We are interested in the numerical simulation of the sedimentation (free fall under gravity) of a ideal knot in a fluid and in particular in the properties of the final steady state movement of the knot (translation, rotation,...).
Physical experiments are made at the Laboratory of the Physics of Living Matter at the IMP-SB-EPFL institute. The results consist in measurements and movies for sedimentation of knots in a tank full of water or oil and will be used for comparison to our model.
For the steady fall of a solid in a Stokes fluid under gravity, the equations for the movement of the solid can be decoupled from those describing the fluid flow; the equations for the dynamic contain coefficients from the so-called resistance matrix obtained by computing six basic Stokes flows [1]. Approximations of this matrix have been computed for ideal knots using the “Rotne-Prager” method [2]; this method allows only the case of the free-space flow, but not in a cylinder tank like in the physical experiment..
Our first goal is to compute the resistance matrix by solving the above Stokes probem with a boundary element method and to compare to the Rotne-Prager approximation for the free space steady flow but also in a cylinder tank.
Next, we will compute the non-steady knot dynamic in the quasi-satic approximation, where the fluid reacts instantaneously to a perturbation (ie the fluid equations are time-independent.
Since some experiments take place in water with a Reynolds number of about 1500, we want to simulate the steady or evolutive fall in a Navier-Stokes fluid. Recently, it was shown that by using a semi-implicit time semi-discretization, the equations for the solid and the fluid can be also decoupled [3]. The simulation can be done with a boundary element or a finite element space discretization.
The first stage consists in simulating the sedimentation in the free space or including the tank, but in the Stokes regime. This has been done with a boundary element method. Movies of the fall can be seen here.
Next, we used a perturbation method for developping a 1D model for the free fall of filaments, eg very slender bodies [4]. This could be the first step towards the simulation of flexible knots sedimentation [5].
G. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications in Handbook of Mathematical Fluid Dynamics, Vol 1, Chap. 7, Elsevier (2002).
O. Gonzalez, A. B. A. Graf and J. H. Maddocks, Dynamics of a rigid body in a Stokes fluid, J. Fluid Mech. 519 (2004), 133-160.
Ph. Caussignac, Numerical simulation of the fall of one solid in a Navier-Stokes liquid, preprint LCVMM-IMB-EPFL (2005).
Ph. Caussignac, A 1D model for a closed rigid filament in Stokes flow, Mathematical Models and Methods in Applied Sciences 19 (2009), 911-937.
Ph. Caussignac, Some possible models for an elastic structure in a 3D Stokes flow, preprint LCVMM-IMB-EPFL (2008).
May 31, 2009