# Node MRSDragNode

Defined in `mime.nodes.hydro.mrs._node.MRSDragNode`.

Algorithm `MIME-NODE-108`.

The resistance of a rigid body in Stokes flow by the method of regularised Stokeslets on its hull kind's samples, with a wall's correction added.

## Governing equations

$$
\mathbf{R} = \mathbf{M}^{\top} (\mathbf{A} + \mathbf{G})^{-1} \mathbf{M}, \quad \mathbf{A}_{mn} = \frac{(r^{2} + 2 \epsilon^{2}) \mathbf{I} + \mathbf{r} \mathbf{r}^{\top}}{8 \pi \mu (r^{2} + \epsilon^{2})^{3/2}}, \quad \mathbf{r} = \mathbf{x}_{m} - \mathbf{x}_{n}, \quad \epsilon^{2} = b^{2} \frac{1}{N} \sum_{n} w_{n}, \quad \mathbf{x}_{n} = \mathbf{p} + R(\mathbf{q}) (\mathbf{P} \mathbf{y}_{n} + \mathbf{c})
$$

## Discretisation

none declared; a node whose order depends on its construction declares it with the `discretization_order()` hook, which is read from a built node.

## Assumptions

- Stokes flow: quasi-steady, the limit of vanishing Reynolds number
- The surface sampled by the kind's quadrature, a point force at each sample, regularised by a blob of the samples' mean spacing times b
- On a separated quadrature: a traction constant over each force point's cell, the cell the points of the kind's finer quadrature nearest it, the wall's correction taken at the force points
- A medium at rest, so the background load is zero

## Limitations

none declared

## Declared regimes (not evidenced here)

none declared

## References

[@Cortez2005], [@Smith2018], [@GallagherSmith2021]

## Symbols

| Symbol | Kind | Name | Units | Implemented by |
|---|---|---|---|---|
| $\mathbf{R}$ | state | `resistance` | N s/m; N s; N m s | `mime.nodes.hydro.mrs._node.MRSDragNode.advance` |
| $\mu$ | parameter | `viscosity` | Pa s | `mime.nodes.hydro.mrs._solve.resistance` |
| $b$ | parameter | `blob` | 1 | `mime.nodes.hydro.mrs._node.MRSDragNode.advance` |
| $\mathbf{c}$ | parameter | `part_centre` | m | `mime.nodes.hydro.mrs._node.MRSDragNode.advance` |
| $\mathbf{P}$ | parameter | `part_rotation` | 1 | `mime.nodes.hydro.mrs._node.MRSDragNode.advance` |
| $\mathbf{p}$ | input | `pose` | m; 1 | `mime.nodes.hydro.mrs._node.MRSDragNode.advance` |
| $\mathbf{q}$ | input | `pose` | m; 1 | `mime.nodes.hydro.mrs._node.MRSDragNode.advance` |

## Built by

- [`drag`](../entries/interaction/drag.md) 0.3.0, rung `mrs`
- [`drag`](../entries/interaction/drag.md) 0.4.0, rung `mrs`