Momentum balance and stresses in a suspension of spherical particles in a plane Couette flow

Abstract

Non-Brownian suspension of monodisperse spherical particles, with volume fractions ranging between ϕ = 0.05 and 0.38 and particle Reynolds numbers ranging between Rep = 0.002 and 20, in plane Couette shear flows is investigated using three-dimensional particle-resolved numerical simulations. We examine the effects of volume fraction and particle Reynolds number on the macroscopic and microscopic stresses in the fluid phase. The effective viscosity of the suspension is in a good agreement with the previous empirical and experimental studies. At Rep = 20, however, the effective viscosity increases significantly compared to the lower particle Reynolds number simulations in the Stokes flow regime. Examining the stresses over the depth of the Couette gap reveals that this increase in wall shear stresses at high particle Reynolds numbers is mainly due to the significantly higher particle phase stress contributions. Next, we examine the momentum balance in the fluid and particle phase for different regimes to assess the significance of particle/particle interaction and fluid and particle inertia. At the highest particle Reynolds number and volume fraction, the particle inertia plays a dominant role in the momentum balance and the fluid inertia is non-negligible, while the short-lived contact forces are negligible compared to these effects. For all other regimes, the fluid inertia is negligible, but the particle inertia and contact forces are important in the momentum balance. Reynolds stresses originated from velocity fluctuations do not contribute significantly to the suspension stresses in any of the regimes we have studied, while the reduction in the shear-induced particle rotation can be a reason for higher wall shear stress at Rep = 20. Finally, we study the kinematics of particles, including their velocity fluctuations, rotation, and diffusion over the depth of the Couette gap. The particle diffusion coefficients in the cross flow direction exhibit an abrupt increase at Rep = 20.

Lamb’s solution and the stress moments for a sphere in Stokes flow

“…In descending order the lines show the collisional contribution n σxz (orange) defined in (103), the gradient dU x /dz of the mean velocity in the x direction (green) and the particle stresslet averaged over thin ''slices'' of the computational domain parallel to the solid boundaries (blue). Aside from the small statistical fluctuations and the steep rise near the boundaries, caused by particle accumulation near the rapidly moving planes (an effect also reported in [39]), the stresslet and velocity gradients are nearly constant in the gap so that one may define a mixture viscosity µ mix according to (see e.g. [1])…”

“…As noted after (39), the explicit expressions of all the moments contain derivatives of the Lamb solution potentials evaluated at the particle center. Thus, only the terms containing the zero power of r will survive in the sums.…”

“…In order to use the recurrence relation (39) for the integral (33) we need to evaluate the integrals with zero and one normal. From the mean value theorem for harmonic functions we immediately have…”

On the dynamics of unconfined and confined vortex rings in dense suspensions

“…The effective viscosity will thus depend on the particle Reynolds number and will increase significantly when compared with empirical relations (e.g. (2.2)) from the Stokes flow regime (Linares-Guerrero, Hunt & Zenit 2017; Rahmani, Hammouti & Wachs 2018). Based on laboratory measurements in a coaxial-cylinder rheometer, Linares-Guerrero et al (2017) observed that the effective viscosity monotonically increased with increasing particle Reynolds number for suspension volume fractions of .…”

Viscoelasticity and the dynamics of concentrated particle suspension in channel flow

“…In the presence of fluid inertia, the effective viscosity of suspensions increases. In the laminar regime, this observation is attributed to multiple effects, including the Reynolds stresses induced by particle fluctuations (Kulkarni & Morris 2008), particle and fluid phase acceleration (Rahmani, Hammouti & Wachs 2018) and anisotropy in microstructure leading to higher effective solid volume fractions (Picano et al 2013). Since viscoelasticity influences both the hydrodynamic forces experienced by particles (Becker et al 1994) and the local microstructure (D'Avino & Maffettone 2015), it is interesting to consider whether and how the increase in effective viscosity is influenced by viscoelastic effects in the carrier fluid.…”