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 0.3.0, rung mrs

  • drag 0.4.0, rung mrs