Session Q34: Micro/Nano Flows: Non-Newtonian/Oscillatory

Structure of the streaming flow generated by a sphere oscillating in a viscous fluid

A body undergoing oscillatory motion in a viscous fluid naturally produces a steady secondary flow due to convective inertia. This is embodied in the streaming flow generated by a sphere executing oscillations in an unbounded fluid. We review the literature on this canonical problem and summarise both exact and asymptotic formulae in the small amplitude limit. These analytical formulae are used to explore the characteristic counter-circulating structure of this flow and clarify some previously unreported features. A single vortex exists regardless of the oscillation frequency, which can drive a counter-circulating flow away from the sphere. The centre of this vortex moves monotonically away from the sphere with decreasing oscillation frequency and engulfs the entire flow domain for β ≡ ωR² / ν < 16.317, where ω is the oscillation frequency, R is the sphere radius, and ν is the fluid kinematic viscosity. This abrupt change in flow structure at the critical frequency, β_critical = 16.317, and its quantification appears to have not been reported previously. We perform a numerical simulation of the Navier-Stokes equations, which reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow.   

Unsteady motion of nearly-spherical particles in viscous fluids

An understanding of how non-spherical particles move in viscous fluids is critical to many applications. The motion of such particles is often studied using the unsteady Stokes equations. Zhang & Stone, J. Fluid Mech. 367, 329-358 (1998) reported an asymptotic theory for nearly-spherical particles, to first order in particle non-sphericity. Importantly, some key physical phenomena are absent at this order, including (i) the coupling between the torque experiences by the particle and its linear translation, (ii) the force it experiences and its rotation, and (iii) the effect of non-sphericity on the orientation-averages of these forces and torques. Here, we accommodate these phenomena through derivation of an asymptotic theory correct to second-order in particle non-sphericity, for the force and torque acting on the particle in a general unsteady Stokes flow. The derived analytical formulae apply to particles of arbitrary shape, giving the leading order theory for the above mentioned phenomena. Several example nearly-spherical particles are considered including a spheroid, a`pear-shaped' particle and a 'spiked' particle. We report independenent simulations of the Navier-Stokes equations that validate the theory.   

Modulating 3D rectified flows via body elasticity

Rectified flows associated with the oscillatory motion of immersed rigid boundaries, known as viscous streaming flows, represent an efficient way of manipulating and controlling fluids via inertial effects. Despite recent progress, little is known about the role of body compliance, particularly when 3D settings are considered, a situation frequently encountered in microfluidics and biology. Motivated by this, we conduct a theoretical investigation into compliance-induced 3D streaming, via the minimal case of an oscillating, hyperelastic sphere immersed in a viscous fluid. We confirm our findings against direct numerical simulations and demonstrate the existence of new elasticity-induced streaming regimes, unavailable in the case of rigid bodies.   

Three-dimensional streaming around an obstacle in a Hele-Shaw cell

The inertial rectification of oscillatory flow driven past an obstacle produces a secondary steady flow called "streaming", which has been used in a variety of microfluidic applications. Understanding of such flows is limited largely to two-dimensional configurations, and typically neglects the high degree of vertical confinement provided by boundary walls in practical microfluidic devices. We develop a three-dimensional streaming theory around an obstruction sandwiched in a microchannel with a Hele-Shaw geometry, in which one dimension (depth) is much shorter than the other two. Applying inertial lubrication theory to solve the incompressible Naiver-Stokes equations for small oscillation amplitudes, we show that the streaming flow has a three-dimensional structure. Notably, the flow changes direction across the depth of channel, distinct from previous observations of three-dimensional microscreaming flows. We show that this vertical flow reversal is supported by our experiments of streaming around short cylinders in microchannels. Our theory also predicts that flow velocity decays as the inverse cube of distance from the cylinder, more rapidly than that expected from two-dimensional approaches. We verify this decay rate quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies. The ability to generate recirculating three-dimensional flows in microconfined geometries is promising for particle trapping and micromixing applications.

    

Time-averaged inertial particle dynamics in oscillatory flow

Oscillatory flows provide a powerful way to manipulate suspended particles in microfluidic settings. We present a comprehensive theory of the dynamics of a compressible spherical particle in oscillatory flow, spanning the range between classical acoustofluidics and streaming regimes. In particular, we study how the density and compressibility contrast between particle and fluid, and the Stokes layer thickness relative to the particle radius, influence the time-averaged motion of the particle. The particle produces an oscillatory disturbance flow, whose inertia drives a secondary disturbance flow. We show that the associated secondary force in turn drives steady particle motion through a Faxén-like relation. By applying the Lorentz Reciprocal Theorem to compressible flows with inertia, we obtain the secondary force analytically while circumventing the need to resolve the complicated secondary flow in detail. We find that the force depends on quadratic combinations of local moments of the ambient oscillatory flow, as well as the physical properties of the fluid and the particle. Our formulation recovers known results for secondary radiation forces in the limit of thin Stokes layers, while predicting qualitative differences at finite Stokes layer thicknesses. In particular, we show that the force's direction can be reversed for certain combinations of frequency, density and compressibility ratio. We demonstrate the application of the theory to efficiently compute time-averaged particle motion in oscillatory flows typical of microfluidic and acoustofluidic applications.   

Reduced modeling of pulsatile flows in compliant microfluidic conduits at arbitrary Womersley number

We investigate the pulsatile fluid--structure interaction (FSI) between a Newtonian fluid and a slender, deformable microtube. We derive a theory for the instantaneous pressure distribution by two-way coupling the pulsatile flow and the tube deformation. The radial displacement is obtained from thin shell theory, assuming axisymmetry and negligible bending at the leading-order in slenderness. The flow rate is related to the pressure gradient (and the tube deformation) via lubrication theory, specifically via the Womersley solution for pulsatile flow. Substituting this relation into the mass conservation equation, we obtain a 1D nonlinear PDE governing the instantaneous pressure distribution along the tube, at arbitrary Womersley number. This PDE is easy to solve numerically to obtain the pressure distribution in the compliant tube. A cycled-average pressure is also computed, which deviates from the steady profile, suggesting a type of FSI-induced streaming. The instantaneous and cycle-averaged pressures are both validated against 3D direct numerical simulations performed with svFSI (part of the open-source software SimVascular). We find good agreement between theory and simulations.   

Liquid beads in an elastic matrix: Rheology of a solid emulsion with phase-changing droplets

Classic viscoelastic systems show correlated storage and loss moduli. To try and decouple them with regards to the structure of the material, we study “solid emulsions” with a crosslinked continuous phase. The rheological properties of emulsions depend on both the continuous and the dispersed phase, as well as the interface between them. Thus, by encapsulating the liquid dispersed phase in a solid elastic matrix, we can create a composite material with tunable rheological properties [1].

The chosen dispersed phase in our “solid emulsion” can solidify at room temperature. This phase change is used to vary the moduli of the dispersed phase. Previous theoretical works have shown that for liquid inclusions we can expect to see different rheological responses depending on the continuous and dispersed phase moduli, the size of the droplets in the matrix and the volume fraction of dispersed phase [2, 3]. However, when the inclusions are solid, the complex modulus should only depend on the continuous phase and volume fraction [4].

We first show the feasibility of a composite system whose storage and loss moduli can be controlled independently, and study the mechanisms for such a decorrelation. For our system, we have observed a decoupling of the storage and loss moduli in the rheological response for liquid insertions in the matrix when varying the volume fraction. When the droplets are solid, we observe however a deviation from the expected behavior solely controlled by the matrix and the volume fraction, that we investigate through rheology and compression-decompression experiments.

 

References:

[1] R. W. Style, R. Tutika, J. Y. Kim, M. D. Bartlett, Advanced Functional Materials, 2020, 31

[2] J. F. Palierne, Rheologica Acta, vol. 29, no 3, p. 204‑214, 1990

[3] R. Pal, Current Opinion in Colloid & Interface Science, vol. 16, no 1, p. 41‑60, 2011

[4] M. Krieger et T. J. Dougherty, Transactions of the Society of Rheology, vol. 3, no 1, p. 137‑152, 1959   

An experimental study of the co-flow of Newtonian and non-Newtonian fluids in a T-shaped microchannel

Microfluidic mixers and separators often concern flow of non-Newtonian fluids through a T-junction. The flow dynamics at the junction plays an important role on the performance of such devices. Some previous studies have focused on the topic with same fluids flowing through the T-junction in an opposing configuration. However, many of the real applications may use fluids with dissimilar rheological properties. In this work, we experimentally investigate the co-flow of Newtonian and non-Newtonian fluids through a planar T-shaped microchannel. Several types of polymer solutions with diverse shear-thinning and elastic properties are tested. Our preliminary results indicate the development of different vortical/nonvortical and unstable flow regimes at the T-junction based on the fluid rheology.

    

Stagnation flow of polymer solutions in a T-shaped microchannel

Viscoelastic flow of polymer solutions opposingly flowing through a T-junction has been a subject of interest for particle manipulation and droplet-based microfluidics lately. Some studies have been conducted on the non-Newtonian fluid dynamics near the stagnation point at the junction, which eventually can dictate the efficacy of those microfluidic operations. However, a systematic understanding of the fluid rheological responses, namely different combinations of shear-thinning and elasticity, is still lacking in such circumstances. In this work, we experimentally investigate the responses of different rheological properties of polymer solutions symmetrically and opposingly flowing through a planar T-junction microchannel. The flow visualization results indicate that the flow regimes can be expected to be either vortex dominated or having unstable streamlines, depending on the strength of elasticity and shear-thinning of the fluid under such flow conditions.