Stokes Flow Limit Research Articles

Steady shape of a miscible bubble rising below an inclined wall at low Reynolds numbers

When a buoyant volume of fluid impinges a horizontal boundary, it spreads symmetrically in the Stokes flow limit. We investigate the low Reynolds number spreading of a low-viscosity volume of fluid, hereafter called a “bubble”, in the limit that the fluids are miscible and interfacial tension can be neglected; the symmetry of the bubble's spreading is broken by allowing the surface to have a finite slope. We use laboratory experiments, parametrized boundary integral numerical calculations, and a scaling argument to show that there exists a steady bubble shape with an aspect ratio (ratio of the semi-axes of the bubble in the horizontal plane) around 1.4–1.6 for slope angles ranging from 5 to 35°. The existence of a steady shape, together with a rapid shift in aspect ratio as the slope angle ϕ increases from 0, suggests a continuous phase transition caused by a loss of symmetry when the finite slope is introduced. We show that the time required for the bubble to reach this steady shape and the constant aspect ratio over the range 10 < ϕ < 35 support the analogy to a phase transition.

FULL NEWTON LATTICE BOLTZMANN METHOD FOR TIME-STEADY FLOWS USING A DIRECT LINEAR SOLVER

A full Newton lattice Boltzmann method is developed for time-steady flows. The general method involves the construction of a residual form for the time-steady, nonlinear Boltzmann equation in terms of the probability distribution. Bounce-back boundary conditions are also incorporated into the residual form. Newton's method is employed to solve the resulting system of non-linear equations. At each Newton iteration, the sparse, banded, Jacobian matrix is formed from the dependencies of the non-linear residuals on the components of the particle distribution. The resulting linear system of equations is solved using a direct solver designed for sparse, banded matrices. For the Stokes flow limit, only one matrix solve is required. Two dimensional flow about a periodic array of disks is simulated as a proof of principle, and the numerical efficiency is carefully assessed. For the case of Stokes flow (Re = 0) with resolution 251×251, the proposed method performs more than 100 times faster than a standard, fully explicit implementation.

Steady Propagation of a Liquid Plug in a Two-Dimensional Channel

In this study, we investigate the steady propagation of a liquid plug within a two-dimensional channel lined by a uniform, thin liquid film. The Navier-Stokes equations with free-surface boundary conditions are solved using the finite volume numerical scheme. We examine the effect of varying plug propagation speed and plug length in both the Stokes flow limit and for finite Reynolds number (Re). For a fixed plug length, the trailing film thickness increases with plug propagation speed. If the plug length is greater than the channel width, the trailing film thickness agrees with previous theories for semi-infinite bubble propagation. As the plug length decreases below the channel width, the trailing film thickness decreases, and for finite Re there is significant interaction between the leading and trailing menisci and their local flow effects. A recirculation flow forms inside the plug core and is skewed towards the rear meniscus as Re increases. The recirculation velocity between both tips decreases with the plug length. The macroscopic pressure gradient, which is the pressure drop between the leading and trailing gas phases divided by the plug length, is a function of U and U2, where U is the plug propagation speed, when the fluid property and the channel geometry are fixed. The U2 term becomes dominant at small values of the plug length. A capillary wave develops at the front meniscus, with an amplitude that increases with Re, and this causes large local changes in wall shear stresses and pressures.

Determination of surface shear viscosity via deep-channel flow with inertia

Results of an experimental and computational study of the flow in an annular region bounded by stationary inner and outer cylinders and driven by the rotation of the floor are presented. The top is a flat air/water interface, covered by an insoluble monolayer. We develop a technique to determine the surface shear viscosity from azimuthal velocity measurements at the interface which extends the range of surface shear viscosity that can be measured using a deep-channel viscometer in the usual Stokes flow regime by exploiting flow inertia. A Navier–Stokes-based model of bulk flow coupled to a Newtonian interface that has surface shear viscosity as the only interfacial property is developed. This is achieved by restricting the flow to regimes where the surface radial velocity vanishes. The use of inertia results in an improved signal-to-noise ratio of the azimuthal velocity measurements by an order of magnitude beyond that available in the Stokes flow limit. Measurements on vitamin K1 and stearic acid monolayers were performed, and their surface shear viscosities over a range of concentrations are determined and found to be in agreement with data in the literature.