The movement of water droplets in clouds around the nose of an atmospheric research aircraft

The dynamic interaction between droplets and the airflow around the hemispherical nose of an aircraft was evaluated. The effect of the aircraft nose on droplet sampling for cloud research is explained. The proportion of different droplet sizes and their concentration at each point around the aircraft nose were determined. In a cloud, interaction between droplets is negligible. Each particle acts, for the calculation of the forces applied to it, as if it is alone in the air. The airflow carrying the droplets, on the average, is not influenced by their presence. The trajectory of each droplet was studied separately after calculating dry airflow. Concentrations were found with a Lagrangian method, using two trajectories computed directly close to one another. Theory confirms that to within 3% experimentally measured concentrations are representative of those in a cloud

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can be seen as a generalization of Allaire's result in [Arch. Rational Mech. Analysis 113 (1991), 209-259] who treated the case of motionless, periodically distributed balls. Our proof is based on slightly simpler, though similar homogenization techniques, except that we avoid the periodicity assumption and use instead the phase-space empirical measure for the particle system. Similar equations are used for describing the fluid phase in various models for sprays

Transport Phenomena and Structuring in Shear Flow of Suspensions near Solid Walls

In this paper we apply the lattice-Boltzmann method and an extension to particle suspensions as introduced by Ladd et al. to study transport phenomena and structuring effects of particles suspended in a fluid near sheared solid walls. We find that a particle free region arises near walls, which has a width depending on the shear rate and the particle concentration. The wall causes the formation of parallel particle layers at low concentrations, where the number of particles per layer decreases with increasing distance to the wall.Comment: 14 pages, 14 figure

Sedimentation and Flow Through Porous Media: Simulating Dynamically Coupled Discrete and Continuum Phases

We describe a method to address efficiently problems of two-phase flow in the regime of low particle Reynolds number and negligible Brownian motion. One of the phases is an incompressible continuous fluid and the other a discrete particulate phase which we simulate by following the motion of single particles. Interactions between the phases are taken into account using locally defined drag forces. We apply our method to the problem of flow through random media at high porosity where we find good agreement to theoretical expectations for the functional dependence of the pressure drop on the solid volume fraction. We undertake further validations on systems undergoing gravity induced sedimentation.Comment: 22 pages REVTEX, figures separately in uudecoded, compressed postscript format - alternatively e-mail '[email protected]' for hardcopies

Numerical calculation of singular integrals related to Hankel transform

AbstractThe singular integral S = ∫0∞ f(x)e−x J0(ωx) dx, related to the Hankel transform of order 0, is calculated numerically by using an integral expression for the Bessel function of order zero, J0. With the assumptions that the function f(x) is bounded and is analytic in some complex domain, the double integral obtained in this way is calculated by a combination of changes of variables and Gauss methods using Laguerre, Chebyshev and Legendre polynomials. The singular integral S′ = ∫0∞ f(x)e−x J1(ωx) dx is derived from S. The subroutines written in FORTRAN run very fast on a personal computer and give a relative precision better than 5 × 10−6

Collision and size evolution of drops in homogeneous isotropic turbulence

International audienceThis paper is devoted to the study of the evolution of the spectrum in a cloud due tocollisions between drops

Interaction of elastic bodies via surface forces. 1. Power-law attraction

We study theoretically the effect of finite elasticity on the attractive power-law interaction of two solids separated by a thin liquid (or gas) film. A new asymptotic technique is developed to determine the deformed shape of the surfaces and to calculate the elasticity contribution to the total force, i.e., an additional term present between the deformed bodies. Both the deformation and the elasticity contribution are found to be nonnegligible well before contact is reached, although they are of much shorter range than the surface force that caused them. This range can be characterized by a reference elasticity length, which depends on elastic constants and size of the solids, as well as on the attractive force that led to deformation. The total force vs separation profile for elastic surfaces is found to depend on how the measurements are made, namely, how the separation is detected: it can lead to either less or more attractive force compared with the case of rigid surfaces

Interaction of elastic bodies via surface forces 2. Exponential decay

Our goal is to study theoretically the effect of deformation on the exponentially decaying interaction of two elastic solids separated by a thin liquid film. The deformed shape of the surfaces and the contribution of the elasticity to the total force, i.e., an additional term present between elastic bodies, are calculated from continuum elastic theory via a new asymptotic technique. Both the deformation and the contribution of the elasticity to the force are found to be significant on the length scale over which the surface force acts. The surface deformation is exponentially decaying with a decay length equal to that of the original surface interaction. It is especially important for large and/or rapidly changing force. The contribution of the elasticity is also exponentially decaying, but with half the decay length. Its strength depends on the elastic constants and size of the solids and on the magnitude and gradient of the original surface force. Depending on how the separation is detected, it can appear either as an attractive or as a repulsive contribution to the force. Our results open the possibility of recalculating the measured force to the interaction free energy