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Three different models, relating to the study of microswimmers immersed in a low Reynolds number fluid, are presented. The underlying, mathematical concepts employed in each are developed using singularity methods of Stokes flow. The first topic concerns the motility of an artificial, three-sphere microswimmer with prescribed, non-reciprocal, internal forces. The swimmer progresses through a low Reynolds number, nonlinear, viscoelastic medium. The model developed illustrates that the presence of the viscoelastic rheology, when compared to a Newtonian environment, increases both the net displacement and swimming efficiency of the microswimmer. The second area concerns biological microswimming, modelling a sperm cell with a hyperactive waveform (vigorous, asymmetric beating), bound to the epithelial walls of the female, reproductive tract. Using resistive-force theory, the model concludes that, for certain regions in parameter space, hyperactivated sperm cells can induce mechanical forces that pull the cell away from the wall binding. This appears to occur via the regulation of the beat amplitude, wavenumber and beat asymmetry. The next topic presents a novel generalisation of slender-body theory that is capable of calculating the approximate flow field around a long, thin, slender body with circular cross sections that vary arbitrarily in radius along a curvilinear centre-line. New, permissible, slender-body shapes include a tapered flagellum and those with ribbed, wave-like structures. Finally, the detailed analytics of the generalised, slender-body theory are exploited to develop a numerical implementation capable of simulating a wider range of slender-body geometries compared to previous studies in the field.

Abstract: In predicting oil and gas recovery, researchers develop models primarily focusing on macro- or milli-scale flows by applying Navier-Stokes types solutions and no-slip boundary conditions. They compare or cross-check their models to known experimental results available in the literature for larger pore scales. In contrast, much less experimental data is available to use in modelling nanoscale flows, which is very important in oil and gas recovery applications. Moreover, porous media flow in smaller scales is also important in other fields of science, for example, in chromatography (with smaller packing particles), nanofluidic diode, fuel cells and other devices. Few theoretical and experimental studies are found in the literature on fluid flow in nanofluidic structures. However, a wide range of flow enhancement results with up to several orders of magnitude discrepancies is present between theoretical and experimental works. Therefore, realizing the need for more experimental data in the case of smaller-scale flows, we focused on studying the flow behaviour of aqueous and organic viscous liquids through silica-based micro and nanoporous structures with the fundamental concern of whether the no-slip boundary condition, a macroscale assumption, can correctly describe the micro and nanoscale flow properties for various combinations of fluid-solid surface interaction. The effect of pore sizes in porous media and the surface interaction between a viscous liquid and porous bed material on micro- and nanofluidic flow phenomena is reported here. We develop micro and nanoporous media by packing nonporous silica microspheres (300-2000 nm particle diameter) inside fused silica microcapillary tubes. The interconnected pores among the silica particles form the micro- or nanofluidic channels for fluid flow. A time-of-flight, photobleaching velocity measurement technique is employed to determine the linear flow velocity of a viscous liquid flowing through a porous media. Throughout the applied pressure range (100-6000 psi), the Newtonian fluids, irrespective of their wetting property, exhibit a laminar flow pattern (Re <

A high-performance Discrete-Vortex Method (DVM) is successfully developed and implemented to directly simulate two-dimensional low to medium-Reynolds number flows around multiple arbitrarily shaped bodies undergoing either prescribed or free motions. The deterministic Viscous-Vortex-Domain (VVD) formulation is adopted to simulate vorticity diffusion. Through the use of CPU and Graphics-Processing-Unit (GPU) parallel computing, significant speedup of the simulation compared to a serial implementation on a CPU is achieved. The validity of the present DVM simulation is confirmed by comparing the present results with published ones for a variety of test cases. The current implementation of DVM has been used to study two novel flow problems of practical interest and has led to significant findings. First, the full and partial ground effects on the lift generation of a flapping (air)foil in normal hovering mode are investigated. To achieve full ground effect, the foil of chord c is made to hover above the center of a finite-sized platform of length 10c. The computed force-enhancement, force-reduction, and force-recovery regimes at low, medium, and high ground clearances are observed to be in line with existing literature. This research puts special focus on partial ground effects when the foil is hovering near the edge of the platform. Lift-modifying mechanisms not previously observed under full ground effect have been discovered. When stroke reversal of the flapping occurs near the edge of the platform, a relatively stationary strong vortex may form above the platform edge. This strong vortex can either increase or decrease the instantaneous lift force on the foil depending on the position of the foil relative to the platform edge. Further, the platform edge may lead to the formation of an additional vortex pair which increases the instantaneous lift force as the foil sweeps past the edge under certain suitable conditions. Lastly, the platform edge can lead to the formation of a reverse von Kármán vortex street that extends well below the stroke plane under suitable geometric arrangements.Second, the flow past a Bach-type vertical-axis wind or current turbine is simulated using the DVM at a Reynolds number of 1,500. The main purpose of the study is to evaluate the suitability of Bach-type turbines for use as micro-scale energy harvesters that can be applied to power, for example, sensor nodes of a Wireless Sensor Network. Through simulations, the maximum power coefficient of the turbine operating at a prescribed constant tip-speed ratio is found to be 0.18, which is comparable to the performance of a turbine of the same geometry at much higher Reynolds numbers. This indicates that there is only minimal performance penalty for miniaturization. The angular velocity of the turbine has a strong influence on the evolution of vortical flow structures. A new wake-capturing mechanism that boosts the performance of the turbine is discovered from the simulations for a certain range of tip-speed ratios where the vortex shed by the advancing blade helps drive the returning blade. In addition to the condition of prescribed rotation, free rotation of a steel Bach-type turbine under a steady current in water is also investigated. Significant fluctuation in angular velocity over one period of rotation is observed. This speed fluctuation is found to be detrimental to energy extraction, reducing the maximum power coefficient to approximately 0.16. This level of power-generation capability implies that such micro-scale turbines can significantly extend the life expectancy of a wireless sensor node or even maintain the node in a low-power state indefinitely.

We analyze the performance of the Method of Regularized Stokeslets (MRS) and the Method of Auxiliary Regularized Stokeslets (MARS) in computing the forces necessary to translate a sphere with unit velocity in Stokes flow. In particular, we explore the dependence of local and global force calculations on various parameters associated with each method. The parameters we varied include the regularization parameter, the discretization of the sphere, and the spread and placement of the auxiliary Stokeslets (MARS only). One challenge when using the MRS is that there is no systematic way to choose the regularization parameter, and the error is sensitive to this choice. In the literature it is stated that, compared to the MRS, the MARS weakens the error dependence on the choice of regularization parameter. We found this to be true in some cases, but not in others. Specifically, the dependence is weakened when comparing the 1-norm of the global force error and 2-norm of the local force error. This behavior is not seen with the max-norm of the local force error. In addition, with the MARS, there is a strong dependence of error on the normalized patch length which controls the spread of the auxiliary Stokeslets. We find the conditions needed to optimize the MARS method to outperform the MRS on the translating sphere problem.

Two-phase flows are ubiquitous, from natural and domestic environments to industrial settings. However, due to their complexity, modeling these fluid systems remains a challenge from both the perspective of fundamental questions on the dynamics of an individual, smooth interface, and the perspective of integral analyses, which involve averaging of the conservation laws over large domains, thereby missing local details of the flow. In this work, we consider a set of five problems concerning the linear and non-linear dynamics of an interface or free surface and the study of cavitation inception. Analyses are carried out by assuming the fluid motion to be irrotational, that is, with zero vorticity, and the fluids to be viscous, although results from rotational analyses are presented for the purpose of comparison. The problems considered here are the following: First, we analyze the non-linear deformation and break-up of a bubble or drop immersed in a uniaxial extensional flow of an incompressible viscous fluid. The method of viscous potential flow, in which the flow field is irrotational and viscosity enters through the balance of normal stresses at the interface, is used in the analysis. The governing equations are solved numerically to track the motion of the interface by coupling a boundary element method with a time-integration routine. When break-up occurs, the break-up time computed here is compared with results obtained elsewhere from numerical simulations of the Navier–Stokes equations, which thus keeps vorticity in the analysis, for several combinations of the relevant dimensionless parameters of the problem. For the bubble, for Weber numbers 3 [less than or equal] We [less than or equal] 6, predictions from viscous potential flow shows good agreement with the results from the Navier–Stokes equations for the bubble break-up time, whereas for larger We, the former underpredicts the results given by the latter. Including viscosity increases the break-up time with respect to the inviscid case. For the drop, as expected, increasing the viscous effects of the irrotational motion produces large, elongated drops that take longer to break up in comparison with results for inviscid fluids. In the second problem, we compute the force acting on a spherical bubble of variable radius moving within a liquid with an outer spherical boundary. Viscous potential flow and the dissipation method, which is another purely irrotational approach stemming from the mechanical energy equation, are both systematically implemented. This exposes the role of the choice of the outer boundary condition for the stress on the drag, an issue not explained in the literature known to us. By means of the well-known “cell-model” analysis, the results for the drag are then applied to the case of a swarm of rising bubbles having a certain void fraction. Computations from the dissipation method for the drag coefficient and rise velocity for a bubble swarm agree with numerical solutions; evaluation against experimental data for high Reynolds and low Weber numbers shows that all the models considered, including those given in the literature, overpredict the bubble swarm rise velocity. In the next two problems, we apply the analysis of viscous potential flow and the dissipation method to study the linear dynamics of waves of “small” amplitude acting either on a plane or on a spherical interface separating a liquid from a dynamically inactive fluid. It is shown that the viscous irrational theories exhibit the features of the wave dynamics by comparing with the exact solution. The range of parameters for which good agreement with the exact solution exists is presented. The general trend shows that for long waves the dissipation method results in the best approximation, whereas for short waves, even for very viscous liquids, viscous potential flow demonstrates better agreement. Finally, the problem of cavitation inception for the flow of a viscous liquid past a stationary sphere is studied by means of the theory of stress-induced cavitation. The flow field for a single phase needed in the analysis is found from three different methods, namely, the numerical solution of the Navier–Stokes equations, the irrotational motion of a viscous fluid, and, in the limit of no inertia, the Stokes flow formulation. The new predictions are then compared with those obtained from the classical pressure criterion. The main finding is that at a fixed cavitation number more viscous liquids are at greater risk to cavitation.