Joel Koplik

Theory of dynamic permeability and tortuosity in fluid-saturated porous media

We consider the response of a Newtonian fluid, saturating the pore space of a rigid isotropic porous medium, subjected to an infinitesimal oscillatory pressure gradient across the sample. We derive the analytic properties of the linear response function as well as the high- and low-frequency limits. In so doing we present a new and well-defined parameter Λ, which enters the high-frequency limit, characteristic of dynamically connected pore sizes. Using these results we construct a simple model for the response in terms of the exact high- and low-frequency parameters; the model is very successful when compared with direct numerical simulations on large lattices with randomly varying tube radii. We demonstrate the relevance of these results to the acoustic properties of non-rigid porous media, and we show how the dynamic permeability/tortuosity can be measured using superfluid 4 He as the pore fluid. We derive the expected response in the case that the internal walls of the pore space are fractal in character.

Pattern selection in fingered growth phenomena

Abstract A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manne...

Molecular Dynamics of Fluid Flow at Solid Surfaces

Molecular dynamics techniques are used to study the microscopic aspects of several slow viscous flows past a solid wall, where both fluid and wall have a molecular structure. Systems of several thousand molecules are found to exhibit reasonable continuum behavior, albeit with significant thermal fluctuations. In Couette and Poiseuille flow of liquids it is found that the no‐slip boundary condition arises naturally as a consequence of molecular roughness, and that the velocity and stress fields agree with the solutions of the Stokes equations. At lower densities slip appears, which can be incorporated into a flow‐independent slip‐length boundary condition. The trajectories of individual molecules in Poiseuille flow are examined, and it is also found that their average behavior is given by Taylor–Aris hydrodynamic dispersion. An immiscible two‐fluid system is simulated by a species‐dependent intermolecular interaction. A static meniscus is observed whose contact angle agrees with simple estimates and, when ...

Capillary displacement and percolation in porous media

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

Boundary conditions at a fluid-solid interface.

We study the boundary conditions at a fluid-solid interface using molecular dynamics simulations covering a broad range of fluid-solid interactions and fluid densities and both simple and chain-molecule fluids. The slip length is shown to be independent of the type of flow, but rather is related to the fluid organization near the solid, as governed by the fluid-solid molecular interactions.

Viscosity renormalization in the Brinkman equation

The Brinkman equation purports to describe low‐Reynolds‐number flow in porous media in situations where velocity gradients are non‐negligible. The equation involves modifying the usual Darcy law by the addition of a standard viscosity term whose coefficient is usually identified with the pure‐fluid viscosity. It is argued instead that the porous medium induces a renormalization of viscosity, which is calculated in the dilute limit and separately in a self‐consistent approximation. The effective Brinkman viscosity is found to decrease from the pore‐fluid value. The calculation fails at low porosity but agrees at least in part with experiment. In addition, the relationship between the Brinkman equation and the phenomenological boundary condition of Beavers and Joseph is discussed and it is pointed out that their experimental configuration provides a simple means of measuring viscosity renormalization.

Lattice Boltzmann method for non-Newtonian (power-law) fluids.

We study an ad hoc extension of the lattice Boltzmann method that allows the simulation of non-Newtonian fluids described by generalized Newtonian models. We extensively test the accuracy of the method for the case of shear-thinning and shear-thickening truncated power-law fluids in the parallel plate geometry, and show that the relative error compared to analytical solutions decays approximately linear with the lattice resolution. Finally, we also tested the method in the reentrant-flow geometry, in which the shear rate is no longer a scalar and the presence of two singular points requires high accuracy in order to obtain satisfactory resolution in the local stress near these points. In this geometry, we also found excellent agreement with the solutions obtained by standard finite-element methods, and the agreement improves with higher lattice resolution.

Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions

The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions. The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients D y and D z , respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ 2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of D y on ϕ, from ϕ 2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for D z but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.

Flow channeling in a single fracture induced by shear displacement

The effect on the transport properties of fractures of a relative shear displacement

Suppression of coalescence by shear and temperature gradients

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