A new resistance function for two rigid spheres in a uniform compressible low-Reynolds-number flow

The pressure moment of a rigid particle is defined as the trace of the first moment of the surface stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication theory. Following the practice established in previous studies on two-body hydrodynamic interactions at low Reynolds number, the results are expressed in terms of a new (stresslet) resistance function

On the bulk viscosity of suspensions

The bulk viscosity of a suspension relates the deviation of the trace of the macroscopic or averaged stress from its equilibrium value to the average rate of expansion. For a suspension the equilibrium macroscopic stress is the sum of the fluid pressure and the osmotic pressure of the suspended particles. An average rate of expansion drives the suspension microstructure out of equilibrium and is resisted by the thermal motion of the particles. Expressions are given to compute the bulk viscosity for all concentrations and all expansion rates and shown to be completely analogous to the well-known formulae for the deviatoric macroscopic stress, which are used, for example, to compute the shear viscosity. The effect of rigid spherical particles on the bulk viscosity is determined to second order in volume fraction and to leading order in the Péclet number, which is defined as the expansion rate made dimensionless with the Brownian time scale. A repulsive hard-sphere-like interparticle force reduces the hydrodynamic interactions between particles and decreases the bulk viscosity

The Bulk Viscosity of Suspensions

Particles suspended in a fluid are known to undergo variations in the local concentration in many flow situations; essentially a compression or expansion of the particle phase. The modeling of this behavior on a macroscopic scale requires knowledge of the effective bulk viscosity of the suspension, which has not been studied before. The bulk viscosity of a pure compressible fluid is defined as the constant of proportionality that relates the difference between the mechanical pressure and the thermodynamic pressure to the rate of compression. The bulk viscosity of a suspension is defined analogous to that for a pure fluid as the constant of proportionality relating the deviation of the trace of the macroscopic stress from its equilibrium value to the average rate of compression. The compression flow drives the suspension microstructure out of equilibrium and the thermal motion of the particles tries to restore equilibrium. The Peclet number (Pe), defined as the expansion rate made dimensionless with the Brownian time-scale, governs the departure of the microstructure from equilibrium. The microstructural forcing in compression is monopolar for small Pe resulting in a significantly slower spatial and temporal response of the microstructure compared to shearing or diffusive motion. We have determined the effective suspension bulk viscosity for all concentrations and all rates of compression, accounting for the full thermodynamic and hydrodynamic interactions that particles experience at the micro-scale. Current simulation techniques were enhanced to enable the dynamic simulation of compression flows in a suspension. A 'compression thinning' of the suspension is observed at small rates of compression and there is some 'compression thickening' at large compression rates. The bulk viscosity diverges as the volume fraction nears maximum packing and is in fact larger than the shear viscosity. Existing models for multiphase flows must therefore include the bulk viscosity term to properly simulate variations in particle concentration. An understanding of bulk viscosity effects in suspensions will enable the modeling of certain aggregation and separation behavior and lead to more accurate models for multiphase flows where there are variations in the particle concentration, such as filtration or fluidization.</p