A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales

Abstract

Abstract. In this work it is shown how the immersed boundary method of (64) for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic (τ −3/2 ) decay for long times (6; 16; 20; 23; 37; 38; 54; 67; 78). A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method.

Simulation of Osmotic Swelling by the Stochastic Immersed Boundary Method

“…We consider physical systems where the Reynolds number is rather small allowing us to neglect the non-linear advection term in the material derivative. However, given the rapid local fluctuations we retain the time derivative term as in the prior works [22,25,42]. This leads us to the stochastic time-dependent, incompressible Stokes equations: ρ ∂u ∂t (x, t) + ∇p (x, t) = µ∆u (x, t) + f total (x, t) , (2.1)…”

“…The mobility M gives the steady-state velocity V in response to an applied force F by V = M F. The operators Γ, Λ have been shown to be related to an immersed boundary particle's mobility by M = ΓL −1 Λ. The L −1 denotes the solution operator for the fluid velocity u of the steady-state incompressible Stokes equations with force density f = ΛF, see [22,25]. An important result is that the particle mobility satisfies V = M F = Γu for the force density f .…”

“…Similar to the mobility calculations, to obtain a finite diffusivity for the particles and membrane we introduce the kernel function δ c and length-scale c, which for isolated particles is closely related to the particle's hydrodynamic radius. More discussion can be found in [22,25].…”

“…In Section 2, we describe the details of our SIBM model and equations of motion for fluid-structure interactions subject to thermal fluctuations based on [22,25]. We then formulate our model for the mechanics of our elastic membrane taking into account the surface tension, a neo-Hookean shear resistance [34], and Helfrich bending rigidity [23].…”

Interactive 3D simulation for fluid–structure interactions using dual coupled GPUs

“…Within the IB method, there too exists different versions. Examples include the original versions [4], the vortex-method version [29], the volume-conserved version [30], the adaptive mesh refinement version [31], the (formally) second-order versions [33,34], the multigrid version [35], the penalty version [36], the implicit versions [37-39, 42, 43], the generalized version for a thick rod [44], the stochastic version [45], the porous media version, the lattice-Boltzmann version [3,48,49,60,61,[53][54][55][56]59,63], the fluid-solute-structure interaction version [50], and the variable viscosity version [52].…”