Theory of rheology in confinement

The viscosity of fluids is generally understood in terms of kinetic mechanisms, i.e., particle collisions, or thermodynamic ones as imposed through structural distortions upon, e.g., applying shear. Often the latter are more relevant, which allows a simpler theoretical description, and, e.g., (damped) Brownian particles can be considered good fluid model systems. We formulate a general theoretical approach for rheology in confinement, based on microscopic equations of motion and classical density functional theory. Specifically, we discuss the viscosity for the case of two parallel walls in relative motion as a function of the wall-to-wall distance, analyzing its relation to the slip length found for a single wall. The previously observed [A. A. Aerov and M. Krüger, J. Chem. Phys. 140, 094701 (2014).] deficiency of inhomogeneous (unphysical) stresses under naive application of shear in confinement is healed when hydrodynamic interactions are included.

Figure 1

Left: Suspension consisting of non-Brownian particles [nBPs; grey, solid arrows] that are controlled from outside and Brownian particles [BPs; blue, dashed arrows]. The goal of this article is to compute the (friction) forces acting on the nBPs, which are a measure for the viscosity of the suspension. In Sec. 2, we give the general Smoluchowski equation for the BPs as well as the forces acting on the nBPs, valid for any shape of the involved particles [Eqs. (8) and (9)]. Right: The specific example case studied in Sec. 4: a suspension of spherical BPs sheared between two walls (where the nBPs play the role of the walls).

Figure 2

(b) Effective viscosity coefficient $\mathrm{\Delta }{\nu }_{\mathrm{eff}}={\nu }_{\mathrm{eff}}-{\nu }_0$ of a suspension sheared between walls, as a function of the distance $d$, normalized to the bulk value. The packing fraction $\mathrm{\Phi }=0.45$, and we consider particles with a small hydrodynamic radius $a_H/2$ and hard interaction radius $a/2, a_H\ll a$. (a) Corresponding equilibrium densities for two exemplary cases, $d=10a$ and $d=3a$ [54]. Dashed vertical lines show the closest approach for particle centers.

Figure 3

(b) Effective viscosity of a hard-sphere fluid ($\mathrm{\Phi }=0.45, a_H=a$) sheared between two walls as a function of the distance $d$ for $\mathrm{Pe}=0$, 6.27, 10.95, and 20 (solid curves; top to bottom). Dashed curves are estimates based on the slip effect, Eq. (28). (I) Examples of density profiles under shear. (II) Enlarged segment of the $\mathrm{Pe}=20$ curve, where discontinuities as a function of $d$ develop. (III) Computer simulation results of Ref. [48] fitted by the curve for $\mathrm{Pe}=0$ assuming a different (negative) slip length (see the text). (a) Corresponding velocity profiles for $d=3a$ and $d=10a$. Curves give the deviation of the flow velocity $V\left(y\right)$ from the flow profile approached for large $d$, denoted $V_{\mathrm{as}}\left(y\right)$, normalized by the velocity of the upper wall $v$. Inset: Construction leading to the estimate of Eq. (28).