Non-Newtonian Flow: Interfacial and Bulk Phenomena

1. Introduction

Nearly all biological fluids and a vast number of non-biological fluids are, to some extent, non-Newtonian, with many also exhibiting viscoelastic responses. Constituents such as colloidal particles, proteins and polymers, surfactants, bubbles, and emulsions (micro-scale drops of other immiscible liquids) can render a fluid non-Newtonian. Examples of biological fluids include blood, which contains cells that act as colloidal particles; the liquid lining of the lung; saliva; tear film; and cerebral spinal fluid. All these liquids contain dissolved proteins and many also contain surfactants. Examples of non-biological fluids that are non-Newtonian include inkjet fluids, paints, crude oil and many petroleum products, and numerous food stuffs. Lastly, the growing number and uses of biologics in pharmaceuticals along with the fundamental challenges in their manufacturing have spurred renewed interest in experimental and theoretical studies of complex fluids and interfaces.

The shear rate dependence of viscosity, the hallmark of non-Newtonian fluids, manifests in bulk fluid due to the interactions between its constituents and the solvent/dispersed fluid. However, other complex fluid responses can arise. For example, at fluid–fluid interfaces, such as gas–liquid or liquid–liquid interfaces, viscoelasticity can arise even if both the bulk fluids and the fluid interface are fully Newtonian. The viscoelasticity of Langmuir monomolecular films is a classic example. When pushed aside, an insoluble monolayer flows back due to the increased surface tension from the stretched side. The resulting elasticity of the interface, along with the coupling between bulk and interfacial flow, can provide a no-slip boundary condition on a monolayer-covered surface of a liquid, akin to a solid wall, and lead to Marangoni flow.

The five contributions to this Special Issue focus on various aspects of complex fluid flows manifesting in non-Newtonian responses in the interior of the fluid and/or its interface. A brief review of each contribution follows.

2. Review of Contributions to This Special Issue

Ligrani et al. (Contribution 1) examined heat transfer in a rheometric flow consisting of a steadily rotating disc atop a concentric cylinder filled with a solution of a viscoelastic polymer at various concentrations. The rotating disc was smaller than the cylindrical container, and the effect of the meniscus contacting the rotating disc was deemed negligible. The system was heated from below, and the heat flux on the container floor was measured. The Oldroyd-B model for viscoelastic fluids was utilized with a relaxation time and separate contributions of solvent (water) and polymer viscosities. The ratio of convection to conduction heat transfer, reported as the Nusselt number, was found to be near unity when the disc was stationary or rotating slowly, consistent with the fact that natural convection was relatively small in that system. The key finding in their study is that, for cases with sufficient polymer concentration, as the disc rotation rate is increased above a certain critical value, the Nusselt number increases monotonically, indicative of elastic instability. This elastic instability, which can manifest without inertial effects, requires a sufficient amount of local shear, resulting in negative normal stress in the flow direction of a viscoelastic fluid. For the case of a solvent alone (zero polymer concentration) at low Reynolds numbers, the Nusselt number was found to be a weak function of the Reynolds number (Re^0.22). Overall, the local shear rate was found to be the primary determinant of the Nusselt number for a given polymer concentration, with a monotonic increase in Nusselt number occurring with increasing concentration. The study shows that the Nusselt number for all cases with the polymer collapses to a single, monotonically increasing function of the square of the Weissenberg number (product of the shear rate and polymer relaxation time) and the weak power of the Reynolds number.

Zhao and Liu (Contribution 2) examined the interfacial instability of vesicles in a time-dependent extensional flow, similar to a planar stagnation flow. The inertialess limit was considered, with the fluid inside the vesicles, the surrounding fluid (with the same viscosity), and the interface taken as incompressible. In practice, this is a suitable model for describing a liposome, a phospholipid bilayer encapsulating an aqueous solution with a therapeutic agent that can be suspended in another aqueous solution. Such bilayers have been widely studied, in part because of their similarity to animal cell membranes. A boundary integral method was used in their simulations, with Newtonian fluids inside and outside the vesicles. A complex fluid response arose in the membrane, which was taken to be inhomogeneous, consisting of two components with different bending rigidities. The two phases of the membrane were advected by the coupling to both the internal flow and the external flow, which were cross-diffused via interfacial diffusivity. Interfacial instability resulting in the wrinkling of the membrane was observed upon flow start-up or reversal of a steady flow. The mechanism that generates the instability appears to be the phase separation of the membrane constituents, where the stiffer component flows to the less curved regions of the membrane and the less stiff component flows to regions with more curvature. The instability is ultimately a result of negative elastic surface tension in some regions of the membrane, a direct consequence of its incompressibility. The simulations were compared favorably against linear analysis for slightly perturbed vesicles.

Zhu (Contribution 3) developed a new analytical solution for the flow drag of a swarm of Newtonian fluid drops through a power law fluid. Specifically, shear-thinning surrounding fluids were considered, and flow regimes were restricted to Stokes flow. For all drop-to-surrounding fluid viscosity ratios, the effective drag for the case of drop swarm, scaled by the Stokes drag, was found to increase with the power law exponent as well as the drop-to-surrounding fluid viscosity ratio. The new solution agrees with the prior analytical results in ref. [1], as well as numerical simulations presented in the paper. A similar increase in drag was observed with increasing drop volume fraction ratio. However for individual drops, the drag was found to decrease with the power law exponent. In the case of drops with viscosity lower than that of the surrounding fluid, the drag became smaller than the Stokes drag. For most cases, the new analytical solution performed on par or better than prior models.

Singh and Narsimhan (Contribution 4) numerically examined film drainage during the collision of two equal-sized drops in an axisymmetric stagnation flow in the Stokes flow regime with the surface of the drops covered by a soluble surfactant. The drainage of the film between drops is a key aspect of drop collision and an impediment to their coalescence. The Boussinesq–Scriven surface model for a Newtonian interface was used, where the stress exerted on the interface by the bulk phases is balanced by the gradient in surface tension, resulting in interfacial elasticity, and the sum of the surface shear and surface dilatational viscosities. The nature of the boundary conditions in this flow is such that the surface shear viscosity and surface dilatational viscosity appear as a sum, similar to that in planar flow problems. Because of the scarcity of consistent data on surface dilatational viscosity, Singh and Narsimhan assumed it to be equal to the surface shear viscosity. The viscoelastic response of the Newtonian interface produced a rich set of phenomena. The simulations were benchmarked against the results for the collision of clean drops in ref. [2], showing agreement for different capillary numbers as well as a wide range of drop-to-surrounding fluid viscosity ratios. The results of Singh and Narsimhan can be extended to viscosity ratios of order one and smaller. For cases with a surfactant, the film drainage time was found to increase with the drop viscosity ratio and surface viscosity. At large drop viscosity ratios, the effect of the surface’s intrinsic viscosity diminishes. Finally, some results were presented for a case where surfactant transport to and from the bulk diminish, essentially modeling an insoluble surfactant.

In the final study, Adam et al. (Contribution 5) numerically examined the axisymmetric flow of a sheared drop in a configuration where the drop is constrained by a stationary spherical cap and sheared by a thin contact ring in the other hemisphere. Several studies have focused on the ring-sheared drop, including experiments conducted in microgravity at the 1 inch (2.54 cm) scale [3]. The fluid in most of these studies is complex, typically containing dissolved proteins. In this system, surface shear viscosity plays a large role in driving the primary flow, i.e., flow in the azimuthal direction, as well as the secondary flow in the meridional plan generated by inertia. The surface tension in the study provides fluid containment, whereas the gradients in surface tension stop any flow in the polar direction, making the interfacial flow purely azimuthal. Adam et al. considered the case where the interfacial flow is non-Newtonian and the bulk flow is Newtonian, the scenario corresponding to cases with relatively low protein concentrations. The numerical model was based on a shear-thinning interfacial response presented in ref. [4] and was two-way-coupled with a Newtonian bulk flow, which included the effects of inertia. Adam et al. show that the parameters describing the interfacial response generally exhibit increased sensitivity at larger Reynolds numbers. Their results also show that the bulk flow pattern is less sensitive than the distribution of the interfacial velocity for determining the interfacial parameters.

3. Conclusions

The papers in this Special Issue, although relatively few, reveal a rich variety of forces and interactions, describing new phenomena and/or furthering our understanding and improving our models. Fluid interfaces were prominently featured in four of them. Only in Ligrani et al. (Contribution 1) was a free surface, although present in the flow apparatus, not an essential aspect of the flow and likely had a negligible effect on the results. The contributions also describe a rich combination of processes: some steady and others transient, some requiring flow inertia whereas others could be captured using Stokes flow, some also including the energy equation but most only involving mass and momentum transport, and some containing non-Newtonian (bulk) liquids; in other studies, either the fluid interface was non-Newtonian or heterogeneous. The results show that, in some cases, the non-Newtonian fluid response has a destabilizing effect on the flow, whereas, in others, flow disturbances are dampened by non-Newtonian effects. In flows that directly involve surface tension effects, either surface tension or its gradient play the key role. In Singh and Narsimhan, Newtonian fluid interfaces were stretched, so the effects of surface dilatational and surface shear viscosity manifested. In Adam et al., the fluid interface was sheared but not dilatated: only the effect of surface shear viscosity influenced the flow, albeit being shear-rate-dependent.

The usual ending of most scientific works that “more study is needed” certainly applies to this Special Issue of Fluids.