Steady Shapes Research Articles

Theoretical and numerical results for spin coating of viscous liquids

A mathematical model is developed for fluid flow in the spin coating process. Spin coating employs centrifugal force to produce coatings of uniform thickness. The long-wave or lubrication approximation is used for the flow of thin liquid layers that are exposed to the air and lie on a spinning horizontal solid substrate. For low rotation rates, steady axisymmetric drop shapes can be found analytically. The stability of these drops is investigated, using an energy method, both with and without the long-wave approximation. For industrially relevant high-speed motions, we formulate and solve a theoretical and numerical model for the three-dimensional time-dependent motion of the deforming drop. We pay particular attention to the formation of “fingers” at the expanding front. The model includes viscous, capillary, gravitational, centrifugal, Coriolis, and finite-contact-angle effects. Both homogeneous and chemically heterogeneous substrates are considered. In agreement with published experiments, the model demonstrates that imperfect wetting behavior is the principal cause of fingering during spin coating. Features of the finger profiles are in close agreement with experimental observations.

A model for large deformation of an ellipsoidal droplet with interfacial tension

A model is developed to predict the transient shape evolution of an ellipsoidal Newtonian droplet with interfacial tension, suspended in another Newtonian fluid with a different viscosity. Using the Eshelby equivalent inclusion theory [Eshelby (1957)], this model adds an approximate interfacial tension term to the exact deformation model of Wetzel and Tucker (2001) for droplets with zero interfacial tension. The model can predict large deformations and rotations of the ellipsoid, it allows any value of viscosity ratio, and it admits arbitrary far-field flows. The model does not incorporate breakup or coalescence. At viscosity ratios less than 0.1, the model blends the Eshelby model (for compact shapes) with a classical slender-body model (for thread-like shapes), providing a smooth transition between these two limits. The model accurately matches many experimental results from the literature, including steady shapes in simple shear, transient shapes during shear reversal, widening in simple shear, relaxation after step shear, and the critical capillary number for breakup in both shear and planar elongation. The model also compares favorably to the ellipsoidal droplet model of Maffettone and Minale (1998).

Transient and steady state three-dimensional drop shapes and dimensions under planar extensional flow

Drop shapes and dimensions in transient and steady states have been studied using a four-roll mill flow apparatus with three-dimensional imaging and real-time derivative edge detection. The deformed drops are approximately ellipsoidal for viscosity ratios of λ > 0.1 with about 4% maximum deviation of actual drop volume from ellipsoidal volume. For λ < 0.1, drops retain their ellipsoidal shape up to deformation of ∼0.5, above which they show progressive deviation from the ellipsoidal shape. The measured length, breadth, and width of deformed drops were compared to predictions of various drop deformation theories. The limit of deformation within which theories predict drop dimensions with 5% accuracy has been tested for 10−4 < λ < 102. The ellipsoidal model of Maffettone and Minale (1998) gives the best prediction for λ < 0.3, while the second order theory of Barthes-Biesel and Acrivos (1973) gives the best overall prediction for λ > 0.3. The application of some theories to the determination of interfacial tension and drop viscosity is discussed.

Experimental studies on the shape and path of small air bubbles rising in clean water

This Letter reports experiments on the shape and path of air bubbles (diameter range 0.1–0.2 cm) rising in clean water. We find that bubbles in this diameter range have two steady shapes, a sphere and an ellipsoid, depending on the size of the capillary tube from which they detach. The spherical bubbles move significantly slower than the ellipsoidal ones of equivalent volume. Bubbles with diameter less than about 0.15 cm rise rectilinearly. The larger spherical bubbles follow zigzag paths while the larger ellipsoidal bubbles follow spiral paths.

Influence of Initial Tension on Out-Plane Vibrations of a Rotating Circular Plate.

A refined solution procedure is presented for analyzing out-plane free vibrations of a rotating circular plate. The equations of motion and boundary conditions are derived by considering the effect of initial tensions due to the rotation. The equations of motion are solved exactly by power series expansions. Based on the numerical calculations, some interesting results, such as in-plane stress distributions in steady rotation, frequencies and mode shapes of the circular plate in out-plane vibration, are presented. The results of the present and classical theory are compared, which verify the advantages of the proposed theory.

Modeling multiphase flows using a novel 3D adaptive remeshing algorithm

A novel three-dimensional adaptive meshing algorithm is presented and applied to finite-element sirnulations of multiphase fluid flows. A threedimensional domain enclosing another phase is discretized by an unstructured mesh of tetrahedra constructed from a triangulated surface of the phase boundaries. Complete remeshing is performed after each time step. The boundary mesh is reconstructed using an existing algorithm employing element addition/subtraction, edge swapping based on Delaunay triangulation and spring-like dynamical relaxation. The volume mesh is then generated from the boundary using the commercial software Hypermesh. The resulting adaptive discretization maintains resolution of prescribed local length scales. We demonstrate our method with finite-element simulation~ of deformable drops ~ubjected to simple shear under Stokes flow conditions. Steady drop shapes in agreement with experimental data as well as the evolution of slender fluid filaments characteristic of drop breakup are accurately described.

Calculation of droplet deformation at intermediate Reynolds number using a Volume of Fluid technique

The deformation of a droplet or bubble in simple extensional flow fields is investigated at intermediate Reynolds number using a Volume-of-Fluid (VOF) numerical algorithm (MAC2) developed by Rudman. The results of an evaluation study of the MAC2 algorithm for such flows are presented. The final steady shapes of droplets in axisymmetric extensional flow fields determined by MAC2 are compared with the published numerical results of Ramaswamy and Leal. This work forms the initial stage of a more comprehensive study of droplet deformation in time-dependent shear fields experienced by drops flowing through devices (e.g.\ homogenisers) designed for droplet breakup.

Shear-induced rupturing of a viscous drop in a Bingham liquid

Deformation and breakup of a viscous drop in a Bingham liquid is investigated numerically with a volume-of-fluid scheme. Initially, a spherical drop is placed between two moving parallel plates. For our parameters, the matrix liquid has yielded. The competing effects driving the motion are the shear and interfacial tension. When interfacial tension effects dominate, the drop evolves to a steady shape which is elongated compared with the case when the outer liquid is Newtonian. Prior to breakup, stress levels are highest at the ends of the elongated drop. When shearing effects dominate, the drop breaks up, again with features that are elongated compared with the Newtonian counterpart. After the initial breakup, the daughter drops assume steady shapes.

Triplet state solvation dynamics: Basics and applications

Applying solvation dynamics experiments to viscous liquids or glassy materials near their glass transition involves long lived triplet probes, whose time dependent phosphorescence signals depend upon the local dipolar orientational dynamics, mechanical responses, and polarities. The current understanding of experimental results regarding steady state and time dependent optical line shapes and positions is reviewed with emphasis on the relation to the macroscopic dielectric properties. Several applications are discussed in detail, where advantage is taken of the spatially local instead of ensemble averaging character of this technique. These examples include studies of dynamical heterogeneity, rotational solute/solvent coupling, secondary relaxations in the glassy state, as well as confinement and interfacial effects.

Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows

The dynamic behaviour and stability of a two-dimensional immiscible droplet subject to shear or pressure-driven flow between parallel plates is studied under conditions of negligible inertial and gravitational forces. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. The boundary-integral method is used to numerically determine the flow along and dynamics of the free surface. For surfactant-free interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid viscosity ratios, droplet sizes and flow type. It is shown that as the capillary number or viscosity ratio or size of the droplet increases, the deformation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary numbers, shear and pressure-driven flows are shown to yield similar steady droplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective–diffusive transport along the interface and no transport to or from the bulk fluids. Increasing the surface Péclet number, the ratio of convective to diffusive transport, leads to an accumulation of surfactant at the downstream end of the droplet and creates Marangoni stresses that immobilize the interface and reduce deformation. The no-slip boundary condition is then relaxed and an integral form of the Navier-slip model is used to examine the effects of allowing the droplet to slip along the solid surface in a pressure-driven flow. For contact angles less than or equal to 90°, a stable droplet spreads along the wall until a steady shape is reached, when the droplet translates across the wall at a constant velocity. The critical capillary number is larger for these droplets compared to those with pinned contact lines. For contact angles greater than 90°, the wetted area between a stable droplet and the wall decreases until a steady shape is reached. The critical capillary number for these droplets is less than that for pinned droplets. Above the critical capillary number the droplet completely detaches for a contact angle of 120°, or part of it is pinched off leaving behind a smaller attached droplet for contact angles less than or equal to 90°.

Pressure-Driven Motion of Drops and Bubbles through Cylindrical Capillaries: Effect of Buoyancy

We examine the motion and deformation of air bubbles and viscous drops through vertical cylindrical capillaries in the presence of an imposed pressure-driven flow. Experimental measurements of the terminal velocity of drops and bubbles are reported for a wide range of drop sizes in a variety of two-phase systems, and the steady drop shapes are quantitatively characterized using digital image analysis. In contrast to the pressure-driven motion of neutrally-buoyant drops, the relative mobility of a buoyant drop is not a monotonically increasing function of capillary number. The relative mobility is enhanced as the buoyancy force becomes more dominant compared to surface tension, or as the drop fluid becomes less viscous relative to the suspending fluid. However, there is a limiting value of the Bond number beyond which the relative mobility becomes insensitive to the value of the Bond number. Similarly, the thickness of the liquid film surrounding large drops increases rapidly with increasing Bond number, but eventually approaches a constant value as the Bond number exceeds the limiting value. This limiting value of the Bond number is found to be a decreasing function of capillary number. When buoyancy and pressure forces act in the same direction, increasing the Bond number is found to delay the formation of a re-entrant cavity at the trailing end of the drop.

Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers

We present the results of numerical simulations of the three-dimensional thermocapillary motion of deformable viscous drops under the influence of a constant temperature gradient within a second liquid medium. In particular, we examine the effects of shape deformations and convective transport of momentum and energy on the migration velocity of the drop. A numerical method based on a continuum model for the fluid–fluid interface is used to account for finite drop deformations. An oct-tree adaptive grid refinement scheme is integrated into the numerical method in order to track the interface without the need for interface reconstruction. Interface deformations arising from the convection of energy at small Reynolds numbers are found to be negligible. On the other hand, deformations of the drop shape due to inertial effects, though small in magnitude, are found to retard the motion of the drop. The steady drop shapes are found to resemble oblate or prolate spheroids without fore and aft symmetry, with the direction of elongation of the drop depending on the value of the density ratio between the two phases. As in the case of a gas bubble, convection of energy is shown to retard the thermocapillary motion of a viscous drop, as the isotherms get wrapped around the front surface of the drop and effectively reduce the surface temperature gradient which drives the motion. The effect of inertia on the mobility of viscous drops is found to be weaker than that in the case of gas bubbles.

Mathematical modelling of industrial aluminium cells with prebaked anodes: Part I: Current distribution and anode shape

In aluminium electrolysis cells with prebaked anodes the anode shape changes with time after a new anode has been set, reaching a steady state profile after several days. Mathematical modelling of the anode consumption, using current densities obtained by solving the Laplace equation in 2D space, showed that a constant shape is reached after 6–8.6 days, depending on the width of the gap to a neighbouring anode or to the sidewall of the cell and on the shape of the frozen sideledge. The calculated steady state shapes were similar to measured shapes of industrial anodes. The current density decreases along the side of the anode from the nominal value at the underside (0.75 A cm−2) to a minimum near the surface of the electrolyte (0.08–0.28 A cm−2) depending on the geometry. The fraction of the current passing through the sides of the anode is of the order of 15%. Two approaches to the calculation of the anode shape are discussed: one method of incremental time steps, and one method using the 'near steady-state shape’ condition.

Irregular eutectic solidification

A numerical calculation is presented for irregular eutectic growth which takes into account the increased difficulty of solute diffusion at a curved interface and the change in temperature of the interface with position. The validity of using a steady state growth model is questioned. The solute distribution for planar and curved interface shapes is calculated using the boundary element method. These distributions are used to calculate interface shapes and self-consistent shapes for steady state growth. Steady state shapes are not found for the experimentally measured average Si flake spacings in the AlSi eutectic. The largest spacing for which a self-consistent interface shape exists is much smaller than the measured average flake spacing. This suggests that a steady state model cannot be used to describe irregular eutectic growth. A time-dependent model must be developed.

Hele–Shaw flows with a free boundary produced by multipoles

We study Hele–Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions.We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the Appendix we discuss a connection between these solutions and a variational problem of potential theory.

The buoyancy-driven motion of a train of viscous drops within a cylindrical tube

The buoyancy-driven motion of a train of viscous drops settling or rising along the axis of a vertical cylindrical tube is investigated. Under the assumption of creeping flow, the evolution of the drops is computed numerically using a boundary integral method that employs the axisymmetric periodic Green's function for flow in a cylindrical tube. Given the drop volume and assuming that the viscosity of the drops is equal to that of the suspending fluid, the motion is studied as a function of the radius of the tube, the separation between the drops, and the Bond number. Two classes of drops are considered: compact drops whose effective radius is smaller than the radius of the tube, and elongated drops whose effective radius is larger than the radius of the tube. It is found that compact drops may have a variety of steady shapes including prolate and oblate, dimpled tops, and shapes containing pockets of entrained ambient fluid. When the surface tension is sufficiently small, compact drops become unstable, evolving to prolate rings with elongated tails. The terminal velocity of compact drops is discussed and compared with that predicted by previous asymptotic analyses for spherical drops. Steady elongated drops assume the shape of long axisymmetric fingers consisting of a nearly cylindrical main body and two curved ends. Relationships between the terminal velocity of elongated drops, the gap between the drops and the wall of the tube, and the Bond number are established. The results are discussed with reference to previous analyses and laboratory measurements for inviscid bubbles.

The influence of viscoelasticity on the existence of steady solutions in two-dimensional rimming flow

The steady, two-dimensional flows and interface shapes of the rimming flow of Newtonian and viscoelastic liquid films are studied by finite-element analysis. The viscoelastic flow calculations are based on the elastic-viscous split stress (EVSS) formulation for differential constitutive models. The EVSS formulation is derived by taking into account the mathematical type of the momentum, continuity and constitutive equations and is extended in this paper to calculation of free-surface flows. Calculations for a viscous Newtonian fluid demonstrate the balance between viscous forces and gravity which sets the shape of the interface of the liquid film coating the inside of the rotating cylinder. The liquid shape evolves from a concentric and circular film at high rotation rates to become thicker on the rising surface as the rotation rate is lowered. No steady flows with continuous films are found to exist below a minimum rotation rate, Ω = Ωc, where the family of flows evolves back toward higher values of Ω. Multiple solutions are predicted for a range of rotation rates, Ω > Ωc, and unstable flows develop a pronounced bulge on the rising side of the film. Asymptotic analysis for a thin film predicts this limiting rotation rate. Adding viscoelasticity to the liquid, as modelled by the Giesekus constitutive equation, leads to the existence of steady solutions at lower rotation rates and causes the bulge to appear on stable films. The minimum rotation rate for steady, viscoelastic flow shifts to lower values as the time constant of the fluid is increased.

A numerical analysis of dendritic and cellular growth of a pure material investigating the transition from “array” to “isolated” growth

A time dependent numerical model has been used to analyse the growth of dendrites into a supercooled liquid bath of a pure material. Both dendritic plates and dendrites having axial symmetry have been treated. Solutions were obtained over nearly twelve orders of magnitude of velocity and two orders of magnitude of undercooling. For a finite surface energy, discrete solutions rather than a continuum of solutions were obtained for steady state dendritic growth at a fixed undercooling. The problem was considered in the limit of a zero specific heat so that the results could be compared with those obtained for the similar Hele-Shaw problem for which analytical solutions are available. It was found that there was an excellent agreement with the earlier work. The steady state results are plotted to show the relationship between the Hele-Shaw and dendrite problem. The transition between constrained and isolated dendrites was examined. It was found that this two parameter problem could be reduced to a single line in the limit of a small thermal diffusion layer and a large dendrite spacing (an isolated dendrite) and also when the thermal layer was larger than the dendrite spacing (a constrained cell or dendrite). Two types of solution, cell-like or dendritic, were found. The steady state dendritic results agreed quantitatively with the prediction of marginal stability but not with physical basis of the model. It has previously been suggested that these steady shapes were not possible unless the surface energy is anisotropic. Possible reasons for the different behaviour found in the present work are discussed. The steady state shapes were tested for stability using time dependent growth.

Statistical-mechanical selection of the shapes of disk galaxies

A new method is proposed for selecting steady state shapes of disk galaxies as 'most probable states' of a large number of stars, given only the total energy and total angular momentum. A partial differential equation is derived for the mean gravitational potential; it is closely related to the 'sinh-Poisson' equation for the mean-field description of a line vortex system or electrostatic guiding-center plasma. A 'water bag' approximation to the distribution function for bound stars renders the equation analytically tractable, but accurate solution of it may require numerical integration, in view of its general nonlinearity.

Steady thermocapillary flows of thin liquid layers. II. Experiment

The steady thermocapillary flow of nonvolatile layers, nonuniformly heated from below, is examined. The interactions among viscous forces, thermocapillarity, and hydrostatic effects give rise to the steady-state dimpling of the interface. The steady dimpling of nonuniformly heated silicone–oil layers with mean thicknesses ranging from 0.125 to 1.684 mm is studied experimentally. The temperature distribution in the substrate is monitored by thermocouples and the interface shapes by a mechanical impedance probe. Measured steady shapes and theoretical predictions agree within 20% for moderate heating when the film is not close to rupture. When the heating rate causes the film to ‘‘dry out’’ above the hottest point on the substrate, the long-wave theory delivers a parametric index, involving thermocapillary and hydrostatic effects, which is an excellent predictor of rupture. Nonlinear long-wave theories of the type discussed here have never been tested experimentally, until now. The confirmation of this thermocapillary theory is suggestive of the validity of the previous long-wave analysis [Phys. Fluids A 2, 313 (1990)] of unsteady, evaporating/condensing liquid layers.