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Concept# Non-Newtonian fluid

Summary

A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are , as are many commonly found substances such as custard, toothpaste, starch suspensions, corn starch, paint, blood, melted butter, and shampoo.
Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of non-Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other non-Newtonian behavior. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity.

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The propagation of fluid driven fractures is used in a number of industrial applications (well stimulation of unconventional reservoirs, development of deep geothermal systems) but also occurs naturally (magmatic dyke intrusion). While the mechanics of hydraulic fractures (HF) in isotropic media is well established, the impact of the anisotropy of natural rocks on HF propagation is still far from being understood. Sedimentary rocks like shales and mudstones are ubiquitous in upper earth crust which are made of fine layers which result in transverse isotropy. In the framework of continuous mechanics, these rocks are commonly modelled as a transverse isotropic media (TI). In addition, a large number of fluids used in HF are non-Newtonian. They typically exhibit a shear-thinning behavior which can be reproduced by different rheological models with varying levels of accuracy (Carreau, power law, Ellis).
In this thesis, we focus on hydraulic fractures in impermeable TI media. We assume that the fracture propagates normal to the isotropy plane without any further assumption on its shape. This configuration is relevant for normal and strike-slip stress regimes where the minimum in-situ stress is horizontal.
We combine a boundary element and a finite volume method with an implicit level set scheme to model the growth of three dimensional planar HF. Both anisotropy of elasticity and fracture energy/toughness are accounted for. This algorithm couples a finite discretization of the fracture with the solution for a steadily moving hydraulic fracture in the tip region. We show that the near tip elastic operator has a similar expression than in the isotropy pending the use of a near-tip elastic modulus which now depends on the local propagation direction with respect to the isotropy plane. Using this numerical model we quantify the fracture elongation as a function of both the elastic and fracture toughness anisotropies. The elongation is maximal in the toughness dominated regime. The transition of the viscosity to the toughness regime occurs faster along the arrester direction, thus promoting fracture elongation.
In parallel, we report laboratory experiments of HF growth in cubic samples of slate Del Carmen under true-triaxial confinement. We were able to propagate a planar HF perpendicular to the bedding planes only when the initial stress normal to bedding was 20 times larger than the other two stresses. For both regimes, the fracture surfaces are very rough with a self-affine behavior in the direction of the bedding and a stationary state in the direction normal to bedding.
We also investigate the effect of a non-Newtonian rheology on HF growth. We solve the problem of steadily moving semi-infinite HF driven by a Carreau fluid. We use a Gauss-Chebyshev method for elasticity combined with finite differences for lubrication flow and solve the resulting non-linear system with the Newton Raphson method. The solution exhibits four asymptotic regions: a linear elastic fracture mechanics (lefm) asymptote near the tip, high-shear rate Newtonian and power law asymptotes in an intermediate region and a low-shear Newtonian asymptote in the far field. For the same dimensionless toughness, the fluid lag is smaller than for a Newtonian fluid of low shear rate viscosity. We show that simpler rheological models (Ellis and power law) cannot capture the complete solution, which accounts for the full rheological behavior.

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Biological microorganisms swim with flagella and cilia that execute nonreciprocal motions for low Reynolds number (Re) propulsion in viscous fluids. This symmetry requirement is a consequence of Purcell's scallop theorem, which complicates the actuation scheme needed by microswimmers. However, most biomedically important fluids are non-Newtonian where the scallop theorem no longer holds. It should therefore be possible to realize a microswimmer that moves with reciprocal periodic body-shape changes in non-Newtonian fluids. Here we report a symmetric 'micro-scallop', a single-hinge microswimmer that can propel in shear thickening and shear thinning (non-Newtonian) fluids by reciprocal motion at low Re. Excellent agreement between our measurements and both numerical and analytical theoretical predictions indicates that the net propulsion is caused by modulation of the fluid viscosity upon varying the shear rate. This reciprocal swimming mechanism opens new possibilities in designing biomedical microdevices that can propel by a simple actuation scheme in non-Newtonian biological fluids.

Locomotion is an essential feature for survival of many organisms, and it is also a technical requirement for untethered biomedical microdevices to operate inside the human body for targeted drug delivery and minimally invasive surgery. The overall goal of this thesis is to develop microdevices that are capable of actively propelling themselves through and performing tasks in complex biological fluids. Two major challenges are encountered: the complex fluidic environment and the small length scale. The first part of the thesis addresses the first challenge: the diverse rheological properties of biological fluids. Most biological fluids are non-Newtonian, and to move in these fluids, the propulsion scheme is highly dependent on the size of the device: 1) When it is comparable or smaller than the mesh size of the biopolymeric network, then the device experiences purely viscous drag, thus the schemes for propulsion in Newtonian fluids will still be effective. To study the active microrheology of porous biological fluids, a magnetic tweezer set-up is constructed to pull magnetic beads of varying size through the porcine vitreous. It is found that for unhindered propulsion through the vitreous, nanodevices with a cross-section of less than 500 nm are needed. 2) When the size of the device is larger than the fluidic mesh size, non-Newtonian properties of the fluid become important and can be utilized for propulsion. We design the first symmetric micro-swimmer actuated by reciprocal motion and demonstrate its propulsion in biologically-relevant fluids. A larger scale low Reynolds number swimmer model is also built and excellent agreement between measurements and theory indicates that the net propulsion is caused by modulation of the fluid viscosity upon varying the shear rate. It opens new possibilities of using a simple actuation scheme for propulsion in non-Newtonian biological fluids. The second challenge is the small size of the device which severely limits the choice of available miniaturized actuators. We explore a novel wireless ultrasonic actuation scheme and a prototype is tested for application in the urinary tract. In order to test the prototype, a general technique is first developed to build a phantom of the human kidney. Based on high resolution medical imaging data, a three dimensional (3D) model is constructed using 3D printing and polymer casting techniques. The phantom not only faithfully reproduces the anatomical structural details, but also mimics the mechanical properties of the real tissue. To address the challenge of powering untethered microdevices, a novel bubble array streaming surface (BASS) actuator is developed. Under ultrasonic excitation, the oscillation of micro gas bubbles results in acoustic streaming and provides a propulsive force that drives the device. Ultrasonic actuators with different bubble sizes are fabricated, and individual driving frequency and propulsive force are measured. The actuator operates as an end-effector of a miniaturized endoscope, which has a cross-sectional side length of only 1 mm, thinner than the endoscopes currently in use. The tip of the end-effector is equipped with a miniaturized camera, its orientation is controlled wirelessly by the actuator, and active cystoscopy is successfully performed inside a rabbit bladder. Miniaturized medical instruments and micro-robots will benefit from the wireless actuation schemes demonstrated herein.