Abstract
We present a multi-scale modeling and simulation framework for low-Reynolds number hydrodynamics of shape-changing immersed objects, e.g., biological microswimmers and active surfaces. The key idea is to consider principal shape changes as generalized coordinates and define conjugate generalized hydrodynamic friction forces. Conveniently, the corresponding generalized friction coefficients can be pre-computed and subsequently reused to solve dynamic equations of motion fast. This framework extends Lagrangian mechanics of dissipative systems to active surfaces and active microswimmers, whose shape dynamics is driven by internal forces. As an application case, we predict in-phase and anti-phase synchronization in pairs of cilia for an experimentally measured cilia beat pattern.Graphic
Highlights
We present a multi-scale modeling and simulation framework for low-Reynolds number hydrodynamics of shape-changing immersed objects, e.g., biological microswimmers and active surfaces
Biological hydrodynamics Biology provides ample examples of active shape changes in fluid environments: Bacteria like E. coli rotate helical prokaryotic flagella to swim [1], other bacteria like Spiroplasma propagate twist waves along their flexible body [2], sperm cells and motile algae posses slender cell appendages termed cilia, whose regular bending waves propel these cells in a fluid [3,4]
Different computational methods of different degrees of approximation have been used in the community, including resistive force theory for slender filaments, which includes short-range, but not longrange hydrodynamic interactions [28,29,30], the more refined method of slender-body theory, which considers a line distribution of hydrodynamic singularities along a filament [31,32,33], or multi-particle collision dynamics, which replaces the continuum description of the Stokes equation by the stochastic dynamics of a large number of “fluid particles” [34,35,36]
Summary
The resultant surface distributions of hydrodynamic friction forces define generalized hydrodynamic friction coefficients by a projection method of Lagrangian mechanics [10,42,43,44,45,46,47,48] These scalar friction coefficients are independent of the velocity of the moving surface. We obtain effective equations of motion for the generalized coordinates from a force balance between these generalized friction forces and active driving forces These active driving forces coarse-grain the internal active processes that drive the active shape changes of the surface (such as the collective dynamics of molecular motors). No-slip boundary condition for an active surface We consider a surface S immersed in the fluid that changes its shape as a function of time. The total force exerted by the surface on the fluid is the surface integral of f (x)
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