J. M. Montanero

Computer simulation of uniformly heated granular fluids

Abstract Direct Monte Carlo simulations of the Enskog-Boltzmann equation for a spatially uniform system of smooth inelastic spheres are performed. In order to reach a steady state, the particles are assumed to be under the action of an external driving force which does work to compensate for the collisional loss of energy. Three different types of external driving are considered: (a) a stochastic force, (b) a deterministic force proportional to the particle velocity and (c) a deterministic force parallel to the particle velocity but constant in magnitude. The Enskog-Boltzmann equation in case (b) is fully equivalent to that of the homogeneous cooling state (where the thermal velocity monotonically decreases with time) when expressed in terms of the particle velocity relative to the thermal velocity. Comparison of the simulation results for the fourth cumulant and the high energy tail with theoretical predictions derived in cases (a) and (b) [T. P. C. van Noije and M. H. Ernst, Gran. Matt. 1, 57 (1998)] shows a good agreement. In contrast to these two cases, the deviation from the Maxwell-Boltzmann distribution is not well represented by Sonine polynomials in case (c), even for low dissipation. In addition, the high energy tail exhibits an underpopulation effect in this case.

Kinetic theory of simple granular shear flows of smooth hard spheres

Steady simple shear flows of smooth inelastic spheres are studied by means of a model kinetic equation and also of a direct Monte Carlo simulation method. Both approaches are based on the Enskog equation and provide for each other a test of consistency. The dependence of the granular temperature and of the shear and normal stresses on both the solid fraction and the coefficient of restitution is analysed. Quite a good agreement is found between theory and simulations in all cases. Also, simplified expressions based on the analytical solution of the model for small dissipation are shown to describe fairly well the simulation results even for not small inelasticity. A critical comparison with previous theories is carried out

Global and local instability of flow focusing: The influence of the geometry

In the flow focusing technique, a liquid flow rate Q is injected through a microcapillary to form a meniscus attached to its edge. The meniscus is stretched until a thin jet tapers from its tip due to the action of a gas stream driven by a pressure drop Δp. Both the liquid jet and the gas stream cross the orifice of a plate located in front of the capillary at a distance H. In the present work, the stability of both the tapering liquid meniscus and the emitted jet is analyzed experimentally. Three regimes are identified: (i) the steady jetting regime, where the liquid meniscus is stable and the jet is convectively unstable; (ii) the local instability regime, where the liquid meniscus is stable and the jet is absolutely unstable; and (iii) the global instability regime, where the liquid meniscus is unstable. The mechanisms responsible for the transitions between those regimes are described. The experiments show the existence of a minimum value Qmin of the flow rate Q below which flow focusing is globally unstable independent of the pressure drop Δp applied to the gas stream. The dependence of the stability threshold Qmin with respect to the capillary-to-orifice distance H is analyzed considering different liquids. If the rest of the geometrical parameters are fixed, there is an optimum value Hopt of the capillary-to-orifice distance H for which the stability threshold Qmin is minimum. We also determine the dependence of Hopt and the corresponding minimum flow rate Qopt with respect to the capillary diameter. In addition, we find that Qmin diverges as the capillary-to-orifice distance H decreases and approaches a certain critical value, at which the transition from flow focusing to “flow blurring” takes place. We confirm our interpretation of the experimental results by conducting numerical simulations for the aforementioned three regimes.In the flow focusing technique, a liquid flow rate Q is injected through a microcapillary to form a meniscus attached to its edge. The meniscus is stretched until a thin jet tapers from its tip due to the action of a gas stream driven by a pressure drop Δp. Both the liquid jet and the gas stream cross the orifice of a plate located in front of the capillary at a distance H. In the present work, the stability of both the tapering liquid meniscus and the emitted jet is analyzed experimentally. Three regimes are identified: (i) the steady jetting regime, where the liquid meniscus is stable and the jet is convectively unstable; (ii) the local instability regime, where the liquid meniscus is stable and the jet is absolutely unstable; and (iii) the global instability regime, where the liquid meniscus is unstable. The mechanisms responsible for the transitions between those regimes are described. The experiments show the existence of a minimum value Qmin of the flow rate Q below which flow focusing is globally u...

Building functional materials for health care and pharmacy from microfluidic principles and Flow Focusing

In this review, we aim at establishing a relationship between the fundamentals of the microfluidics technologies used in the Pharmacy field, and the achievements accomplished by those technologies. We describe the main methods for manufacturing micrometer drops, bubbles, and capsules, as well as the corresponding underlying physical mechanisms. In this regard, the review is intended to show non-specialist readers the dynamical processes which determine the success of microfluidics techniques. Flow focusing (FF) is a droplet-based method widely used to produce different types of fluid entities on a continuous basis by applying an extensional co-flow. We take this technique as an example to illustrate how microfluidics technologies for drug delivery are progressing from a deep understanding of the physics of fluids involved. Specifically, we describe the limitations of FF, and review novel methods which enhance its stability and robustness. In the last part of this paper, we review some of the accomplishments of microfluidics when it comes to drug manufacturing and delivery. Special attention is paid to the production of the microencapsulated form because this fluidic structure gathers the main functionalities sought for in Pharmacy. We also show how FF has been adapted to satisfy an ample variety of pharmaceutical requirements to date.

Transport coefficients of a heated granular gas

The Navier–Stokes transport coefficients of a granular gas are obtained from the Chapman–Enskog solution to the Boltzmann equation. The granular gas is heated by the action of an external driving force (thermostat) which does work to compensate for the collisional loss of energy. Two types of thermostats are considered: (a) a deterministic force proportional to the particle velocity (Gaussian thermostat), and (b) a random external force (stochastic thermostat). As happens in the free cooling case, the transport coefficients are determined from linear integral equations which can be approximately solved by means of a Sonine polynomial expansion. In the leading order, we get those coefficients as explicit functions of the restitution coefficient α. The results are compared with those obtained in the free cooling case, indicating that the above thermostat forces do not play a neutral role in the transport. The kinetic theory results are also compared with those obtained from Monte Carlo simulations of the Boltzmann equation for the shear viscosity. The comparison shows an excellent agreement between theory and simulation over a wide range of values of the restitution coefficient. Finally, the expressions of the transport coefficients for a gas of inelastic hard spheres are extended to the revised Enskog theory for a description at higher densities.

An experimental analysis of the linear vibration of axisymmetric liquid bridges

The linear vibration of axisymmetric liquid bridges of millimetric size was analyzed experimentally. The equilibrium shape and the evolution of the interface position were recorded by a high-speed video camera. To calculate the surface tension, the Theoretical Image Fitting Analysis method was adapted to process images of liquid bridges at equilibrium. An image processing method was developed to obtain both the position of the solid supports delimiting the liquid bridge and the interface deformation as a function of time. The former allows one to accurately measure the inertial force acting on the liquid bridge. The interface deformation was determined at the subpixel level, so that vibrations of very small amplitude could be analyzed. The results are compared with various theoretical approaches in the linear regime.

The minimum or natural rate of flow and droplet size ejected by Taylor cone-jets: physical symmetries and scaling laws

We aim to establish the scaling laws for both the minimum rate of flow attainable in the steady cone-jet mode of electrospray, and the size of the resulting droplets in that limit. Use is made of a small body of literature on Taylor cone-jets reporting precise measurements of the transported electric current and droplet size as a function of the liquid properties and flow rate. The projection of the data onto an appropriate non-dimensional parameter space maps a region bounded by the minimum rate of flow attainable in the steady state. To explain these experimental results, we propose a theoretical model based on the generalized concept of physical symmetry, stemming from the system time invariance (steadiness). A group of symmetries rising at the cone- to-jet geometrical transition determines the scaling for the minimum flow rate and related variables. If the flow rate is decreased below that minimum value, those symmetries break down, which leads to dripping. We find that the system exhibits two instability mechanisms depending on the nature of the forces arising against the flow: one dominated by viscosity and the other by the liquid polarity. In the former case, full charge relaxation is guaranteed down to the minimum flow rate, while in the latter the instability condition becomes equivalent to

Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas

Motivated by the disagreement found at high dissipation between simulation data for the heat flux transport coefficients and the expressions derived from the Boltzmann equation by the standard first Sonine approximation [J.J. Brey, M.J. Ruiz-Montero, Phys. Rev. E 70 (2004) 051301, J.J. Brey, M.J. Ruiz-Montero, P. Maynar, M.I. Garcia de Soria, J. Phys. Condens. Matter 17 (2005) S2489], we implement in this paper a modified version of the first Sonine approximation in which the Maxwell–Boltzmann weight function is replaced by the homogeneous cooling state (HCS) distribution. The structure of the transport coefficients is common in both approximations, the distinction appearing in the coefficient of the fourth cumulant a2. Comparison with computer simulations shows that the modified approximation significantly improves the estimates for the heat flux transport coefficients at strong dissipation. In addition, the slight discrepancies between simulation and the standard first Sonine estimates for the shear viscosity and the self-diffusion coefficient are also partially corrected by the modified approximation. Finally, the extension of the modified first Sonine approximation to the transport coefficients of the Enskog kinetic theory is presented.

Theoretical and experimental analysis of the equilibrium contours of liquid bridges of arbitrary shape

The equilibrium shape of the liquid bridge interface is analyzed theoretically and experimentally. Both axisymmetric and nonaxisymmetric perturbations are considered. The axisymmetric deviations are those related to volume effects, the difference between the radii of the disks, and the axial forces acting on the liquid bridge. The nonaxisymmetric deviations are those due to the eccentricity of the disk and the action of lateral forces. The theoretical study is performed using three different techniques: (i) an analytical expansion around the cylindrical solution, (ii) a finite difference scheme, and (iii) an approximate numerical approach valid only for slight nonaxisymmetric deviations. The results of the three methods are compared systematically. There is a very good agreement between the analytical and the numerical approaches for contours which are close to cylindrical, and the agreement extends to configurations with only moderate deviations from cylindrical. Experiments are performed using the so-called neutral buoyancy or plateau technique. Theoretical and experimental contours are compared considering a wide range of values for the parameters characterizing the perturbations. In general, the finite difference method provides reasonably accurate predictions even for large deviations of the liquid bridge contour from cylindrical.

Diffusion of impurities in a granular gas

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann-Lorentz equation by means of the Chapman-Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient D is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate D up to the second order in the Sonine expansion and get explicit expressions for D in terms of the coefficients of restitution for the impurity-gas and gas-gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo method. In the simulations, the diffusion coefficient is measured via the mean-square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In these cases, the second Sonine approximation to D improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.