Solving microscopic flow problems using Stokes equations in SPH

Abstract

Starting from the Smoothed Particle Hydrodynamics method (SPH), we propose an alternative way to solve flow problems at a very low Reynolds number. The method is based on an explicit drop out of the inertial terms in the normal SPH equations, and solves the coupled system to find the velocities of the particles using the conjugate gradient method. The method will be called NSPH which refers to the non-inertial character of the equations. Whereas the time-step in standard SPH formulations for low Reynolds numbers is linearly restricted by the inverse of the viscosity and quadratically by the particle resolution, the stability of the NSPH solution benefits from a higher viscosity and is independent of the particle resolution. Since this method allows for a much higher time-step, it solves creeping flow problems with a high resolution and a long timescale up to three orders of magnitude faster than SPH. In this paper, we compare the accuracy and capabilities of the new NSPH method to canonical SPH solutions considering a number of standard problems in fluid dynamics. In addition, we show that NSPH is capable of modeling more complex physical phenomena such as the motion of a red blood cell in plasma.

Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results

“…Tanaka et al [242] and later Hosseini et al [196] coupled the SPH Navier-Stokes solver to a 2D deformable cell model to simulate fluid flow associated with red blood cells. Van Liedekerke et al [141] further extended the SPH method to solve the overdamped (Stokes) equations for a fluid, thereby drastically reducing the computational time.…”

“…27, we arrive at a linear problem described by a sparse symmetric matrix, which can be solved efficiently by a Conjugate Gradient method [140,141]. Note that the above system of equations of motion (Eq.…”

Simulation of micro-scale porous flow using Smoothed Particle Hydrodynamics

“…To integrate the system of ordinary differential equations that is generated by sph, an explicit scheme can be used with a sufficiently small time-step [15,20]. Implicit sph schemes also exist [11,7] but, for simplicity, we use an explicit scheme in this work.…”

“…To avoid summing unnecessarily many minuscule contributions from far away particles, kernels with compact support are preferred [13]. The most common are cubic and quintic B-splines that approximate the Gaussian [16,20]. Lower order splines have Fourier transforms that decrease slowly and so are vulnerable to instability from transverse modes [14].…”

Everything you always wanted to know about SDPD⋆ (⋆but were afraid to ask)

“…Also, novel splitting-implicit strategies have been introduced in SDPD for the description of the solvent which bypass the viscous stability time step condition (∆t r 2 c /ν), particularly strict under microflow conditions [13] . In the context of SPH, highly viscous flows in the Stokes approximation have been recently solved by using conjugate gradient method [101] . As mentioned, all these new approaches involve some kind of non-local effect which shows up numerically as a number of sub-time steps iterations required.…”