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Register a new userEffective rheology of two-phase flow in a capillary fiber bundle model
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We investigate the effective rheology of two-phase flow in a bundle of parallel capillary tubes carrying two immiscible fluids under an external pressure drop. The diameter of each tube varies along its length and the corresponding capillary threshold pressures are considered to be distributed randomly according to a uniform probability distribution. We demonstrate through analytical calculations that a transition from a linear Darcy regime to a non-linear behavior occurs while decreasing the pressure drop $Delta P$, where the total flow rate $langle Q rangle$ varies with $Delta P$ with an exponent $2$. This exponent for the non-linear regime changes when a lower cut-off $P_m$ is introduced in the threshold distribution. We demonstrate analytically that, in the limit where $Delta P$ approaches $P_m$, the flow rate scales as $langle Q rangle sim (|Delta P|-P_m)^{3/2}$. We have also provided some numerical results in support to our analytical findings.
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A concentrated, vertical monolayer of identical spherical squirmers, which may be bottom-heavy, and which are subjected to a linear shear flow, is modelled computationally by two different methods: Stokesian dynamics, and a lubrication-theory-based method. Inertia is negligible. The aim is to compute the effective shear viscosity and, where possible, the normal stress differences as functions of the areal fraction of spheres $phi$, the squirming parameter $beta$ (proportional to the ratio of a squirmers active stresslet to its swimming speed), the ratio $Sq$ of swimming speed to a typical speed of the shear flow, the bottom-heaviness parameter $G_{bh}$, the angle $alpha$ that the shear flow makes with the horizontal, and two parameters that define the repulsive force that is required computationally to prevent the squirmers from overlapping when their distance apart is less than a critical value $epsilon a$, where $epsilon$ is very small and $a$ is the sphere radius. The Stokesian dynamics method allows the rheological quantities to be computed for values of $phi$ up to $0.75$; the lubrication-theory method can be used for $phi> 0.5$. A major finding of this work is that, despite very different assumptions, the two methods of computation give overlapping results for viscosity as a function of $phi$ in the range $0.5 < phi < 0.75$. This suggests that lubrication theory, based on near-field interactions alone, contains most of the relevant physics, and that taking account of interactions with more distant particles than the nearest is not essential to describe the dominant physics.
Experimental and numerical investigations are performed to provide an assessment of the transport behavior of an ultrasonic oscillatory two-phase flow in a microchannel. The work is inspired by the flow observed in an innovative ultrasonic fabric drying device using a piezoelectric bimorph transducer with microchannels, where a water-air two-phase flow is transported by harmonically oscillating microchannels. The flow exhibits highly unsteady behavior as the water and air interact with each other during the vibration cycles, making it significantly different from the well-studied steady flow in microchannels. The computational fluid dynamics (CFD) modeling is realized by combing the turbulence Reynolds-averaged Navier-Stokes (RANS) k-${omega}$ model with the phase-field method to resolve the dynamics of the two-phase flow. The numerical results are qualitatively validated by the experiment. Through parametric studies, we specifically examined the effects of vibration conditions (i.e., frequency and amplitude), microchannel taper angle, and wall surface contact angle (i.e., wettability) on the flow rate through the microchannel. The results will advance the potential applications where oscillatory or general unsteady microchannel two-phase flows may be present.
We present a theoretical framework for immiscible incompressible two-phase flow in homogeneous porous media that connects the distribution of local fluid velocities to the average seepage velocities. By dividing the pore area along a cross-section transversal to the average flow direction up into differential areas associated with the local flow velocities, we construct a distribution function that allows us not only to re-establish existing relationships between the seepage velocities of the immiscible fluids, but also to find new relations between their higher moments. We support and demonstrate the formalism through numerical simulations using a dynamic pore-network model for immiscible two-phase flow with two- and three-dimensional pore networks. Our numerical results are in agreement with the theoretical considerations.
Harnessing fluidic instabilities to produce structures with robust and regular properties has recently emerged as a new fabrication paradigm. This is exemplified in the work of Gumennik et al. [Nat. Comm. 4:2216, DOI: 10.1038/ncomms3216, (2013)], in which the authors fabricate silicon spheres by feeding a silicon-in-silica co-axial fiber into a flame. Following the localized melting of the silicon, a capillary instability of the silicon-silica interface induces the formation of uniform silicon spheres. Here, we try to unravel the physical mechanisms at play in selecting the size of these particles, which was notably observed by Gumennik et al. to vary monotonically with the speed at which the fiber is fed into the flame. Using a simplified model derived from standard long-wavelength approximations, we show that linear stability analysis strikingly fails at predicting the selected particle size. Nonetheless, nonlinear simulations of the simplified model do recover the particle size observed in experiments, without any adjustable parameters. This shows that the formation of the silicon spheres in this system is an intrinsically nonlinear process that has little in common with the loss of stability of the underlying base flow solution.
In this investigation we revisit the concept of effective free surfaces arising in the solution of the time-averaged fluid dynamics equations in the presence of free boundaries. This work is motivated by applications of the optimization and optimal control theory to problems involving free surfaces, where the time-dependent formulations lead to many technical difficulties which are however alleviated when steady governing equations are used instead. By introducing a number of precisely stated assumptions, we develop and validate an approach in which the interface between the different phases, understood in the time-averaged sense, is sharp. In the proposed formulation the terms representing the fluctuations of the free boundaries and of the hydrodynamic quantities appear as boundary conditions on the effective surface and require suitable closure models. As a simple model problem we consider impingement of free-falling droplets onto a fluid in a pool with a free surface, and a simple algebraic closure model is proposed for this system. The resulting averaged equations are of the free-boundary type and an efficient computational approach based on shape optimization formulation is developed for their solution. The computed effective surfaces exhibit consistent dependence on the problem parameters and compare favorably with the results obtained when the data from the actual time-dependent problem is used in lieu of the closure model.
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