Inertial effects on the rheology of a dilute emulsion

Abstract

The behaviour of an isolated nearly spherical drop in an ambient linear flow is examined analytically at small but finite Reynolds numbers, and thereby the first effects of inertia on the bulk stress in a dilute emulsion of neutrally buoyant drops are calculated. The Reynolds numbers, Re = a2ρ/μ and $\hat{\Rey} \,{=}\, \dot{\gamma}a^2\rho/\hat{\mu}$, are the relevant dimensionless measures of inertia in the continuous and disperse(drop) phases, respectively. Here, a is the drop radius, is the shear rate, ρ is the common density and and μ are, respectively, the viscosities of the drop and the suspending fluid. The assumption of nearly spherical drops implies the dominance of surface tension, and the analysis therefore corresponds to the limit of the capillary number(Ca) based on the viscosity of the suspending fluid being small but finite; in other words, Ca ≪ 1, where Ca = μa/T, T being the coefficient of interfacial tension. The bulk stress is determined to O(φRe) via two approaches. The first one is the familiar direct approach based on determining the force density associated with the disturbance velocity field on the surface of the drop; the latter is determined to O(Re) from a regular perturbation analysis. The second approach is based on a novel reciprocal theorem formulation and allows the calculation, to O(Re), of the drop stresslet, and hence the emulsion bulk stress, with knowledge of only the leading-order Stokes fields. The first approach is used to determine the bulk stress for linear flows without vortex stretching, while the reciprocal theorem approach allows one to generalize this result to any linear flow. For the case of simple shear flow, the inertial contributions to the bulk stress lead to normal stress differences(N1, N2) at O(φRe), where φ(≪1) is the volume fraction of the disperse phase. Inertia leads to negative and positive contributions, respectively, to N1 and N2 at O(φRe). The signs of the inertial contributions to the normal stress differences may be related to the O(ReCa) tilting of the drop towards the velocity gradient direction. These signs are, however, opposite to that of the normal stress differences in the creeping flow limit. The latter are O(φCa) and result from an O(Ca2) deformation of the drop acting to tilt it towards the flow axis. As a result, even a modest amount of inertia has a significant effect on the rheology of a dilute emulsion. In particular, both normal stress differences reverse sign at critical Reynolds numbers(Rec) of O(Ca) in the limit Ca ≪ 1. This criterion for the reversal in the signs of N1 and N2 is more conveniently expressed in terms of a critical Ohnesorge number(Oh) based on the viscosity of the suspending fluid, where Oh = μ/(ρaT)1/2. The critical Ohnesorge number for a sign reversal in N1 is found to be lower than that for N2, and the precise numerical value is a function of λ. In uniaxial extensional flow, the Trouton viscosity remains unaltered at O(φRe), the first effects of inertia now being restricted to O(φRe3/2). The analytical results for simple shear flow compare favourably with the recent numerical simulations of Li & Sarkar (J. Rheol., vol. 49, 2005, p. 1377).

Heterogeneous dispersions as microcontinuum fluids

“…To validate the model, we apply it to the case of viscous droplets and Navier-slip spheres. The results correct and extend several prior results for constitutive equations of rigid spheres (Einstein 1906(Einstein , 1911Prosperetti 2004;Prosperetti et al 2006;Jin et al 2018;Zhou & Prosperetti 2020) and viscous droplets (Taylor 1932;Raja, Subramanian & Koch 2010), and are shown to be consistent with experimental observations. The study is concluded with a discussion of results and future prospects.…”

Effects of inertia and viscoelasticity on sedimenting anisotropic particles

“…Further, unlike polymeric solutions, the normal stress differences for many of these systems are comparable in magnitude. Restricting ourselves to cases where analytical expressions for the normal stress differences are available: examples of such fluids include non-dilute suspensions of Brownian spherical particles (Brady & Vicic 1995), dilute suspensions of Brownian spheroids (Brenner & Condiff 1974), and dilute emulsions of surfactant-free (Raja, Subramanian & Koch 2010), surfactant-covered (Vlahovska, Blawzdziewicz & Loewenberg 2002) spherical drops and vesicles (Vlahovska & Garica 2007). Consideration of these examples below shows that varies over a significantly wider range when compared to polymer solutions alone.…”

“…For a dilute emulsion of surfactant-free spherical drops, the normal stress differences are O(µγ Caφ), where Ca = µγ a/Γ , Γ being the interfacial tension coefficient, plays the role of De. The microstructural anisotropy is associated with drop deformation, and the ratio of normal stress differences is now a function of the viscosity ratio (Schowalter, Chaffey & Brenner 1968;Raja et al 2010), being given by:…”

On the macroscopic modelling of dilute emulsions under flow

“…This pioneering work by Einstein in the dilute limit of solid particle suspensions has been extended to describe higher order concentration effects and/or droplet particles with finite internal viscosity (Taylor 1932(Taylor , 1934Batchelor 1970;Happel and Brenner 1983). However, even for dilute suspensions, there are higher order effects that give rise to normal stress contributions (Schowalter, Chaffey & Brenner 1968;Frankel & Acrivos 1970;Raja, Subramanian & Koch 2010), which therefore cannot be described by a modified viscosity corresponding to a generalized Newtonian model. Moreover, for suspensions of deformable liquid particles, i.e.…”