Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids

Abstract

The Prandtl model is certainly the simplest and most generic microscopic model describing solid friction. It consists of a single, thermalized atom attached to a spring, which is dragged past a sinusoidal potential representing the surface energy corrugation of a counterface. While it was primarily introduced to rationalize how Coulomb’s friction law can arise from small-scale instabilities, Prandtl argued that his model also describes the shear thinning of liquids. Given its success regarding the interpretation of atomic-force-microscopy experiments, surprisingly little attention has been paid to the question how the Prandtl model relates to fluid rheology. Analyzing its Langevin and Brownian dynamics, we show that the Prandtl model produces friction–velocity relationships, which, converted to a dependence of effective (excess) viscosity on shear rate η ( γ ˙ ) , is strikingly similar to the Carreau–Yasuda (CY) relation, which is obeyed by many non-Newtonian liquids. The two dimensionless parameters in the CY relation are found to span a broad range of values. When thermal energy is small compared to the corrugation of the sinusoidal potential, the leading-order γ ˙ 2 corrections to the equilibrium viscosity only matter in the initial part of the cross-over from Stokes friction to the regime, where η obeys approximately a sublinear power law of 1 / γ ˙ .

Probing the Rheological Properties of Liquids Under Conditions of Elastohydrodynamic Lubrication Using Simulations and Machine Learning

“…These rearrangements could be dominated by shear-assisted thermal activation [62,63]. Other mechanisms such as those originating from the elasticity of the squalane molecules as represented in the Prandtl model may also explain the origins of the shear thinning behavior at high pressures [64]. New investigations utilizing different types of high-dimensional data are required to make progress toward a more precise diagnosis.…”

Contributions of Molecular Dynamics Simulations to Elastohydrodynamic Lubrication

“…The Carreau equation contains many other models as limiting cases [12], including the Eyring model [13]. While the Eyring and Carreau models predict similar shear stress (or viscosity) at low shear rates, they diverge at high shear rates [3].…”

Thermal Effects on Pull-Off Force in the Johnson–Kendall–Roberts Model

“…Additionally, in athermal case, the elastic body jumps out of contact when the spring stiffness k is less than the maximum curvature of the potential energy V ′′ max . This mechanism is analogous to that of the friction-velocity relation in the Prandtl model [14][15][16], in which a particle of mass m is dragged through a sinusoidal potential with a driving spring with velocity v. On the other hand, several experimental studies have demonstrated that the velocity can significantly affect the pull-off force [17][18][19]. In light of this fact, we expect that this model to also accurately describe the relation between pull-off force and velocity and further predict the contribution of thermal effects on this relation.…”