Free surface deformation due to a source or a sink in Stokes flow

Free surface shape and cusp formation are analyzed by considering two-dimensional viscous flow due to a line source or a line sink below the free surface where the strength of source/sink is given arbitrarily. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are shown and the formation of a cusp on the free surface is discussed. As the capillary number decreases in negative, the free surface shape becomes singular and in a real fluid a cusp should form on the free surface below some negative critical capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the small capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Free-surface deformation due to spiral flow owing to a source/sink and a vortex in Stokes flow

The free-surface shape and cusp formation are analyzed by considering a viscous flow arising from the superposition of a source/sink and vortex below the free surface where the strength of the source and vortex are arbitrary. In the analysis, Stokes’ approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are obtained, and the formation of a cusp on the free surface is discussed. Above some critical capillary number with a sink, the free-surface shape becomes singular and an apparent cusp should form on the free surface below a real fluid. On the other hand, no cusp would occur for sources of zero or positive strength. Typical streamline patterns are also shown for some capillary numbers. As the capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Cusped ripples at the plane surface of a viscous liquid

Summary We study the time-evolution of periodical ripples of a viscous liquid at the plane free surface under the action of a distant pure straining flow. We neglect inertial forces (Stokes flow) and include surface tension effects. The solutions for a contracting surface and constant strain rate show that the ripples may develop near-cusps during a stage of the evolution, though later the free surface inevitably asymptotically tends to a smooth plane with vanishing ripples due to the action of capillarity. We obtain the condition for cusp formation in this intermediate stage in terms of the initial capillary number and aspect ratio. If the capillary number is kept constant, the surface tends to shrink through a succession of self-similar trochoidal shapes, whose aspect ratio is given by the capillary number. Received 23 March 1998, accepted for publication 23 July 1998     

Surface tension and buoyancy-driven flow in a non-isothermal liquid bridge

The Navier–Stokes–Boussinesq equations governing the transport of momentum, mass and heat in a non-isothermal liquid bridge with a temperature-dependent surface tension are solved using a vorticity-stream-function formulation together with a non-orthogonal co-ordinate transformation. The equations are discretized using a pseudo-unsteady semi-implicit finite difference scheme and are solved by the ADI method. A Picard-type iteration is adopted which consists of inner and outer iterative processes. The outer iteration is used to update the shape of the free surface. Two schemes have been used for the outer iteration; both use the force balance normal to the free surface as the distinguished boundary condition. The first scheme involves successive approximation by the direct solution of the distinguished boundary condition. The second scheme uses the artificial force imbalance between the fluid pressure, viscous and capillary forces at the free surface which arises when the boundary condition for force balance normal to the surface is not satisfied. This artificial imbalance is then used to change the surface shape until the distinguished boundary condition is satisfied. These schemes have been used to examine a variety of model liquid bridge situations including purely thermocapillary-driven flow situations and mixed thermocapillary- and bouyancy-driven flow.     

Growth and structure of nematic spherulites under shallow thermal quenches

The growth kinetics, shape, interfacial and internal orientation texture of a submicron nematic spherulite arising during the isotropic-to-nematic liquid crystal phase transformation under shallow thermal quenches is analyzed using theory, scaling, and numerical simulations based on the Landau – de Gennes model (The Physics of Liquid Crystals, 2nd edn. Clarendon, Oxford). The numerical computations from this model yield interfacial cusp formation that relaxes through the nucleation of two disclination lines of topological charge +1/2 and subsequently leads to intra-droplet texturing and a net topological charge within the spherulite of +1. The timing of these events suggests that cusp formation at the interface is intimately associated with the interfacial defect shedding mechanism (J. Chem. Phys. 124:244902, 2006) for shallow quenches. These results are different than predictions for deep quenches (J. Chem. Phys. 124:244902, 2006) where interfacial defect shedding leads to four defects and a net topological charge of +2. A liquid crystal dynamic shape equation is derived from the Landau – de Gennes model to account for the interface shape changes in terms of surface viscosity, the driving forces due to the uniaxial nematic-isotropic free energy difference, capillary forces, and friction forces, and used to semi-quantitatively show that during cusp formation and defect shedding, gradient elasticity, capillary forces and friction play significant roles in decelerating and accelerating the surface. An interfacial eigenvalue analysis shows that during the shallow quench, disclination lines nucleate within the interface itself and then texturize the nematic droplet as they migrate from within the interface to the bulk of the growing nematic droplet. After defect shedding, the spherulite is nearly circular and grows with constant velocity, in agreement with experiments. The results shed new light on intra-spherulite texturing mechanisms in phase ordering under weak driving forces.      

Two regimes of liquid film flow on a rotating cylinder

A steady flow of a thin film of a viscous incompressible liquid on a rotating cylinder (the cylinder axis is perpendicular to the direction of the force of gravity) is considered. Capillary effects are taken into account on the free surface. Thin-layer equations derived by Pukhnachov, which depend on the Galileo number and capillary number, are solved. If the first parameter equals zero, the force of gravity also equals zero. If the second parameter equals zero, the surface-tension coefficient also equals zero. The values of these parameters that ensure the solution existence and the number of solutions are determined by the method of collocations. One more solution corresponding to the drop-shaped free surface is found numerically. Variations of flow parameters caused by variations of the Galileo number and capillary number are considered. Branching of the solutions is examined numerically. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 68–78, January–February, 2007.     

Possible classification of the self-oscillatory spouting regimes of plane vertical submerged jets of a heavy fluid

New results of an experimental investigation of self-oscillatory regimes of plane vertical jet spouting from beneath the free surface of a heavy incompressible fluid are discussed. The experiments were performed on a setup with discharge over a weir. The range of dimensionless jet submergence values on which bifurcation change of spouting regime is observable is studied. It is established that on the Froude number and dimensionless jet submergence ranges considered in the study six characteristic spouting regimes differing in free surface shape and self-oscillation frequency can exist. It is shown that these regimes can be subdivided into three typical groups with respect to the dependence of the self-oscillation period on the jet flow rate. A dimensionless parameter that makes it possible to identify the boundaries of the bifurcation change in spouting regimes is obtained for each of these groups. For certain spouting regimes without the formation of free jets numerical calculations are carried out using the STAR-CD software package; the calculated results are in good agreement with experimental data.     

Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.     

Translational motion of a cylinder below the free surface of a fluid

A method of solving the problem of the translational motion of a cylinder of given shape below the free surface of an infinitely deep heavy fluid is developed. As distinct from existing techniques, the method permits the obtaining of a solution which becomes exact as the Froude number increases without bound. The solution of the problem of the motion of a circular cylinder is considered in detail. Suggestions are made concerning the characteristic properties of an exact solution of the general problem.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 9–22, November–December, 1996.     

Flow of a fluid across an obstacle with breaking of the wave front

The phenomenon of breaking of surface gravity waves on the sea in a coastal zone has much in common with a steady hydraulic jump. When surface waves break a vortex forms which rolls down along the leading slope of the wave, as in the case of the formation of a hydraulic jump. Up to the present time a number of theoretical flow models have been proposed for the region in which a wave front breaks. Many of these studies are discussed in [1, 2]. However, the questions of the origin and dynamics of the vortex and its effect on the flow remain open. With the object of studying these questions in greater detail, the results are given below of numerical and experimental simulation of a hydraulic jump. The results of numerical calculations of the shape of the free surface and the velocity profiles in the various sections agree fairly well with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 103–106, May–June, 1985.     

Experimental free coating flows at high capillary and Reynolds number

 Experimental studies are carried out to enhance the fundamental understanding of coating processes over a broad parametric range. Experiments herein identify the phenomena leading to the formation of an asymptotic meniscus profile, which eventually develops a cusp at the interface. The non-dimensional parameters that describe these phenomena are identified. In addition, flow visualization is carried out to reveal the entire flow structure using a visible laser. Two phenomena of free coating are identified depending on a parameter called the property number(Po). When Po is larger than about 0.5, the non-dimensional final film thickness (T ₀) becomes constant beyond the capillary number(Ca) of about unity. When Po is less than about 0.1, T ₀ depends on Ca and the Reynolds number(Re) but it becomes constant beyond the Weber number(=Ca Re) of about 0.2. In both cases T ₀ becomes constant as the effect of surface tension on the meniscus becomes relatively unimportant. The cusp formation is due to the effect of inertia(Re). The effect of applicator dimensions on T ₀ is also investigated for large Re flows. Received: 12 May 1998/Accepted: 19 January 1999     

The Unsteady Wake of a Circular Cylinder near a Free Surface

The behaviour of the wake Strouhal number for flow past a cylinder close to a free surface at both low and moderate Froude numbers is investigated numerically. For the low Froude number case (i.e., gravity-dominated), the results obtained are similar to those for flow past a cylinder close to an adjacent no-slip boundary. As the distance between the wall and the cylinder is reduced, the Strouhal number, as measured from the time varying lift, increases to a maximum at a gap ratio of 0.70. Further gap reduction leads to a rapid decrease in the Strouhal number, with shedding finally ceasing altogether at gap ratios below 0.16. The agreement between the results for a free surface and a no-slip boundary suggests that the mechanism behind the suppression of vortex shedding is common. For flow at a fixed gap ratio and a moderate Froude number, two distinctly different wake states are observed with the flow passing over the cylinder tending to switch from a state of attachment to the free surface, to one of separation from it, and then back again in a pseudo-periodic fashion. Even though there is a significant difference in Reynolds number, the predicted numerical two-dimensional behaviour is found to compare favourably with the experimental observations at higher Reynolds number.     

AN ALGORITHM FOR SIMULATION OF STEADY FREE SURFACE FLOWS

An algorithm to simulate steady, viscous free surface flows is presented in this paper. A Picard-type approach wherein the flow and free surface updates are performed alternately is utilized to iterate for a solution. The procedure is intended for large-scale two- or three-dimensional problems. A surface-intrinsic co-ordinate system which facilities representation of general free surface shapes is used. Using a Galerkin finite element method (GFEM), two free surface updates, namely kinematic and normal stress updates are formulated. It is shown that the effects of surface tension, surface tension gradients and imposition of contact angles can be simulated elegantly within the framework of the GFEM. A novel feature of the updates is that the deformations are sought in a direction normal to the current iterate free surface shape, with the result that the method is ideally suited when used in conjunction with an automatic mesh generator. With the normal stress update a volume constraint can also be imposed. A segregated method is utilized to solve iteratively one degree of freedom at a time for the solution of the flow variables. As a result, the memory and disc space requirements are minimal. Sample problems in extrusion, coating and crystal growth are presented to clearly illustrate the convergence behaviour and accuracy of the algorithm.     

A fluid-saturated porous medium under the action of a moving concentrated load

The problem of motion of a concentrated load along the surface of a fluid-saturated porous medium is studied for a subsonic range of speeds. An analytical solution is found. It is shown that there exists a critical speed equal to the speed of the Rayleigh-type surface waves in a porous elastic medium. If this critical speed is exceeded, then the behavior of the solution and the free surface shape are changed. The free surface shape is analyzed at different speeds.     

Primary and Secondary Infiltration of Wetting Liquid Sessile Droplet into Porous Medium

The infiltration of a wetting droplet into the porous medium is a two-step process referred to as primary and secondary infiltration. In the primary infiltration there is a free liquid present at the porous medium surface, and when no fluid is left on the surface, the secondary infiltration is initiated. In both situations the driving force is the capillary pressure that is influenced by the local medium heterogeneities. A capillary network model based on the micro-force balance is developed with the same formulation applied to both infiltrations. The only difference between the two is that the net liquid flow into the porous medium in the secondary infiltration is equal to zero. The primary infiltration starts as a single-phase (fully saturated) flow and may proceed as a multiphase flow. The multiphase flow emerges as the interface (flow front) becomes irregular in shape. The immobile clusters of the originally present phase can be locally formed due to entrapment. Throughout the infiltration, it was found that portions of the liquid phase can be detached from the main body of the liquid phase forming some isolated liquid ganglia that increase in number and decrease in size. The termination of the secondary infiltration occurs once the ganglia become immobile due to their reduction in size. From the transient solution, the changes in the liquid saturation and capillary pressure during the droplet infiltration are determined. The solution developed in this study is used to investigate the droplet infiltration dynamics. However, the solution can be used to study the flow in fuel cell, nano-arrays, composites, and printing.     

Evolution of the free surface of a plane channel during filling with a viscous fluid

The behavior of the free surface of a viscous incompressible fluid is studied in the process of filling of a plane channel inclined to the gravity direction. The problem is numerically solved within the framework of the Navier-Stokes equations under the assumption that the fluid is Newtonian and capillary effects can be neglected. The numerical technique is based on the joint use of the SIMPLE and invariant methods. Two filling regimes are found to exist, namely, that with a steady shape of the free surface without the formation of internal cavities and that with cavity formation in the initial filling stage characterized by spouting behavior of the free boundary.     

Gas displacement of viscoplastic liquids in capillary tubes

The displacement of viscoplastic liquids in capillary tubes by gas injection is examined. The viscoplasticity alters the flow kinematics and changes dramatically the amount of mass left attached at the tube wall as compared to the Newtonian case, studied experimentally by G.I. Taylor in 1961 [G.I. Taylor, Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech. 10 (1961) 161–165]. Experiments with Carbopol aqueous solutions were performed for different flow rates. A recently proposed viscosity function for viscoplastic liquids was fitted to the rheological data of the Carbopol solutions. A new dimensionless rheological property – the jump number – arises in the dimensionless version of this viscosity function. The results show the effect of the viscoplastic character of the liquid on the free surface shape and on the thickness of the film of liquid left attached to the wall. This thickness decreases with the jump number and increases with the flow rate. It is also observed that there is a critical dimensionless flow rate below which the displacement is apparently perfect, i.e. there is no observable liquid left attached to the wall. This behavior is shown to be directly related to the fully developed flow far ahead the air–liquid interface.     

The development of Marangoni convection induced by local addition of a surfactant

The development of concentration convection induced by local addition of a surfactant solution onto a horizontal free surface of water is studied experimentally and theoretically. The experiment revealed that the capillary motion develops in a threshold manner, with the threshold value depending on the degree of purification of the fluid, the initial concentration of the surfactant, and the area of the free surface. To describe the threshold mechanism of the concentration convection, a number of theoretical models is considered. Different rheological properties of the surface phase, including the nonlinear dependence of the surface shear stress on the surface velocity, are examined. In the numerical experiment, the convective-flow patterns are calculated for different free-surface boundary conditions, and the time dependence of the flow intensity is investigated.     

Modeling of convective structures in a thin layer of silicone oil in air flow under heating

Numerical simulation was performed to study convective structures in a thin silicone oil layer heated from below, whose free surface is exposed to air flow generating drift flow. The basic equations are transformation to a form suitable for spectral methods. The steady flow velocity profile obtained in a laboratory experiment is calculated. It is shown that increasing the Reynolds number leads to the transition from polygonal convective cells to longitudinal rolls (elongated along the flow). The dependence of the transition Reynolds number on the temperature on the lower boundary of the layer is obtained. The calculation results are compared with experimental data.     

Sessile drop deformations under an impinging jet

The problem of steady axisymmetric deformations of a liquid sessile drop on a flat solid surface under an impinging gas jet is of interest for understanding the fundamental behavior of free surface flows as well as for establishing the theoretical basis in process design for the Aerosol \({{\rm Jet}^{\circledR}}\) direct-write technology. It is studied here numerically using a Galerkin finite-element method, by computing solutions of Navier–Stokes equations. For effective material deposition in Aerosol \({{\rm Jet}^{\circledR}}\) printing, the desired value of Reynolds number for the laminar gas jet is found to be greater than ~500. The sessile drop can be severely deformed by an impinging gas jet when the capillary number is approaching a critical value beyond which no steady axisymmetric free surface deformation can exist. Solution branches in a parameter space show turning points at the critical values of capillary number, which typically indicate the onset of free surface shape instability. By tracking solution branches around turning points with an arc-length continuation algorithm, critical values of capillary number can be accurately determined. Near turning points, all the free surface profiles in various parameter settings take a common shape with a dimple at the center and bulge near the contact line. An empirical formula for the critical capillary number for sessile drops with \({45^{\circ}}\) contact angle is derived for typical ranges of jet Reynolds number and relative drop sizes especially pertinent to Aerosol \({{\rm Jet}^{\circledR}}\) printing.