# Meshless methods

Complex fluids are fluids which advect a structure. For example, liquid crystals can be modelled as a continuum of microscopic rods immersed in a Newtonian fluid. In order to alter the orientation of these rods, one must consider the velocity field and its spatial gradient. At a point, this data is known as a 1-jet of the velocity and is represented by a finite dimensional group. In [1] we have verified the computational tractability of this group. The resulting integrator satisfies a variant of the circulation theorem, and could be competitive with traditional vortex methods . This is found in work on particle methods done in [2]. Recent work is allowing these ideas to come to fruition [3], as shown in the movies below. Finally, the notion of jets can be leveraged to augment the traditional vortex blob method, and permit nontrivial dynamics below the regularization length-scale [4]

[1] L. Colombo, H.O. Jacobs. *Lagrangian mechanics on centered semi-direct products*, 13 pages, to appear in “Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden”, Fields Institute Communications Series (2015)

[2] H.O. Jacobs, T.S. Ratiu, M. Desbrun. *On the coupling between an ideal fluid and immersed particles*, *Physica D,* vol. 265, pp. 40–56 (2013)* *

[3] C.J. Cotter, D.D. Holm, H.O. Jacobs, D.M. Meier.* The jetlet hierarchy of ideal fluid dynamics*, Journal of Physics A:47 (2014) (arXiv:1402.0086)

[4] C.J. Cotter, J. Eldering, D.D. Holm, H.O. Jacobs, D.M. Meier. *Weak Dual Pairs and *Jetlet* Methods for Ideal Incompressible Fluid Models*. J. Nonlinear Science (2016)

[5] D.D. Holm, H.O. Jacobs. *Multiple Vortex Blobs: Symplectic Geometry and Dynamics*. J. Nonlinear Science (accepted Dec. 2016) (arXiv:1505.05950)