On the macroscopic modelling of dilute emulsions under flow

Abstract

A new macroscopic model describing the rheology and microstructure of dilute emulsions with droplet morphology is developed based on an internal contravariant conformation tensor variable which is physically identified with the deformed ellipsoidal geometry of the dispersed phase. The model is consistent with existing first-order capillary number, $O(Ca)$, theory describing the microstructure as well as $O(Ca^{2})$ theory describing the emulsion-contributed extra stress. These asymptotic solutions are also used to determine all of the model parameters, making it the only macroscopic emulsion model that is consistent with all available asymptotic theories in the limit of small $Ca$. The governing equations are obtained from the Poisson and dissipation brackets, as developed for an incompressible fluid system endowed with an internal contravariant second-order tensor, subject to the imposition of the constraint of a unit determinant. First proposed by Maffettone & Minale (J. Non-Newtonian Fluid Mech., vol. 78, 1998, pp. 227–241), this constraint physically corresponds to conservation of the volume of the dispersed phase in the emulsion. The Hamiltonian of the emulsion is expressed through the surface energy of the dispersed phase, in addition to the kinetic energy, following previous work by Grmela et al. (J. Non-Newtonian Fluid Mech., vol. 212, 2014, pp. 1–12), but employing a more accurate evaluation of the surface area in terms of the internal contravariant conformation tensor. Structural predictions of the ellipsoid droplet morphology obtained with the new model are compared with classic experiments by Torza et al. (J. Colloid Interface Sci., vol. 38, 1972, pp. 395–411), showing good agreement.

The relationship between viscoelasticity and elasticity

“…Only in the cases where a 2 = 1 do the viscoelastic constitutive relations exhibit a well-defined elastic limit, in the sense that their behaviours converge to that of an elastic solid in the limit of λ → ∞. The very same conclusions were reached in the context of emulsions, whose drops deform into ellipsoids—in that case the eigenvalues of A represent the square of the semi-axes of the deformed droplets [59,60]. Any other time derivative, which implies non-affine motion, does not correspond to any limit of the theory of elasticity.…”

“…Any isotropic contribution to the stress can be written as part of the pressure p. The non-Newtonian contribution σ p originates from the presence of the microstructure inside the fluid. Though we will refer to σ p as the 'polymeric stress', having in mind dilute polymer suspensions, the concept equally applies to emulsions whose microstructure is described by droplet deformations [59,60]. The stress σ p is governed by a separate evolution equation, the constitutive equation, which encodes the non-Newtonian properties of the fluid.…”

“…In § 5 c, we supply a list of various models. Apart from the question of frame invariance, models have to be consistent with the requirements of thermodynamics [ 6 , 7 , 59 , 60 , 63 ].…”

Elucidating the rheological implications of adding particles in blood

“…In this work, we will employ non-equilibrium thermodynamics (NET) to develop a sophisticated mathematical model addressing the rheological behavior of nanoblood. To accomplish this, we will develop the model using the generalized bracket formalism (Beris and Edwards 1994 ) of NET (Beris and Edwards 1994 ; Grmela and Öttinger 1997 ; Öttinger and Grmela 1997 ; Edwards et al 2003 ; Öttinger 2005 ), by means of which several microstructured systems have been addressed up to date, such as, but not limited to, immiscible complex fluids (Edwards and Dressler 2003 ; Dressler and Edwards 2004 ; Dressler et al 2008 ; Grmela et al 2014 ; Mwasame et al 2017 ), polymer melts and solutions (Beris and Edwards 1994 ; Grmela and Öttinger 1997 ; Öttinger and Grmela 1997 ; Öttinger 2005 ; Stephanou et al 2009 , 2016 , 2020b ), polymer nanocomposites (Rajabian et al 2005 ; Eslami et al 2010 ; Stephanou et al 2014 ; Stephanou 2015 ), micellar systems (Germann et al 2013 ; Stephanou et al 2020a ), blood (Tsimouri et al 2018 ; Stephanou 2020 ; Stephanou and Tsimouri 2020 ), drilling fluids (Stephanou 2018 ), and thixotropic fluids (Stephanou and Georgiou 2018 ). The use of a NET formalism has the compelling advantage, over other approaches, that the constitutive model as a whole is guaranteed consistency with the laws of thermodynamics (as extended for beyond equilibrium systems).…”

The relationship between viscoelasticity and elasticity

“…The non-Newtonian contribution σ p originates from the stretching of the microstructure inside the fluid. Though we will refer to σ p as the "polymeric stress", having in mind dilute polymer suspensions, the concept equally applies to emulsions whose microstructure is described by droplet deformations [20,21]. The non-Newtonian contribution σ p is governed by a separate evolution equation, describing the state of the component governed by a long time scale λ, for example a polymer.…”