Hydrodynamic Friction of a Filament Resistive Force Theory

Hydrodynamic Friction of a Filament Resistive Force Theory 

Dynamic equations for the elastic filament need a proper account of the frictional forces with the surrounding fluid. Within resistive force theory, they are proportional to the local velocities of the filaments. Furthermore, inthelow Reynolds number limit they have to balance the bending and stretching forces introduced in Eqs. (9.5) and (9.9). Using the two friction coefficients per unit length, Yy and Yj., one thus arrives at the highly nonlinear dynamic equations  

The dyadic product t t projects the velocity dr/dt on its component parallel to t [ t t)a = tt a] and the projector 1—f f extracts the component perpendicular to t. The dynamic Eqs. (9.13) have to be supplemented by the inextensibility constraint dr/ds = 1. Assuming free boundary conditions for the ends of the filament, which means that in the surface terms of Eqs. (9.4) and (9.8) the variations br and 6dr/ds are arbitrary, results in the conditions 

The first and second conditions mean that the respective external forces and torques at the free ends of the filament vanish. 

To characterize the elastohydrodynamic properties of an elastic filament of length L, we define the characteristic number 

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