The motion of solid particles in fluids plays an important role in sedimentation, crystal growth, filtration, suspension rheology, microfluidic devices such as cell lysis method and several other natural and industrial applications. In mechanical cell-lysis method, interaction and collision between cells and glass beads in liquids are used to break cells and release DNA as a part of sample-to-answer nucleic acid analysis.
In order to study this complex problem which involves particle-cell interaction in a viscous fluid, initially the effect of cells is neglected and the particle interaction in a particulate flow is investigated. First, the theoretical analysis of dilute particulate flow is carried out. Using the method of reflections in Laplace space, the equation of motion for two particles moving in a Stokes flow is explicitly derived. The results indicate that the Basset force corresponding to the motion of two spheres is larger than the solitary-particle Basset force .
To accurately predict the behavior of particulate flows, fundamental knowledge of the mechanisms of single-particle collision is required. More specifically, the study of particle-wall collision provides deeper insight into modeling particle-laden flows when particle interaction is important. In order to elucidate the basic physics of the particle interaction and collision processes, a Distributed-Lagrange-Multiplier-Based computational method for colliding particles in a solid-fluid system is developed. The Navier-Stokes equations are directly solved and no model is used for solid phase. The collision between particles is simulated by taking into account the effect of particles roughness and the Stokes number . Comparison of the present methodology with experimental studies for the bouncing motion of a spherical particle onto a wall shows very good agreement and validates the collision strategy (figure 1). This collision strategy is extended for systems of multi-particle and general shape objects in a viscous fluid . There has been a need for a good collision scheme for particulate flow simulations. This approach can be used in engineering applications and commercial CFD codes in addition to academic use.
Figure 1 - Coefficient of restitution normalized by that for dry collision as a function of St. Present results are shown using solid circles and experimental measurements for different materials by Gondret et al. (2002) are shown using open symbols. Lubrication theory of Davis et al. (1986)(-).
Movie 1 - Collision of a sphere onto a wall.
In the above investigations, particle interaction and collision in a Newtonian fluid was considered whereas the suspensions of cells and DNA molecules exhibit both viscoelastic and shear thinning characteristics. Thus, particle-wall interaction in viscoelastic fluids is experimentally investigated. A sphere is released in a tank filled with polyethylene-oxide (PEO) mixed with water with varying concentrations up to 1.5%. The effect of Stokes and Deborah numbers on the rebound velocity when a spherical particle collides onto a wall is considered . Higher rebound occurs for higher poly(ethylene-oxide) concentration at the same stokes number (figure 2a). However, the results for the coefficient of restitution in polymeric liquids can be collapsed together with the Newtonian fluid behavior if one defines the Stokes number based on the local shear rate (figure 2b). This implies that the shear-thinning behavior of these liquids is more important than their viscoelasticity during the collision process.
Figure 2 - Collision in a polymeric liquid. Present results are shown by solid symbols. Experimental measurements for Newtonian liquids by Gondret et al. (2002) are shown by open symbols. Lubrication theory of Davis et al. (1986) (-).
Particle-particle interactions in viscoelastic fluids are dramatically different than in Newtonian fluids: particles disperse in the flow of Newtonian fluids and they aggregate in the flow of viscoelastic fluids. Our analyses based on second-order fluid model suggest that the chaining of particles in viscoelastic liquids is the result of local effects and is due to three fundamental causes: (1) a viscoelastic ``pressure'' generated by normal stresses due to shear; (2) the total time derivative of the Stokes pressure; and (3) the change in the sign of the normal stress, which is a purely extensional effect. In order to study the effect of viscoelasticity of the fluid on the particle interaction, the following problems are studied: 1) The motion of a sphere normal to a wall in a second-order fluid is investigated. The normal stress at the surface of the sphere is calculated and the viscoelastic effects on the normal stress for different separation distances are analyzed . For small separation distances, when the particle is moving away from the wall, a tensile normal stress exists at the trailing edge if the fluid is Newtonian, while for a second-order fluid a larger tensile stress is observed. When the particle is moving towards the wall, the stress is compressive at the leading edge for a Newtonian fluid whereas a large tensile stress is observed for a second-order fluid. The contribution of the second-order fluid to the overall force applied to the particle is an attractive force towards the wall in both situations. 2) The forces acting on two fixed spheres in a second-order uniform flow are investigated . For flow along the line of centers or perpendicular to it, the net force is in the direction that tends to decrease the particle separation distance. For the case of flow at arbitrary angle, unequal forces are applied to the spheres perpendicularly to the line of centers. These forces result in a change of orientation of the sedimenting spheres until the line of centers aligns with the flow direction. The results explain the experimentally observed chaining of sedimenting particles in a viscoelastic fluid.
Finally, we have developed numerical tools to understand the interaction of rigid and deformable droplets (simplified cell) in a viscous fluid. A level-Set method is used to represent the interface and surface tension. The results show that the presence of particles leads to larger droplet deformation, and a perforation in the center of the droplet which facilitates droplet breakup (figure 3). It is found that the critical Stokes number above which a perforation occurs increases linearly with the inverse of the capillary number and viscosity ratio (figure 4) .
Figure 3 - Deformation and breakup of a droplet in a particulate shear flow. a) Ca=30, Re=100. b) Ca=0.025, Re=80. A perforation in the center of droplet is generated as the particles collide.
Movie 2 - Deformation and breakup of a droplet in a particulate shear flow at Ca=30, Re=100.
Figure 4 - The critical Stokes number above which a perforation occurs linearly changes with the inverse of the capillary number and the viscosity ratio.
Funded by NSF grant No. CBET-0828104