec{f_k} \cdot (\vec{x} - \vec{x_k})}{2 \pi (\vec{x} - \vec{x_k})^2}$$

Am I misunderstanding something?


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