Simulations of deformable systems in fluids under shear flow using an arbitrary Lagrangian Eulerian technique

Highlights

• We study the deformation of soft systems in fluids under unbounded shear flow.

• The study is done with a finite element method based code for viscoelastic fluids.

• To track the interface, a finite element method is defined on it.

• Deformed shape of drops and elastic particles are nearly ellipsoidal.

• Results on elastic particles in viscoelastic fluids are shown for the first time.

Abstract

An arbitrary Lagrangian Eulerian finite element method based numerical code for viscoelastic fluids using well-known stabilization techniques (SUPG, DEVSS, log-conformation) is adapted to perform a 3D study of soft systems (drops, elastic particles) suspended in Newtonian and viscoelastic fluids under unbounded shear flow. Since the interface between the suspended objects and the matrix needs to be tracked, a finite element method with SUPG stabilization and second-order time discretization is defined on the interface, with the normal velocity of the interface equal to the normal component of the fluid velocity and a tangential velocity such that the elements on the interface are evenly distributed. This allows the mesh to get rid of the tank-treading motion of the particle. Both drops and elastic particles deform because of the flow and attain stationary deformed shape and orientation with respect to the flow direction. The effects of the physical parameters of the system on the phenomenon are investigated. The code is validated for drops and elastic particles in a Newtonian fluid through comparison with data from literature. New results on the deformation of elastic particles in an Upper Convected Maxwell fluid and a Giesekus fluid are presented.

Introduction

Microfluidics, i.e. the processing of fluids in channels which have a characteristic length in the order of microns, has gained an increasing importance over the last 15 years. In several microfluidic systems, as cytometric, sorting and diagnostic devices, suspensions of soft particles flow in miniaturized channels [1], [2]: in such a situation, a wide range of phenomena can arise due to the deformability and the elasticity of the suspended objects, the geometry of the channels, the interactions between the objects and the walls of the flow cells, and eventually the complexity of the suspending media.

Among deformable systems, drops have been extensively investigated since the 1930s, when Taylor experimentally studied the deformation of Newtonian drops in a Newtonian fluid under simple shear flow in the low capillary number regime [3], [4]. For drops, the capillary number, which is the ratio between viscous and interfacial forces, is defined as $\mathit{Ca}=\frac{{\eta }_m{R}_0\dot{\gamma }}{\Gamma }$, where ${\eta }_m$ is the ambient fluid viscosity, $R_0$ is the radius of the undeformed drop, $\dot{\gamma }$ is the shear rate and $\Gamma$ is the interfacial tension between the two fluids. The drops are found to deform and assume an ellipsoidal shape with a fixed orientation with respect to the flow direction, the deformation at the steady state linearly increasing with Ca according to the law:

$$D=\frac{19\lambda +16}{16\lambda +16}\mathit{Ca}$$

where D is the Taylor deformation parameter, defined as the ratio of the difference and the sum of the major and minor semiaxes of the ellipsoid in the shear plane, and $\lambda$ is the ratio between the viscosities of the drop and the suspending matrix. During both the transient and steady deformation, the fluid bulk of the drops circulates around the center of mass. Taylor’s results have been afterwards confirmed theoretically, experimentally and computationally [5], [6], [7]. Kennedy and co-authors [8] showed that if the viscosity ratio $\lambda$ is less than about 4, when Ca reaches a critical value, the interfacial tension is no more capable of contrasting the deformation and the drop breaks up. On the contrary, above such critical $\lambda$-value, the drop does not break up even at high Ca-values. Several papers focus on identifying $\lambda -\mathit{Ca}$ break-up conditions for drops in shear flow [6], [7], [9]. The presence of solid walls in the vicinity of a drop can substantially influence its deformation: in confined shear flow, very elongated non-ellipsoidal deformed shapes have been detected [10]. If one considers fluid complexity and in particular viscoelasticity, a new parameter has to be taken into account: the Deborah number De, which is the ratio between the fluid and the flow characteristic times. On whatever side viscoelasticity acts (i.e. drop, matrix or both), in shear flow the stationary orientation angle of a drop is lower than in the corresponding Newtonian/Newtonian case [11]. A phenomenological model is available for drops under all flow conditions in the small deformation regime, which can handle viscoelasticity in both the matrix and the dispersed phase [12].

Elastic particles are quite interesting from both a scientific and a technological point of view, since they are present, for example, in filled polymers and are also models for more complicated systems. Moreover, not much attention has been devoted to such objects and a wide comprehension of their behaviors in flow is still lacking. In 1946, Fröhlich and Sack [13] were the first to investigate the behavior of a suspension of elastic spheres in a Newtonian fluid undergoing extensional flow, linking the extensional stress of the suspension to the strain rate. In 1967, Roscoe [14] studied the behavior of a dilute suspension of viscoelastic spheres in a Newtonian fluid subjected to shear flow, obtaining from a theoretical point of view the deformation of the objects, the stress and the viscosity of the suspension. He showed that the particles attain an ellipsoidal steady deformed shape with fixed orientation, and that the material within the ellipsoids continuously rotates. In the same year, Goddard and Miller [15] derived a constitutive equation for dilute suspensions of slightly deformed elastic spheres. In 1981, Murata [16] studied the deformation of a spherical elastic particle in an arbitrary flow field by using a perturbation analysis. Since then, almost nothing has been done on elastic particles until the end of last decade, when Gao and Hu [17] performed a 2D perturbation analysis from which they derived the relationship between the steady elliptical deformation of an elastic particle, expressed by the Taylor deformation parameter D, and the dimensionless parameter which measures the relative weights of the viscous and elastic forces in the system (the elastic capillary number Ca). The authors showed that for small Ca-values, at the stationary $D=\mathit{Ca}$, and validated such result by means of 2D Arbitrary Lagrangian Eulerian Finite Element Method (ALE FEM) simulations. In 2011, Gao et al. [18] studied the behavior of an initially spherical elastic particle suspended in a Newtonian fluid in shear flow through a polarization method [19], [20], coming to a validation of Roscoe’s results.

More complicated soft systems are vesicles and capsules, that are constituted by liquid drops wrapped in thin elastic membranes. Such objects, which are often non-spherical at rest, are widely used in the biotechnological field as drug deliverers and models for cells (e.g. Red Blood Cells). Several theoretical, experimental, and numerical works are available on the deformation of vesicles and capsules in Newtonian fluids under unconfined shear flow, as reported in their review by Finken and co-authors [21]. Depending on the physical parameters, different regimes of motion are detected, such as stationary tank-treading (TT), where the orientation angle with respect to the flow direction is fixed and the elastic membrane rotates around the internal fluid like the treads of a tank, tumbling (TU), where the whole object periodically rotates around the vorticity axis, vacillating–breathing (VB) (also known as oscillating (OS)), where the orientation angle oscillates around a mean value, and some intermittent regimes, that consist in an evolution of the motion regime in time (e.g., $\mathit{TU}\to \mathit{VB}\to \mathit{TT}$). Stationary TT and TU being the two extreme behaviors, an increase in $\lambda$ is found to promote TU, whereas an increase in the flow intensity promotes TT.

In this paper, an arbitrary Lagrangian Eulerian finite element method based code for viscoelastic fluids using well-known stabilization techniques such as SUPG [22], DEVSS [23], [24], and log-conformation [25], [26] is adapted and extended to study the behavior of drops and elastic particles suspended in Newtonian and viscoelastic fluids under unbounded shear flow in 3D. To the best of our knowledge, we show the deformation of elastic particles in unbounded shear flow of Upper Convected Maxwell (UCM) and Giesekus (Gsk) fluids for the first time.

The paper is organized as follows: in Section 2, the governing equations for the systems considered are presented; in Section 3, the numerical technique adopted to solve the model equations is explained in detail; in Section 4, numerical convergence tests are shown; in Section 5, results are presented; finally, in Section 6, some conclusions are drawn.