Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 188–194. Introduction The purposes of this paper are to present theoretical and experimental evidence for occurrence of macroscopic instabilities in displacement of one viscous fluid by another immiscible with it through a uniform porous medium and to compare available experimental data with some predictions of a theory of instability developed by the first author. The instabilities are referred to as macroscopic in the sense that spatially quasi-sinusoidal, growing fingers of the displacing liquid are formed, the width and peak-to-peak separation (wavelength) of which is large relative to a characteristic length of the particular permeable medium such as grain size. Visual models of two kinds have been used to obtain observations: displacement of oil by water-glycerine solutions through the flow channel formed by closely spaced parallel plates and displacement of oil by water with and without initial interstitial water through unconsolidated glass powder packs, employing the technique of matching indices of refraction. In all cases we have observed macroscopic instabilities or fingers under conditions predicted by the theory to be favorable for their occurrence. The phenomenon discussed here is not the production of streamers due to gross inhomogeneities such as permeability stratification of the porous medium.It is our object to show, on the contrary, that a quantitative prediction of finger spacing is possible in a porous medium known to be macroscopically homogeneous and isotropic throughout. The importance of the phenomenon in its influence on the configuration of oil and water with respect to oil production behavior was noted earlier by Engelberts and Klinkenberg who coined the term "viscous fingering". Theory Necessary and Sufficient Conditions for Instability and Initial Kinematics There are several levels of increasing complexity in the theoretical description of instability of fluid displacements in permeable media. Of these, the simplest description, adapted to low permeability systems, is selected for presentation. More inclusive descriptions are reserved for separate publication.

We study a genuine Brownian motor by hard disk molecular dynamics and calculate analytically its properties, including its drift speed and thermal conductivity, from microscopic theory.

We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced by C. Van

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We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals.Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale δn > 0, which converges to zero as the number of particles n tends to infinity.We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on both the theory of λ-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δn-regularised version of the Groma-Balogh equations, and a priori estimates for the Groma-Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n → ∞ we require δn to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δn converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations.These results show how the validity of the Groma-Balogh equations as the limit of manyparticle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.