G. M. Homsy

Two-phase displacement in Hele Shaw cells: theory

A theory describing two-phase displacement in the gap between closely spaced planes is developed. The main assumptions of the theory are that the displaced fluid wets the walls, and that the capillary number Ca and the ratio of gap width to transverse characteristic length e are both small. Relatively mild restrictions apply to the ratio M of viscosities of displacing to displaced fluids; in particular the theory holds for M = o ( Ca −1/3 ). We formulate the theory as a double asymptotic expansion in the small parameters e and Ca 1/3 . The expansion in e is uniform while that in Ca 1/3 is not, necessitating the use of matched asymptotic expansions. The previous work of Bretherton (1961) is clarified and extended, and both the form and the constants in the effective boundary condition of Chouke, van Meurs & van der Poel (1959) and of Saffman & Taylor (1958) are determined.

Stokes flow through periodic arrays of spheres

We treat the problem of slow flow through a periodic array of spheres. Our interest is in the drag force exerted on the array, and hence the permeability of such arrays. It is shown to be convenient to formulate the problem as a set of two-dimensional integral equations for the unknown surface stress vector, thus lowering the dimension of the problem. This set is solved numerically to obtain the drag as a function of particle concentration and packing characteristics. Results are given over the full concentration range for simple cubic, body-centred cubic and face-centred cubic arrays and these agree well with previous limited experimental, asymptotic and numerical results.

Stability of miscible displacements in porous media: Rectilinear flow

A theoretical treatment of the stability of miscible displacement in a porous medium is presented. For a rectilinear displacement process, since the base state of uniform velocity and a dispersive concentration profile is time dependent, we make the quasi‐steady‐state approximation that the base state evolves slowly with respect to the growth of disturbances, leading to predictions of the growth rate. Comparison of results with initial value solutions of the partial differential equations shows that, excluding short times, there is good agreement between the two theories. Comparison of the theory with several experiments in the literature indicates that the theory gives a good prediction of the most dangerous wavelength of unstable fingers. An approximate analysis for transversely anisotropic media has elucidated the role of transverse dispersion in controlling the length scale of fingers.

Simulation of nonlinear viscous fingering in miscible displacement

The nonlinear behavior of viscous fingering in miscible displacements is studied. A Fourier spectral method is used as the basic scheme for numerical simulation. In its simplest formulation, the problem can be reduced to two algebraic equations for flow quantities and a first‐order ordinary differential equation in time for the concentration. There are two parameters, the Peclet number (Pe) and mobility ratio (M), that determine the stability characteristics. The result shows that at short times, both the growth rate and the wavelength of fingers are in good agreement with predictions from our previous linear stability theory. However, as the time goes on, the nonlinear behavior of fingers becomes important. There are always a few dominant fingers that spread and shield the growth of other fingers. The spreading and shielding effects are caused by a spanwise secondary instability, and are aided by the transverse dispersion. It is shown that once a finger becomes large enough, the concentration gradient of...

Evaporating menisci of wetting fluids

Abstract We model the transport processes occurring in a horizontal evaporating meniscus which is affected by both capillarity and by multilayer adsorption. Perturbation theory is used to describe the profile change relative to the static isothermal profile. We solve for this deviation profile, as well as for the evaporative flux from the interface, for a range of values appropriate to the nondimensional parameters in the model. The results clearly demonstrate that large heat fluxes can occur in the transition region between the capillary meniscus and the adsorbed layer. Moreover, the deviation profile predicts a thinning in the meniscus region with a shift in the apparent zero toward the gap, a much thinner nonevaporating adsorbed film, and an increase in apparent contact angle.

Thermophoretic deposition of small particles in laminar tube flow

Abstract The deposition of small particles due to a migration velocity dependent upon temperature gradients (thermophoresis) is studied theoretically for laminar tube flow. The prototype boundary value problem is that in which the wall temperature is suddenly decreased at a given axial position. Because of the ultimate relaxation of the temperature gradient, only some fraction of the particles initially present will deposit on the walls. A Leveque solution for short distances is used to establish a scaling for the deposition efficiency. The effects of weak Brownian diffusion are treated rigorously, and limiting efficiencies for long tubes are determined numerically. Some interesting contrasts between this problem and that for aerosol deposition due solely to diffusion are discussed.

High Marangoni number convection in a square cavity

Steady thermocapillary flow is examined in a square two‐dimensional cavity with a single free surface and differentially heated side walls. The numerical solutions are obtained with a finite difference method applied to a streamfunction‐temperature formulation. This work investigates the Prandtl number dependence, structure, and stability of high Marangoni number flow. It is found that the character of thermocapillary flow is highly sensitive to the value of the Prandtl number over a range of Marangoni numbers exceeding 1×105 for 1≤Pr≤50, the magnitude of the flow showing nonmonotonic dependence on the Marangoni number for Pr≤∼10. A complete structural analogy is observed between flow in a cavity driven by a moving lid and thermocapillary flow in the boundary layer limit, and it is found that all the solutions, spanning a wide range of Marangoni and Prandtl numbers, are linearly stable to a restricted class of disturbances.

Stability of Newtonian and viscoelastic dynamic contact lines

The stability of the moving contact line is examined for both Newtonian and viscoelastic fluids. Two methods for relieving the contact line singularity are chosen: matching the free surface profile to a precursor film of thickness b, and introducing slip at the solid substrate. The linear stability of the Newtonian capillary ridge with the precursor film model was first examined by Troian et al. [Europhys. Lett. 10, 25 (1989)]. Using energy analysis, we show that in this case the stability of the advancing capillary ridge is governed by rearrangement of fluid in the flow direction, whereby thicker regions develop that advance more rapidly under the influence of a body force. In addition, we solve the Newtonian linear stability problem for the slip model and obtain results very similar to those from the precursor film model. Interestingly, stability results for the two models compare quantitatively when the precursor film thickness b is numerically equal to the slip parameter α. With the slip model, it is ...

HINDERED SETTLING AND HYDRODYNAMIC DISPERSION IN QUIESCENT SEDIMENTING SUSPENSIONS

Abstract The motion of an individual sphere settling in the midst of a suspension of like spheres has been examined experimentally for suspensions with volume concentrations, φ, of 2.5–10%, under creeping flow conditions and in the absence of Brownian motion. In the experiments, silvered glass spheres were tracked in optically transparent suspensions of glass beads. Arrival times measured at a series of horizontal planes were converted into average settling speeds. These average speeds yield the hindered settling speed as a function of concentration. The hindered settling speed, normalized by the isolated sphere settling speed, exhibits a 1 − 4φ + 8φ 2 dependence for the range of concentrations investigated. The settling speed fluctuations are quite large, ranging up to 46% of the average, and have long-time (large settling distance) behavior characteristic of a Fickian diffusion process. Dispersion coefficients have therefore been determined from the asymptotic dependence upon settling distance of the variance in settling speed. These coefficients scale with the product of the hindered settling speed and the sphere radius. The dimensionless dispersion coefficients, all O(1) , increase with concentration for φ , then slightly decrease at higher concentrations. Verification of the scaling through the use of two particle sizes, care taken to mix the suspensions to random, uniform initial conditions, and the robustness of the statistics over many realizations preclude the possibility of this phenomenon being an experimental artifact and support the hypothesis that hydrodynamic dispersion of suspended particles will result from viscous interactions between the particles.

Nonlinear analysis of buoyant convection in binary solidification with application to channel formation

We consider the problem of nonlinear thermal-solutal convection in the mushy zone accompanying unstable directional solidification of binary systems. Attention is focused on possible nonlinear mechanisms of chimney formation leading to the occurrence of freckles in solid castings, and in particular the coupling between the convection and the resulting porosity of the mush. We make analytical progress by considering the case of small growth Peclet number, δ, small departures from the eutectic point, and infinite Lewis number. Our linear stability results indicate a small O (δ) shift in the critical Darcy-Rayleigh number, in accord with previous analyses. We find that nonlinear two-dimensional rolls may be either sub- or supercritical, depending upon a single parameter combining the magnitude of the dependence of mush permeability on solids fraction and the variations in solids fraction owing to melting or freezing. A critical value of this combined parameter is given for the transition from supercritical to subcritical rolls. Three-dimensional hexagons are found to be transcritical, with branches corresponding to upflow and lower porosity in either the centres or boundaries of the cells. These general results are discussed in relation to experimental observations and are found to be in general qualitative agreement with them.