Hydrodynamics and Heat Transfer Associated with Condensation on a Moving Drop: Solutions for Intermediate Reynolds Numbers

The hydrodynamics and heat/mass transport associated with condensation on a moving drop have been investigated for the intermediate Reynolds-number range of drop motion (Re = O(100)). The drop environment is a mixture of saturated vapour and a non-condensable. The formulation entails a simultaneous solution of the quasi-steady elliptic partial differential equations that describe the flow field and transport in the gaseous phase, and the motion inside the liquid drop. The heat transport inside the drop is treated as a transient process. Results are reported for the interfacial velocities, drag, external and internal flow structure, heat flux, drop growth rate and temperature-time history inside the drop. The results obtained here have been compared with experimental data where available, and these show excellent agreement. The results reveal several novel features. The surface-shear stress increases with condensation. The pressure level in the rear of the drop is higher. As a consequence, the friction drag is higher and the pressure drag is lower. The total drag coefficient increases with condensation rate for small values of drop size or temperature differential, and it decreases for large values of these parameters. The volume of the separated-flow region in the rear of the drop decreases with condensation. At very high rates of condensation, the recirculatory wake is completely suppressed. Condensation also delays the appearance of the weak secondary internal vortex motion in the drop. The heat and mass fluxes are significantly affected by the presence of the non-condensable in the gaseous phase and by the circulation inside the drop

Flow Past a Liquid Drop with a Large Non-uniform Radial Velocity

In this analysis, the translation of a liquid drop experiencing a strong non-uniform radial velocity has been investigated. The situation arises when a moving liquid drop experiences condensation, evaporation or material decomposition at the surface. By simultaneously treating the flow fields inside and outside the drop, we have obtained physical results relevant to the problem. The magnitude of the radial velocity is allowed to be very large, but the drop motion is restricted to slow translation. The solution to the problem has been developed by considering a uniform radial flow with the translatory motion introduced as a perturbation. The role played by the inertial terms due to the strong radial field has been clearly delineated. The study has revealed several interesting features. An inward normal velocity on a slowly moving drop increases the drag. An increasing outward normal velocity decreases the drag up to a minimum beyond which it increases. The total drag force not only consists of contributions from the viscous and the form drags but also from the momentum transport at the interface. Since the liquid drop admits a non-zero tangential velocity, the tangential momentum convected by the radial velocity forms a part of this drag force. The circulation inside the drop decreases (increases) with an outward (inward) normal velocity. A sufficiently large non-uniform outward velocity causes the circulation to reverse. In the limit of the internal viscosity becoming infinite, our analysis collapses to the simple case of a translating rigid sphere experiencing a large non-uniform radial velocity. By letting the radial velocity become vanishingly small the Stokes-flow solution is recovered. An important contribution of the present study is the identification of a new singularity in the flow description. It accounts for both the inertial and the viscous forces and displays Stokeslet-like characteristics at infinity

Linear Stability of a Viscous-Inviscid Interface

In this paper the stability of the interface separating fluids of widely differing viscosities has been examined. It is shown that a viscous-inviscid (V-I) model offers a consistent zeroth-order approximation to the stability problem. The zeroth-order solution is obtained by neglecting the smallest-order effect, viz., viscosity on the less viscous side of the interface. In this sense, the V-I model significantly differs from the Kelvin-Helmholtz (K-H) approach where both the viscosities are dropped in a single step. A closed form solution for the stability criterion governing the V-I model has been obtained, and a novel instability mechanism is described. It is shown that the V-I model is also a consistent zeroth-order approximation for the Rayleigh-Taylor problem of a viscous-viscous, nonflowing interface when the viscosity ratio tends to zero. For the interface separating two viscous, nonflowing, incompressible fluids, exact solutions for the velocities, pressures, and interface displacement for a disturbance of a given wavelength have been provided for the stable (lighter fluid on top) wave motion. By discussing the roles played by the dynamic and kinematic viscosities, it is made clear why neither the V-I nor the K-H model should apply to the air-water interface. The results of the V-I model compare well with experimental observations. The V-I model serves as an excellent basis for comparison in detailed numerical studies of the viscous-viscous interface

Charged Particle Distributions and Heat Transfer in a Discharge Between Geometrically Dissimilar Electrodes: From Breakdown to Steady State

The low-current electric discharge from a fine wire anode to a planar cathode in atmospheric pressure air is numerically simulated from high-voltage prebreakdown through electron temperature growth, then ionization and consequent current growth to steady state, limited by a ballast resistor in the external circuit. Conservation of number ~mass! for ions and electrons, Gauss’ law for the self-consistent electric field, and energy conservation for electrons have been solved from breakdown to steady state in a body fitted coordinate system generated specifically for these two geometrically dissimilar electrodes. To facilitate the discussion of the results, the discharge has been categorized under ~a! electron acceleration period, ~b! charged particle generation period, ~c! current increase and voltage drop period, and ~d! current and voltage stabilization period. Results are given for transient electron, ion, and temperature distributions in the gap as well as current growth and voltage drop across the gap. Heat flux from the discharge to the wire is calculated. The numerical simulations were compared with experiments performed under the same conditions on a wire bonding machine with very close correspondence

The Dynamics of Two Spherical Particles in a Confined Rotating Flow: Pedalling Motion

We have numerically investigated the interaction dynamics between two rigid spherical particles moving in a fluid-filled cylinder that is rotating at a constant speed. The cylinder rotation is about a horizontal axis. The particle densities are less than that of the fluid. The numerical procedure employed to solve the mathematical formulation is based on a three-dimensional arbitrary Larangian–Eulerian (ALE), moving mesh finite-element technique, described in a frame of reference rotating with the cylinder. Results are obtained in the ranges of particle Reynolds number, 1\u3cRep≤60, and shear Reynolds number, 1≤Res \u3c10. Two identical particles, depending on initial conditions at release, approach each other (‘drafting’ and ‘kissing’), tumble in the axial direction, and axially migrate towards opposing transverse planes on which they ‘settle’ (settling planes). Under some other initial conditions, the particles migrate directly onto their settling planes. For two identical particles, the settling planes are equidistant from the mid-transverse plane of the cylinder and the locations of the planes are determined by particle–particle and particle–wall force balances. Furthermore, for identical particles and given values of Rep and Res, the locations of such settling planes remain the same, independent of the initial conditions at release. While located on these settling planes, as viewed in an inertial frame, the particles may attain three possible distinct states depending on the values of the Reynolds numbers. In one state (low Rep, high Res ), the particles attain and remain at fixed equilibrium points on their settling planes. In the second (all Rep, low Res ), they execute spiralling motions about fixed points on their respective settling planes. These fixed points coincide with the locations of the equilibrium point which would occur on the mid-axial plane in the case of a single particle. In the third state (low Rep, moderate Res or high Rep, moderate to high Res ), they execute near-circular orbital motion on their respective settling planes, again about fixed points. These fixed points also coincide with the locations of the equilibrium points corresponding to single-particle dynamics. Both the spiral and near-circular motions of the particles occur in an out-of-phase manner with regard to their radial positions about the fixed point; the near-circular out-of-phase motion resembles bicycle pedalling. Also, in the second and third states, the particles simultaneously experience very weak axial oscillations about their settling planes, the frequency of such oscillations coinciding with the frequency of rotation of the circular cylinder. The behaviours of two non-identical particles (same density but different sizes, or same size but different densities) are different from those of identical particles. For example, non-identical particles may both end up settling on the mid-axial plane. This occurs when the locations of their corresponding single-particle equilibrium points are far apart. When such points are not far apart, particles may settle on planes that may not be symmetrical about the mid-axial plane. While located on their settling planes, their equilibrium states may not be similar. For example, for particles of the same density but of different sizes, the smaller of the two may execute a spiralling motion while the larger is in near-circular orbital motion. With particles of the same size but of different densities, while the lighter of the two approaches its equilibrium point on the mid-axial plane, the heavier one experiences a circular motion on the same plane about its equilibrium point. A major reason for the eventual attainment of these various states is noted to be the interplay between the particle–particle and particle–wall forces

Two-Dimenslonal Analysis of Electrical Breakdown in a Nonuniform Gap Between a Wire and a Plane

Electrical breakdown of a gap between a wire (modeled as a hyperboloid) and a plane has been investigated numerically by solving the two-dimensional form of the diffusion flux equations for the charged particle number densities and Poisson\u27s equation for the self-consistent electric field. Electron impact ionization, thermal ionization, and three-body recombination have been considered as the charged particle production and loss mechanisms. The electrode surfaces are considered to be absorbing and the initial density of the particles is small, but nonzero, A gap length of 0.5 mm is investigated and the gas medium is air or argon at atmospheric pressure. The temporal development of the profiles of ion and electron number densities, potential and electric field, and current growth on both the electrodes are presented when the applied voltage is 1500 and 2500 V for both positive and negative wires. When the wire is negatively biased, the peaks in the radial distribution of both of the charged particle densities near the wire occur off the axis except during the very early part of the breakdown. With positive polarity, the electron density maximum always occurs on the discharge axis, while for ions it moves away from the axis, later in the transient, due to the reverse particle drift in the electric field from the negative polarity case, The discharge spreads farther out into the ambient (almost two times the gap length) when the wire is negatively biased than with positive polarity. The effect of charge separation on the externally applied electric field is significant at voltages 2500 V and higher. Ionization is greater in argon than in air for a fixed potential difference between the electrodes

Effect of Negative Ions on Electrical Breakdown in a Nonuniform Air Gap Between a Wire and a Plane

Electrical breakdown of an axisymmetric, atmospheric pressure air gap between a wire and a plane has been investigated for a gap length of 0.5 mm. 0- and 02- have been identified as the negative ions affecting the discharge development in air, besides electrons and positive ions, and have been included in the electrical breakdown model. Five coupled two-dimensional transient partial differential equations describing the discharge evolution in the air gap have been solved using a finite difference algorithm developed earlier. Temporal development of the charged particle number densities, electrostatic potential, electric field, and current at both the electrodes is presented when the wire is negatively biased at 2500 V. The impact of negative ions on gap breakdown has been assessed by comparing the results of analyses with and without negative ions. It is concluded that the negative ions have negligible effect during the early stages of the discharge development. However, as the discharge evolves, the negative ions cause a net loss of electrons from the discharge. The effect is most pronounced away from the discharge axis, where peaks in the electron density occur as breakdown proceeds. Radial spread of discharge and current growth rate are relatively unaffected by the presence of negative ions, but the magnitude of total current at the electrodes has been found to decrease by a decade when the negative ions are present

Laminar Condensation on a Moving Drop. Part 2. Numerical Solutions

In this paper, we investigate the problem of transient laminar condensation on a moving drop by the semianalytical series-truncation method. The objectives are to assess the validity and the accuracy of the matched-asymptotic method employed in Part 1 . The fluid flow and thermodynamic variables are expanded as complete series of Legendre polynomials. The resulting transient momentum, energy and species equations are integrated numerically. The numerical scheme basically involves a three-point central difference for the spatial derivatives and a backward difference expression for the temporal derivatives. The finite-difference equations have been solved by the strongly implicit procedure. Good agreement of the fully transient numerical results with the singular perturbation approximation results of Part 1 lends credibility to a quasi-steady treatment of the continuous phase. The computational time requirements for the fully numerical solutions increase with decreasing non-condensable gas mass fraction in the bulk environment

Thermal and electrical characteristics of a two‐dimensional tanh‐conductivity arc

The two-dimensional variable-property arc has been studied through the use of the tanh-conductivity model. Results that describe the thermal and electric arc characteristics for various values of the electrode temperatures and aspect ratios are given. The numerical evaluation is carried out by the use of a Galerkin technique. The results exhibit several novel and interesting features depending on the arc parameters. For large aspect ratios (ratio of the interelectrode distance to that between the bounding walls) and small electrode temperatures, the current---electric-field characteristics tend toward those of a slender arc. However, at a given aspect ratio with large enough electrode temperatures, the distinct minimum noted in the slender-arc characteristics does not occur. Also, for a given aspect ratio and large enough differences in electrode potential, the electric-field-current characteristic is nearly linear and is independent of the electrode temperature. The transverse electrostatic potential is found to have no significant variation in cross-sectional planes. The qualitative nature of the thermal characteristics are similar to those of a constant-property arc although significant differences in quantitative results exist. Wall and electrode heat transfer rates are provided

Finite-sized gas bubble motion in a blood vessel: Non-Newtonian effects

We have numerically investigated the axisymmetric motion of a finite-sized nearly occluding air bubble through a shear-thinning Casson fluid flowing in blood vessels of circular cross section. The numerical solution entails solving a two-layer fluid model - a cell-free layer and a non-Newtonian core together with the gas bubble. This problem is of interest to the field of rheology and for gas embolism studies in health sciences. The numerical method is based on a modified front-tracking method. The viscosity expression in the Casson model for blood (bulk fluid) includes the hematocrit [the volume fraction of red blood cells (RBCs)] as an explicit parameter. Three different flow Reynolds numbers, Reapp=ΡlUmaxd/µapp, in the neighborhood of 0.2, 2, and 200 are investigated. Here, Ρl is the density of blood, Umax is the centerline velocity of the inlet Casson profile, d is the diameter of the vessel, and µapp is the apparent viscosity of whole blood. Three different hematocrits have also been considered: 0.45, 0.4, and 0.335. The vessel sizes considered correspond to small arteries, and small and large arterioles in normal humans. The degree of bubble occlusion is characterized by the ratio of bubble to vessel radius (aspect ratio), λ, in the range 0.9 ≤ λ≤1.05. For arteriolar flow, where relevant, the Fahraeus-Lindqvist effects are taken into account. Both horizontal and vertical vessel geometries have been investigated. Many significant insights are revealed by our study: (i) bubble motion causes large temporal and spatial gradients of shear stress at the endothelial cell (EC) surface lining the blood vessel wall as the bubble approaches the cell, moves over it, and passes it by; (ii) rapid reversals occur in the sign of the shear stress (+ → - → +) imparted to the cell surface during bubble motion; (iii) large shear stress gradients together with sign reversals are ascribable to the development of a recirculation vortex at the rear of the bubble; (iv) computed magnitudes of shear stress gradients coupled with their sign reversals may correspond to levels that cause injury to the cell by membrane disruption through impulsive compression and stretching; and (v) for the vessel sizes and flow rates investigated, gravitational effects are negligible