Computational Rheology of Complex Fluids: Polymeric & Wormlike Micellar Solution Cases

ABSTRACT

In this work, the essence of Non- Newtonian Fluid Mechanics and Computational Rheology is presented through three examples applied to the rheological characterisation of polymeric solutions using the SwanINNFM(q) family-of-fluids (1-5), and of worm-like micellar solutions using the BMP + _τ_p rheological equation-of-state (6-8), within the computational modelling of two benchmark flows in Non-Newtonian Fluid Mechanics: contraction-expansion flow geometries and flow past a sphere. The predictive capabilities of our computational tools are demonstrated, where mathematical models derived from conservation principles are solved (9-11) alongside the construction of constitu- tive equations from theoretical rheology (1-11). These mathematical models are solved using a computational algorithm based on a hybrid formulation of spatial discretisation in the form of finite elements for the mass and momentum balance equations, and finite volumes for the constitutive equation (1-8, 12-15). In contraction-expansion type bench- mark flows, firstly for polymeric fluids, experimental pressure-drop measure- ments were reproduced quantitatively using the SwanINNFM family-of-fluids (1-5). We were able, for the first time, to predict quantitatively and explain long-standing augmented excess pres- sure-drops and highly-dynamic vortex structures observed in the flow of polymeric Boger fluids (16-21). Building upon contraction-expansion flows of thixo- viscoelastoplastic concentrated worm-like micellar solutions, the effects of considering extreme shear thinning and flow segregation through yield stress and shear banding were demonstrated (6-8, 22). Using the BMP + _τ_p constitu- tive model (8), shear bands are predicted in fully-developed flow zones away from the constriction, and their interaction with the complex deformation imposed by the contraction is reported. For the flow-past-sphere benchmark flow (7), numerical solutions obtained with the BMP + _τ_p model qualitatively reproduce features reported experimentally for the descent of spheres in worm-like micellar solutions, i.e., a flow instability associated with oscillations in the sphere settling velocity and negative wakes (22), and, for relatively concentrated micellar solu- tions, asymmetrical yield fronts.

INTRODUCTION

One of the fundamental contributions of rheology is the identification of diverse materials as Newtonian (those that follow Newton’s Law of Viscosity, i.e., those which display a constant vis- cosity at constant temperature and pres- sure), and as non-Newtonian, i.e., those that do not comply with the Newtonian definition. The latter manifest non-linear flow properties through a variable appar- ent viscosity with deformation rate, time of an imposed flow, and even displaying simultaneous liquid and solid properties in the form of viscoelasticity and yield stress, to name a few typical rheological responses (9-11).

In its practice, rheology divides its study into four main areas (9-11): (i) rheometry, which spans over material-property measurement, e.g., fundamentally viscosity, elastic modulus, relaxation time; (ii) constitutive modelling, through which constitutive equations seek to reproduce and explain the material properties of complex fluids; (iii) non-Newtonian fluid mechanics, which studies the flow of non-Newtonian materials in complex geometries, whose essence lies in inhomogeneous deformations (deformations that combine shear and extension simultaneously in the flow field) and are reflected in physical arrangements with diverse geometric changes observed in nature and in technological applications, such as contractions and expansions, and flows around objects, among others; and

(iv) computational rheology, which focuses its efforts in obtaining approximate numerical solutions to the flows studied in non-Newtonian fluid mechanics.

Complex fluids are materials with non- linear rheological characteristics derived from their microstructure, which may be classified as soft matter (9-11). Complex fluids are found in countless techno- logical applications, e.g., cements, paints, toothpaste, foams, crude oil and its heavy fractions, drilling muds in oil extraction, foodstuff, mayonnaise, plastics, reactive mixtures, and cosmetics (9-11, 22-31). In addition, many biological fluids, such as blood, mucus, saliva and tissues, may display non-linear rheological properties (32-36).

The combination of: (i) the non-linear rheological properties of complex fluids, (ii) the conservation equations, i.e. of mass, momentum and thermal energy, and (iii) the simultaneous non-homoge- neous shear and extensional deformations imposed in complex flows, result in mathematical problems of the highest complexity when attempting to describe, understand, and theoretically predict the experimental manifestations in non- Newtonian Fluid Mechanics (37-40). The interest of Computational Rheology is the prediction of complex flows of non- Newtonian materials. It bases its action on the development and application of advanced numerical techniques to the highly non-linear partial-differential- equation systems that represent flow problems whose solution is practically unattainable by exact methods (37-40).

There is a plethora of numerical algorithms for solving computational rheology problems (37-41). In general, their formulation has as a basis on Eulerian or Lagrangian frames of reference. The most popular Eulerian algorithms are based on finite-element and finite-volume methods (6-8, 37-40), devised to cover the mixed parabolic-hyperbolic nature of the mass-momentum-energy bal- ance and constitutive equations. On the side of Lagrangian algorithms, particle dynamics methods (Smoothed Particle Hydrodynamics, Dissipative Particle Dynamics and lubrication dynamics methods), are among the most widely used (41), and represent a suitable option for the computational prediction of the rheology of suspensions and particulate systems (42).

Polymeric materials (melts and solutions) are made up of long-chain molecules, which interact closely through entanglement and reptation in molten and dissolved states. These interactions are the origin of their characteristic non-Newtonian features, in the form of marked shear thinning, and viscoelasticity through significantly-augmented normal-stress differences (10-11). The reflection of such rheological response in complex deformations has been a matter of extensive research (16-21). Studies on many benchmark flows have focused on their kinematic and dynamic response, for which augmented pressure drops and diverse vortex-enhancement mechanisms occupy a central role (16-21). In fact, the theoretical prediction and under- standing of such features remain an open research topic to date, where efforts are still being concentrated in elucidating how polymeric materials respond under inhomogeneous deformations (1-6).

Worm-like micellar solutions (WLMs) are complex fluids composed of dispersions of elongated micelles that interact essentially through relatively weak entanglements; these physical interactions promote their thixotropic, viscoelastic and plastic properties (22, 25-31). WLMs are also known as living polymers, due to their ability to restructure when flowing by two mechanisms, i.e., (i) reptation, as polymers do, and (ii) construction and destruction of micellar structures (22, 25-31). For these reasons and their var- ied rheological properties, these complex thixo-viscoelastoplastic materials are used in a wide range of applications, such as in cleaning and home and health-care products (shampoos, soaps, detergents, drug carriers); in the petroleum industry, as drilling and well-stimulation fluids; in pumping systems, lubricants and emulsifiers (22, 25-31).

The diversity of rheological properties of polymers and WLMs is a chal- lenge for the development of constitutive equations capable of describing their experimental manifestations in simple and complex flows (37-40). Polymers and WLMs generally display shear thinning, extensional hardening and softening, viscoelasticity, thixotropy (16-21, 22, 25-31) and, in the specific case of WLMs, flow segregation in the form of yield stress (27) and banding (35). All of these responses occur simultaneously and manifest across diverse spatial-temporal scales (22, 25-31, 35).

Constitutive equations for polymeric materials are diverse and numerous, some coming from microscopic arguments and others based on continuum approaches (11). Among the most widely- used constitutive-equation approaches of differential nature are those of the FENE type, where the Peterlin and the Chilcott-Rallison closures dominate (11, 43-44), and the Phan-Thien-Tanner paradigm (45), which have been successful in reproducing and explaining the response of a wide range of polymer melts and solutions.

For WLMs, constitutive equations are still being developed (6-8, 22, 46-50). There are two main theoretical frame- works, namely, (i) theories based on structural variables, and (ii) microscopic theories. The former are the most popular, as they portray the evolution of the internal WLM structure, explicitly related to material functions (6-8, 46, 48-49). Among these models are those in the Bautista- Manero-Puig (BMP) (6-8) and de Souza- Mendes (48-49) families. The constitutive equations in the BMP framework predict key properties of WLMs and other complex fluids, and have been successfully used to study the flow of WLMs in complex deformations (6-8). Microscopic theories study the interaction of micelles in their construction/destruction dynamics in flow, via kinetic equations whose solution is related to material properties through averages (47, 50).

Experimental studies on the flow of polymeric fluids and WLMs in complex geometries reveal rich features, with dynamic vortex-enhancement mechanisms and pressure drops. These act as alternative energy-dissipation mechanisms in contraction flows, and through drag coefficients in sphere settling, revealing instabilities manifested in par- ticle oscillations and negative wakes (16- 22, 51).

In benchmark contraction and con- traction-expansion flows, complex vortex dynamics have been recorded experimentally. At low volumetric flow rates, symmetric kinematic structures are observed, similar to those observed in the contraction flow of Newtonian fluids. At high volumetric flow rate, asymmetric vortices, promoted by viscoelasticity, lead to time-dependent chaotic flows (16- 22, 51).

In the sedimentation of smooth spheres in semi-dilute WLMs, oscillations in the particle descent velocity have been reported. These are caused by strong negative wakes behind the sphere; for polymeric liquids, a similar response is recorded as velocity overshoots (16-21, 51). These phenomena have been studied as flow instabilities with respect to the steady rate of descent characteristic of Newtonian fluids (22). In WLMs, these findings have been attributed to the complex dynamics of structure con- struction-destruction of the elongated micelles (leading to thixotropy) and the viscoelasticity of the micellar solution (6-8, 22). For concentrated mixtures, these thixo-viscoelastoplastic solutions form gels that display markedly-asymmetric yield fronts around the sphere (22, 51), as previously reported by Holenberg et al. (52) and Putz et al. (53).

One of the iconic manifestations of WLMs is a type of flow segregation called shear-banding, which is characterised by a spontaneous separation of the solution into two or more shear bands of material that coexist, supporting a constant shear stress, but with distinct apparent viscosity (8, 22, 35). This paper presents a compendium of research work conducted by the author on computational predictions of the response of polymeric and WLM solutions in complex benchmark flows (1-8). These works illustrate the use of computational rheology in the numerical solution of two typical problems of non-Newtonian fluid mechanics: flows past a sphere (7), and flows through contractions and contraction-expansions (6,8). These benchmark flows have industrial and technological applicability; (i) flow around spheres is applied in particle suspension in medi- cine and the food industry (shelf life), and is also an approximation for clay trans- port in enhanced oil extraction fluids (37- 40); whilst (ii) contraction-expansion flow is found in industrial equipment with pipe and fitting changes (37-40), and lies at the heart of polymer and food process- ing operations (22, 54).

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