Numerical simulation of the horizontal Bridgman growth. Part I: Two-dimensional flow

We study the generation of periodic velocity and temperature fields in a plane horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals, although the present paper is limited to two-dimensional flows. We use transient finite difference and finite element algorithms which lead to identical results. We demonstrate the oscillatory mechanism in two different geometries.     

Numerical simulation of the horizontal Bridgman growth. Part II: Three-dimensional flow

We study the steady-state three-dimensional flow which occurs in a horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals. We find that three-dimensional effects can have a dramatic influence upon the flow, which, before the onset of periodic disturbances, differs appreciably from its two-dimensional counterpart. We also investigate the sensitivity of the flow to non-symmetric disturbances.     

Simulation of polymeric flows in the injection moulding process

Recent progress in the simulation of polymeric flows of two key problems in the injection moulding process, carried out by a team at Cornell University, is briefly described. For the filling of cooled thin cavities, the fluid is characterized by a power-law viscosity with exponential temperature dependence, and interaction between the transient thermal boundary-layer and the core flow in a domain with moving boundary is essential. The earlier procedure of Hieber and Shen is modified in two aspects: a boundary-integral formulation replaces the finite-element treatment of the pressure, and an ‘energy integral’ approach is used for the transient temperature. The second problem is the steady visco-elastic flow in the juncture region where sudden changes of the geometry and large strain rates occur. The constitutive equation is postulated according to the Leonov model. The main features in the numerical implementation are: integration along a streamline to determine the elastic deformation tensors for a given velocity field, and finite-element treatment (in time-dependent form) of the pressure and fields for given stresses. In an example where the contraction ratio is 7:1, results for nominal Deborah number exceeding 100 show no numerical instability. (However, for this problem, the true Weissenberg number, i.e. the ratio of local first-normal-stress difference to shear stress turns out to be generally O(10).) The predictions also correlate very well with experimental birefringence measurements.     

Magnetohydrodynamic flow in a rectangular duct

The magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow, has been investigated. One of the duct's boundaries which is perpendicular to the magnetic field is taken partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 100. All the infinite series obtained are transformed to infinite integrals first and then to finite integrals which contain modified Bessel functions of the second kind. In this way, the difficulties associated with the computation of infinite integrals with oscillating integrands and slowly converging infinite series, the convergence of which is further affected for large values of M, have been avoided. It is found that, as M increases, boundary layers are formed near the non-conducting boundaries and in the interface region, and a stagnant region is developed in front of the conducting boundary for velocity field. The maximm value of magnetic field takes place on the conducting part. These behaviours are shown on some graphs.     

Surface tension driven flows in micro-gravity conditions

The importance of convective flows generated by surface tension gradients, in comparison with the ones generated by other driving forces, has been investigated in connection with space technological applications involving fluid processes. A theoretical model of the boundary conditions at the interface, considered free and diffusive, has been derived in general tensor form to allow for the use of non orthogonal curvilinear co-ordinates. For the study of flow fields contained in enclosures, these co-ordinates are more suitable to fit all teh boundaries, in particular near the contact angle between the interface and the solid walls, thus giving more accurate numerical solutions. A computational procedure to solve the complete set of bulk and surface equations is proposed and applied to a simplified two dimensional flow in a rectangular enclosure with a temperature gradient between the lateral walls. The numerical results show the importance of considering the interface to be deformable and diffusive for an accurate evaluation of the convective flow in the fluid bulk.     

Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

String flash-boiling in gasoline direct injection simulations with transient needle motion

A computational study was performed to investigate the influence of transient needle motion on gasoline direct injection (GDI) internal nozzle flow and near-field sprays. Simulations were conducted with a compressible Eulerian flow solver modeling liquid, vapor, and non-condensable gas phases with a diffuse interface. Variable rate generation and condensation of fuel vapor were captured using the homogeneous relaxation model (HRM). The non-flashing (spray G) and flashing (spray G2) conditions specified by the Engine Combustion Network were modeled using the nominal spray G nozzle geometry. Transient needle lift and wobble were based upon ensemble averaged X-ray imaging preformed at Argonne National Lab. The minimum needle lift simulated was 5 µm and dynamic mesh motion was achieved with Laplacian smoothing. The results were qualitatively validated against experimental imaging and the experimental rate of injection profile was captured accurately using pressure boundary conditions and needle motion to actuate the injection. Low needle lift is shown to result in vapor generation near the injector seat. Finally, the internal injector flow is shown to be highly complex, containing many transient and interacting vortices which result in perturbations in the spray angle and fluctuations in the mass flux. This complex internal flow also results in intermittent string flash-boiling when a strong vortex is injected and the resulting swirling spray contains a thermal non-equilibrium vapor core.     

Deterministic physical influence control of interfacial motion in thermal processing of solidified materials

In this paper, a new experimental method of phase interface motion control with time dependent boundary cooling is presented for ice–water solidification problems. A numerical method for inverse heat transfer problems was developed to predict the transient boundary conditions, which produce a prescribed phase interface motion. In the experimental study, the predicted boundary temperatures from the numerical simulation were used to control the ice–water interface movement for various specified interface motions. Two cases of different phase interface velocities were considered. Water supercooling was observed during each experiment. A time delay in the thermal control was calculated based on an analytical solution. Close agreement between measured data and specified interface motion was achieved for the ice–water solidification problems.     

The Interface Between Fresh and Salt Groundwater in Horizontal Aquifers: The Dupuit–Forchheimer Approximation Revisited

We analyze the motion of a sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend. The analysis is based on earlier results of De Josselin de Jong (Proc Euromech 143:75–82, 1981). Parameterizing the height of the interface along the horizontal base of the aquifer and assuming the validity of the Dupuit–Forchheimer approximation in both the fresh and saltwater, he derived an approximate interface motion equation. This equation is a nonlinear doubly degenerate diffusion equation in terms of the height of the interface. In that paper, he also developed a stream function-based formulation for the dynamics of a two-fluid interface. By replacing the two fluids by one hypothetical fluid, with a distribution of vortices along the interface, the exact discharge field throughout the flow domain can be determined. Starting point for our analysis is the stream function formulation. We derive an exact integro-differential equation for the movement of the interface. We show that the pointwise differential terms are identical to the approximate Dupuit–Forchheimer interface motion equation as derived by De Josselin de Jong. We analyze (mathematical) properties of the additional integral term in the exact interface motion formulation to validate the approximate Dupuit–Forchheimer interface motion equation. We also consider the case of flat interfaces, and we study the behavior of the toe of the interface. In particular, we give a criterion for finite or infinite speed of propagation.     

Turbulence structure and heat and mass transfer mechanism at a gas-liquid interface in a wind-wave tunnel

Turbulence structure and heat and mass transfer mechanism across a wavy sheared gas-liquid interface are fluid-mechanically investigated in a wind-wave tunnel. Heat and mass transfer velocities are reported and the relationship between the scalar transfer velocities and the turbulence structure is discussed. In addition, three-dimensional direct numerical simulation is carried out to investigate the flow structure over a rigid-wavy wall similar to that over the wave. The results show that the organized motion in the air flow intermittently appears on the front side of the wave crest, and its structure is rather similar to the flow structure over the rigid-wavy wall. The organized motion in the air flow induces the organized motion in the water flow and the organized motion renews the air-water interface. The scalar transfer across a wavy sheared gas-liquid interface is controlled by the organized surface-renewal motion in the water flow     

Analysis of forced-convection boiling flow instabilities in a single-channel upflow system

A numerical model is developed to predict the steady-state and transient behaviour of forced-convection boiling two-phase flow in a single channel. The model is based on the assumption of homogeneous two-phase flow and thermodynamic equilibrium of the phases. Compressibility effects in the two-phase region, motion of the bulk boiling interface and the thermal capacity of the heater wall have been included in the analysis. The model is used to study the effects of heat input, inlet subcooling and flow rate on the system behaviour. For comparison purposes, an experimental investigation was conducted using a single-channel, electrically heated, forced-convection upflow system. Steady-state operating characteristics, and stable and unstable regions, are determined as a function of heat flux, inlet subcooling and mass flow rate. Different modes of oscillation and their characteristics have been investigated. The model's predictions are in good agreement with the experimental results.     

The effect of far field flow on a spherical crystal growth in the undercooled melt

The effect of convective flow on a spherical crystal growth in the undercooled melt with a moderate far field flow is studied. The asymptotic solution of the evolution of the interface of the spherical crystal growth is obtained by the matched asymptotic expansion method. The analytic result shows that the convective flow in the undercooled melt has a strong effect on the evolution of spherical crystal growth. The convective flow induced by the far field flow makes the interface of the growing spherical crystal enhance its growth velocity in the upstream direction of the far field flow and inhibit growth in the downstream direction, and the interface of the decaying spherical crystal further decay in the upstream direction and inhibit decay in the downstream direction. The maximum growth velocity of the interface of the spherical crystal influenced by the far field flow is obtained.     

Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow

A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both the fluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwise and spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. The numerical result shows that the streamwise electric field suppresses transient growth of the disturbance with spanwise uniform wave number. The spanwise electric field diminishes transient growth of the disturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in the wave number plane. The peak at relatively large spanwise wave number is dominated by the lift-up mechanism and little influenced by electric field. Differently, the peak at relatively small wave number is associated with the characteristic of the interface and possibly influenced by electric field. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth is studied as well. A scaling law is obtained for relatively small Weber numbers and relatively large Reynolds numbers.     

Effect of an insoluble surfactant film on the stability of the concentration-driven Marangoni flow

The stability and the structure of the concentration and capillary driven Marangoni flow from a localized source is experimentally investigated in the presence of an adsorbed layer of an insoluble surfactant. It is found that the presence of the surfactant on the interface leads to the instability of the main axisymmetric flow with the result that a secondary azimuthally-periodic flow with a multivortex structure is developed. The structure of the convective motion on the interface is studied as a function of the Marangoni flow intensity and the surface density of the surfactant. The azimuthal wavenumber is shown to increase with the Marangoni number and to decrease with increase in the surface density of the surfactant. It is established that there exists a threshold value of the surface density of the surfactant at which the surface flow does not occur.     

The circular disk straddling the interface of a two-phase flow

The motion is determined for a thin circular disk straddling the plane interface of an immiscible two phase creeping flow and moving parallel or perpendicular to the interface. Expressions are derived for the drag coefficient on the disk.     

Coating flow theory by finite element and asymptotic analysis of the navier-stokes system

Coating flows are laminar free surface flows, preferably steady and two-dimensional, by which a liquid film is deposited on a substrate. Their theory rests on mass and momentum accounting for which Galerkin's weighted residual method, finite element basis functions, isoparametric mappings, and a new free surface parametrization prove particularly well-suited, especially in coping with the highly deformed free boundaries, irregular flow domains, and the singular nature of static and dynamic contact lines where fluid interfaces intersect solid surfaces. Typically, short forming zones of rapidly rearranging two-dimensional flow merge with simpler asymptotic regimes of developing or developed flow upstream and downstream. The two-dimensional computational domain can be shrunk in size by imposing boundary conditions from asymptotic analysis of those regimes or by matching to one-dimensional finite element solutions of asymptotic equations. The theory is laid out with special attention to conditions at free surfaces, contact lines, and open inflow and outflow boundaries. Efficient computation of predictions is described with emphasis on a grand Newton iteration that converges rapidly and brings other benefits. Sample results for curtain coating and roll coating flows of Newtonian liquids illustrate the power and effectiveness of the theory.     

On a family of Stokes flows

A family of simple Stokes flows involving sliding surfaces adjacent to surfaces at rest is considered. Principally, two specific flow configurations are investigated: (i) that arising when parts of the boundary of an infinitely long circular cylinder are rotating about the axis while other parts of the boundary are at rest, and (ii) the flow produced when a cap of a sphere is held at rest while the remainder of the sphere rotates about the symmetry axis. In each case computer plots of streamlines or constant velocity lines are presented to give a general impression of the resulting flow pattern.     

Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth

We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses.     

The effect of hydrodynamic friction on the stability of a wall layer in annular two-phase flow

This note is concerned with annulat flow in a vertical tube when the gas stream in the central portion of the tube is separated from the wall by an annular layer of liquid. The friction at the interface may have either a stabilizing or destabilizing effect on such a flow regime with respect to small disturbances of the interface. The liquid-layer thickness is assumed small, which permits direct use of the results of [1] in studying the stability. The analysis is applicable to laminar and developed turbulent gas motions; in both cases the motion in the liquid layer is assumed laminar.     

On the non-persistence of irrotational motion in a viscous heat-conducting fluid

We consider the possibility of irrotational flow in a fluid exterior to a moving rigid obstacle, or interior to a moving rigid shell. Observations show that when a rigid body is impulsively set into motion an irrotational flow may exist initially but does not persist. The breakup of this irrotational flow and the associated phenomenon of generation of vorticity at the wall are generally attributed to the condition of adherence at the fluid-solid interface. Since this condition itself is derived from observation, one can ask whether there is another explanation for the phenomenon. The purpose of this paper is to show that a persistent irrotational flow is incompatible with the second law of thermodynamics.