A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Analysis of Self-Organization

The dissertation is devoted to the study of problems in calculus of variation, free boundary problems and gradient flows with respect to the Wasserstein metric. More concretely, we consider the problem of characterizing the regularity of minimizers to a certain interaction energy. Minimizers of the interaction energy have a somewhat surprising relationship with solutions to obstacle problems. Here we prove and exploit this relationship to obtain novel regularity results. Another problem we tackle is describing the asymptotic behavior of the Cahn-Hilliard equation with degenerate mobility. By framing the Cahn-Hilliard equation with degenerate mobility as a gradient flow in Wasserstein metric, in one space dimension, we prove its convergence to a degenerate parabolic equation under the framework recently developed by Sandier-Serfaty.

Transient Flows in Low Permeability Porous Media with Composite Start-Up Pressure Gradient

A new mathematical model for the transient flow in the composite low permeability is established. It is solved by FEM with different boundary conditions such as infinite, circular closed and constant pressure boundary conditions. The typical curves for transient wellbore pressure have been presented. It is shown that the pressure and pressure derivative curves with composite start-up pressure gradients have different slopes which are depended on the start-up pressure gradients and the mobility radios in different regions. The boundary effects are the same as the normal reservoirs without start-up pressure gradients. The study provides a new tool to analyze the transient pressure test data in the low permeability reservoir.

Finite Element Solution Of Surface-Tension Driven Flows In Laser Surface-Melting

Lasers are very efficient in heating localized regions and hence they find a wide application in surface treatment processes. The surface of a material can be selectively modified to give superior wear and corrosion resistance. In laser surface-melting and welding problems, the high temperature gradient prevailing in the free surface induces a surface-tension gradient which is the dominant driving force for convection (known as thermo-capillary or Marangoni convection). It has been reported that the surface-tension driven convection plays a dominant role in determining the melt pool shape. In most of the earlier works on laser-melting and related problems, the finite difference method (FDM) has been used to solve the Navier Stokes equations [1]. Since the Reynolds number is quite high in these cases, upwinding has been used. Though upwinding gives physically realistic solutions even on a coarse grid, the results are inaccurate. McLay and Carey have solved the thermo-capillary flow in welding problems by an implicit finite element method [2]. They used the conventional Galerkin finite element method (FEM) which requires that the pressure be interpolated by one order lower than velocity (mixed interpolation). This restricts the choice of elements to certain higher order elements which need numerical integration for evaluation of element matrices. The implicit algorithm yields a system of nonlinear, unsymmetric equations which are not positive definite. Computations would be possible only with large mainframe computers.Sluzalec [3] has modeled the pulsed laser-melting problem by an explicit method (FEM). He has used the six-node triangular element with mixed interpolation. Since he has considered the buoyancy induced flow only, the velocity values are small. In the present work, an equal order explicit FEM is used to compute the thermo-capillary flow in the laser surface-melting problem. As this method permits equal order interpolation, there is no restriction in the choice of elements. Even linear elements such as the three-node triangular elements can be used. As the governing equations are solved in a sequential manner, the computer memory requirement is less. The finite element formulation is discussed in this paper along with typical numerical results.

Unsteady Incompressible Two-Dimensional and Axisymmetric Turbulent Boundary Layer Flows

The unsteady turbulent incompressible boundary-layer flow over two-dimensional and axisymmetric bodies with pressure gradient has been studied. An eddy-viscosity model has been used to model the Reynolds shear stress. The unsteadiness is due to variations in the free stream velocity with time. The nonlinear partial differential equation with three independent variables governing the flow has been solved using Keller's Box method. The results indicate that the free stram velocity distribution exerts strong influence on the boundary-layer characteristics. The point of zero skin friction is found to move upstream as time increases.

Experimental evidence of the kinematic Cosserat effect in dense granular flows

To capture shear localization in the flow of dense granular materials in a continuum description, it has earlier been proposed that granular materials be treated as Cosserat, or micropolar, continua. Here, we provide experimental verification of the kinematic Cosserat effect, or the deviation of the particle spin from the material spin induced by the velocity gradient. Contrary to earlier belief, we find this effect to be sizable even outside the shear layers. Remarkably, the particles and material elements spin in opposite directions in flow through a hopper.

Dynamics of dense sheared granular flows. Part 1. Structure and diffusion

Shear flows of inelastic spheres in three dimensions in the Volume fraction range 0.4-0.64 are analysed using event-driven simulations.Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution e(n) (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution e(t) (negative of the ratio of post- and pre-collisional velocities perpendicular to the line Joining the centres). Here, we have considered both e(t) = +1 and e(t) = e(n) (rough particles) and e(t) =-1 (smooth particles), and the normal coefficient of restitution e(n) was varied in the range 0.6-0.98. Care was taken to avoid inelastic collapse and ensure there are no particle overlaps during the simulation. First, we studied the ordering in the system by examining the icosahedral order parameter Q(6) in three dimensions and the planar order parameter q(6) in the plane perpendicular to the gradient direction. It was found that for shear flows of sufficiently large size, the system Continues to be in the random state, with Q(6) and q(6) close to 0, even for volume fractions between phi = 0.5 and phi = 0.6; in contrast, for a system of elastic particles in the absence of shear, the system orders (crystallizes) at phi = 0.49. This indicates that the shear flow prevents ordering in a system of sufficiently large size. In a shear flow of inelastic particles, the strain rate and the temperature are related through the energy balance equation, and all time scales can be non-dimensionalized by the inverse of the strain rate. Therefore, the dynamics of the system are determined only by the volume fraction and the coefficients of restitution. The variation of the collision frequency with volume fraction and coefficient of estitution was examined. It was found, by plotting the inverse of the collision frequency as a function of volume fraction, that the collision frequency at constant strain rate diverges at a volume fraction phi(ad) (volume fraction for arrested dynamics) which is lower than the random close-packing Volume fraction 0.64 in the absence of shear. The volume fraction phi(ad) decreases as the coefficient of restitution is decreased from e(n) = 1; phi(ad) has a minimum of about 0.585 for coefficient of restitution e(n) in the range 0.6-0.8 for rough particles and is slightly larger for smooth particles. It is found that the dissipation rate and all components of the stress diverge proportional to the collision frequency in the close-packing limit. The qualitative behaviour of the increase in the stress and dissipation rate are well Captured by results derived from kinetic theory, but the quantitative agreement is lacking even if the collision frequency obtained from simulations is used to calculate the pair correlation function used In the theory.

Nonsimilar incompressible laminar boundary-layer flows in micropolar fluids

The solution of the steady laminar incompressible nonsimilar boundary-layer problem for micropolar fluids over two-dimensional and axisymmetric bodies has been presented. The partial differential equations governing the flow have been transformed into new co-ordinates having finite range. The resulting equations have been solved numerically using implicit finite-difference scheme. The computations have been carried out for a cylinder and a sphere. The results indicate that the separation in micropolar fluids occurs at earlier streamwise locations as compared to Newtonian fluids. The skin friction and velocity profiles depend on the shape of the body and are almost insensitive to microrotation or coupling parameter, provided the coupling parameter is small. On the other hand, the microrotation profiles and microrotation gradient depend on the microrotation parameter and they are insensitive to the coupling parameter.

Determination of skin friction in strong pressure-gradient equilibrium and near-equilibrium turbulent boundary layers

The conventional Clauser-chart method for determination of local skin friction in zero or weak pressure-gradient turbulent boundary layer flows fails entirely in strong pressure-gradient situations. This failure occurs due to the large departure of the mean velocity profile from the universal logarithmic law upon which the conventional Clauser-chart method is based. It is possible to extend this method,even for strong pressure-gradient situations involving equilibrium or near-equilibrium turbulent boundary layers by making use of the so-called non-universal logarithmic laws. These non-universal log laws depend on the local strength of the pressure gradient and may be regarded as perturbations of the universal log law.The present paper shows that the modified Clauser-chart method, so developed, yields quit satisfactory results in terms of estimation of local skin friction in strongly accelerated or retarded equilibrium and near-equilibrium turbulent boundary layers that are not very close to relaminarization or separation.

Absolute/Convective Instability Transition in a Backward Facing Step Combustor: Fundamental Mechanism and Influence of Density Gradient

Hydrodynamic instabilities of the flow field in lean premixed gas turbine combustors can generate velocity perturbations that wrinkle and distort the flame sheet over length scales that are smaller than the flame length. The resultant heat release oscillations can then potentially result in combustion instability. Thus, it is essential to understand the hydrodynamic instability characteristics of the combustor flow field in order to understand its overall influence on combustion instability characteristics. To this end, this paper elucidates the role of fluctuating vorticity production from a linear hydrodynamic stability analysis as the key mechanism promoting absolute/convective instability transitions in shear layers occurring in the flow behind a backward facing step. These results are obtained within the framework of an inviscid, incompressible, local temporal and spatio-temporal stability analysis. Vorticity fluctuations in this limit result from interaction between two competing mechanisms-(1) production from interaction between velocity perturbations and the base flow vorticity gradient and (2) baroclinic torque in the presence of base flow density gradients. This interaction has a significant effect on hydrodynamic instability characteristics when the base flow density and velocity gradients are colocated. Regions in the space of parameters characterizing the base flow velocity profile, i.e., shear layer thickness and ratio of forward to reverse flow velocity, corresponding to convective and absolute instability are identified. The implications of the present results on understanding prior experimental studies of combustion instability in backward facing step combustors and hydrodynamic instability in other flows such as heated jets and bluff body stabilized flames is discussed.

HYDROMAGNETICS OF ADVECTIVE ACCRETION FLOWS AROUND BLACK HOLES: REMOVAL OF ANGULAR MOMENTUM BY LARGE-SCALE MAGNETIC STRESSES

We show that the removal of angular momentum is possible in the presence of large-scale magnetic stresses in geometrically thick, advective, sub-Keplerian accretion flows around black holes in steady state, in the complete absence of alpha-viscosity. The efficiency of such an angular momentum transfer could be equivalent to that of alpha-viscosity with alpha = 0.01-0.08. Nevertheless, the required field is well below its equipartition value, leading to a magnetically stable disk flow. This is essentially important in order to describe the hard spectral state of the sources when the flow is non/sub-Keplerian. We show in our simpler 1.5 dimensional, vertically averaged disk model that the larger the vertical-gradient of the azimuthal component of the magnetic field is, the stronger the rate of angular momentum transfer becomes, which in turn may lead to a faster rate of outflowing matter. Finding efficient angular momentum transfer in black hole disks via magnetic stresses alone, is very interesting when the generic origin of alpha-viscosity is still being explored.

ABSOLUTE/CONVECTIVE INSTABILITY TRANSITION IN A BACKWARD FACING STEP COMBUSTOR: FUNDAMENTAL MECHANISM AND INFLUENCE OF DENSITY GRADIENT

Hydrodynamic instabilities of the flow field in lean premixed gas turbine combustors can generate velocity perturbations that wrinkle and distort the flame sheet over length scales that are smaller than the flame length. The resultant heat release oscillations can then potentially result in combustion instability. Thus, it is essential to understand the hydrodynamic instability characteristics of the combustor flow field in order to understand its overall influence on combustion instability characteristics. To this end, this paper elucidates the role of fluctuating vorticity production from a linear hydrodynamic stability analysis as the key mechanism promoting absolute/convective instability transitions in shear layers occurring in the flow behind a backward facing step. These results are obtained within the framework of an inviscid, incompressible, local temporal and spatio-temporal stability analysis. Vorticity fluctuations in this limit result from interaction between two competing mechanisms - (1) production from interaction between velocity perturbations and the base flow vorticity gradient and (2) baroclinic torque in the presence of base flow density gradients. This interaction has a significant effect on hydrodynamic instability characteristics when the base flow density and velocity gradients are co-located. Regions in the space of parameters characterizing the base flow velocity profile, i.e. shear layer thickness and ratio of forward to reverse flow velocity, corresponding to convective and absolute instability are identified. The implications of the present results on prior observations of flow instability in other flows such as heated jets and bluff-body stabilized flames is discussed.

Influences of slope gradient on soil erosion

The main factors influencing soil erosion include the net rain excess, the water depth, the velocity, the shear stress of overland flows, and the erosion-resisting capacity of soil. The laws of these factors varying with the slope gradient were investigated by using the kinematic wave theory. Furthermore, the critical slope gradient of erosion was driven. The analysis shows that the critical slope gradient of soil erosion is dependent on grain size, soil bulk density, surface roughness, runoff length, net rain excess, and the friction coefficient of soil, etc. The critical slope gradient has been estimated theoretically with its range between 41.5 degrees similar to 50 degrees.

Studies on two-phase co-current air/non-Newtonian shear-thinning fluid flows in inclined smooth pipes

In this work. co-current flow characteristics of air/non-Newtonian liquid systems in inclined smooth pipes are studied experimentally and theoretically using transparent tubes of 20, 40 and 60 turn in diameter. Each tube includes two 10 m lone pipe branches connected by a U-bend that is capable of being inclined to any angle, from a completely horizontal to a fully vertical position. The flow rate of each phase is varied over a wide range. The studied flow phenomena are bubbly, plug flow, slug flow, churn flow and annular flow. These are observed and recorded by a high flow. stratified flow. -speed camera over a wide range of operating conditions. The effects of the liquid phase properties, the inclination angle and the pipe diameter on two-phase flow characteristics are systematically studied. The Heywood-Charles model for horizontal flow was modified to accommodate stratified flow in inclined pipes, taking into account the average void fraction and pressure drop of the mixture flow of a gas/non-Newtonian liquid. The pressure drop gradient model of Taitel and Barnea for a gas/Newtonian liquid slug flow was extended to include liquids possessing shear-thinning flow behaviour in inclined pipes. The comparison of the predicted values with the experimental data shows that the models presented here provide a reasonable estimate of the average void fraction and the corresponding pressure drop for the mixture flow of a gas/ non-Newtonian liquid. (C) 2007 Elsevier Ltd. All rights reserved.