Ricardo Cortez

Ricardo Cortez

We present a variation of the method of regularized Stokeslet (MRS) specialized for the case of forces and torques distributed over filaments in three dimensions. The new formulation is based on the exact solution of Stokes equation generated by a linear continuous distribution of regularized forces along a line segment. Therefore, a straight filament with linearly varying forces does not require discretization. A general filament is approximated by a piecewise linear curve in three dimensions where the length of each line segment is chosen only based on the variation of the force field and the desired accuracy of its piecewise linear approximation. The most significant advantage of this formulation is that the values of the regularization parameter $ε$ and the length of the segments $h$ are decoupled as long as $ε<h$ so that $ε$ can be selected as a proxy for the radius of the filament and $h$ is chosen to discretize the forces and torques. We analyze the performance on test problems and present biological applications of sperm motility based on existing models of swimming flagella in open space and near a plane wall. The results show, for example, that because the forces along the flagellum vary mildly, a flagellum can be approximated with as few as 11 segments of length $h$ while fixing the regularization parameter to $ε=h/30$, overcoming the need for hundreds of discretization nodes required by the MRS when $ε$ is small. The filament behaves like a slender cylindrical tube of radius $\approx 0.97ε$ so that the value of $ε$ influences the flagellum's swimming speed. For fixed regularization, doubling the number of line segments does not affect the results significantly as long as the force field is resolved. Examples that require rotlets and potential dipoles along the filament are also presented.

John LaGrone, Ricardo Cortez, Wen Yan et al.

The macroscopic properties of polymeric fluids are inherited from the material properties of the fibers embedded in the solvent. The behavior of such passive fibers in flow has been of interest in a wide range of systems, including cellular mechanics, nutrient aquisition by diatom chains in the ocean, and industrial applications such as paper manufacturing. The rotational dynamics and shape evolution of fibers in shear depends upon the slenderness of the fiber and the non-dimensional "elasto-viscous" number that measures the ratio of the fluid's viscous forces to the fiber's elastic forces. For a small elasto-viscous number, the nearly-rigid fiber rotates in the shear, but when the elasto-viscous number reaches a threshhold, buckling occurs. For even larger elasto-viscous numbers, there is a transition to a "snaking behavior" where the fiber remains aligned with the shear axis, but its ends curl in, in opposite directions. These experimentally-observed behaviors have recently been characterized computationally using slender-body theory and immersed boundary computations. However, classical experiments with nylon fibers and recent experiments with actin filaments have demonstrated that for even larger elasto-viscous numbers, multiple buckling sites and coiling can occur. Using a regularized Stokeslet framework coupled with a kernel independent fast multipole method, we present simulations that capture these complex fiber dynamics.