Transport and clogging dynamics of flexible rods in pore constrictions

The transport and clogging behavior of flexible particles in confined flows is a complex interplay between elastic and hydrodynamic forces and wall interactions. While the motion of non-spherical particles in unbounded flows is well understood, their behavior in confined spaces remains less explored. This study introduces a coupled computational fluid dynamics-discrete element method (CFD-DEM) approach to investigate the transport and clogging dynamics of flexible rod-shaped particles in confined pore constrictions. The spatio-temporal analysis reveals the influence of the rod's initial conditions and flexibility on its transport dynamics through a pore constriction. The simulation results demonstrate an increase in the lateral drift of the rod upon exiting the pore that can be scaled with channel height confinement. The clogging dynamics are explored based on hydrodynamic and mechanical forces, unveiling conditions for mechanical clogging through sieving. The developed method allows for the deconvolution of the forces that contribute to particle trajectories in confined flow, which is highly relevant in particle separation processes, fibrous-shaped virus filtration, biological flows, and related applications. The method is embedded into the open-source CFDEM framework, facilitating future extensions to explore multiple particle dynamics, intermolecular forces, external influences, and complex geometries.


Fluid flow calculation
Fluid flow in the fluid domain Ω f is solved using the Navier-Stokes equation for an incompressible fluid (1) and the continuity equation (2): Here, ρ f , u f , and µ are the density, the velocity, and the viscosity of the fluid, respectively, and p denotes the pressure.The coupling of the fluid phase with the particulate phase is realized with the following boundary conditions: Equation ( 3) is responsible for transferring the particle's velocity u p to the fluid velocity field u f inside the domain of the particle Ω s .The interface condition in equation ( 4) describes the fluid stress acting on the particle's interface Γ s , where σ is the stress tensor of the fluid field, n is the outer normal vector, and t Γ s denotes the traction vector of the fluid acting on the particle.Integration over the particle's interface Γ s is performed to transfer equation ( 4) into a force term F f that considers pressure gradient force and viscous forces of the fluid.In our case, other fluid forces, such as Basset, Saffman, and Magnus forces and gravity effects, are small and therefore neglected.

Bonded particle model
For the representation of a flexible rod, we use a multi-sphere model with virtual elastic bonds for connection that was developed by Guo et al. 1 and Schramm et al. 2 .The bond forces are calculated incrementally as follows: F b n and F b t are the normal and tangential forces of the virtual elastic bonds connecting the individual spheres.and  v r t denote the normal and tangential bending stiffness, the normal and tangential (shear) deformation, and the normal and tangential relative velocity between two bonded spheres, respectively.The Young's modulus E b and the shear modulus G b of the bond are correlated through the Poisson's ratio ν : The bond is assumed to be cylindrical with a radius r b , a length l b , and a cross-sectional area A = πr 2 b .Here, the bond radius r b is assumed to be the same as the radius of an individual sphere r p , and the length l b is the distance between the centers of two connected spheres, which is twice the radius 2r p .To account for energy dissipation in the elastic wave propagation through the bonds, a normal and tangential damping force F b damp,n and F b damp,t are added: where β damp is the bond damping coefficient, and m p is the mass of an individual sphere.In the DEM framework, the trajectory of each sub-sphere within the multi-sphere rod is calculated individually based on the sum of the forces acting on it: a Chemical Process Engineering AVT.CVT, RWTH Aachen University, Forckenbeckstraße 51, 52074 Aachen, Germany.E-mail: manuscripts.cvt@avt.rwth-aachen.deb DWI -Leibniz Institute for Interactive Materials e.V., Forckenbeckstraße 50, 52074 Aachen, Germany.
where F n and F t denote the normal and tangential particle-particle and particle-wall contact forces and are calculated with the Hertz contact model 3 .F b sums up possible body forces, such as gravity, which is neglected in this work.F f is the particle-fluid interaction force.In addition to the Hertz model, we consider rolling friction and tangential sliding.Rolling friction arises from the small-scale non-sphericity of the particles and is taken into account by using a constant directional torque (CDT) model.A history spring model is chosen to account for the resistance from tangential sliding. 4The calculation of the moments acting on the spheres and bonds is performed analogously to the force calculation.

Stability criteria
In this work, four numerical stability criteria were used to determine the time steps of CFD and DEM and the coupling interval.For the CFD part, the Courant-Friedlich-Lewy (CFL) condition reads 5 : For the DEM part, the time step is taken as a fraction of 15 % of the Rayleigh time ∆t R , which is defined by: where d p and ρ p are the diameter and density of the particle, G denotes the shear modulus and ν is the Poisson ratio.
The coupling interval CI resulting from the stability criteria for the CFD and DEM timestep was checked to fulfill 6 : where ε f , ρ f , and u are the void fraction, the density, and the velocity vector of the fluid.ρ p and v i are the density and the velocity vector of the particle.C d denotes the fluid drag force and is simplified to Stokes flow with C D = 24 Re p .

Experimental observations: Materials and methods
The microfluidic chips for the validation experiments were fabricated as described elsewhere 7 .Briefly, a casting mold for channels with a width of 200µm and a height of 20µm was printed with a direct laser writing setup, the Nanoscribe Professional GT+ (Nanoscribe GmbH, Germany).The molds were replicated via soft lithography with Polydimethylsiloxane (Sylgard 184, Mavom GmbH, Germany) at a 10:1 ratio of monomer to crosslinker.The microfluidic chips were bonded to cover slips and coated with an MTS:TEOS coating 8 .
For the experiments, the channels were filled with the photoresist IP-L 780 Photoresist (Nanoscribe GmbH, Germany) and loaded to the Nansocribe.The pore constriction was created using in-chip direct laser writing, as described by Lüken et al. 7 .After printing of the pore constriction, tubing was filled with the photoresist IP-L 780 and connected to the chip.Hydrostatic pressure was used to induce a controlled flow of resin through the pore.Particles were fabricated in-situ at the desired location and orientation.Each particle consists of an individual laser trajectory written with a laser power of 50 mW and a scanspeed of 25000µm.