An Approximate Lie Group Investigation into the Spreading of a Liquid Drop on a Slowly Dropping Flat Plane II: Small Surface Tension Effects

The spreading of a thin liquid drop under gravity and small surfacetension on a slowly dropping flat plane is investigated. The initialslope of the flat plane is assumed to be small. By considering astraightforward forward perturbation, the fourth-order nonlinear partialdifferential equation modelling the spreading of the liquid drop reducesto a second-order nonlinear partial differential equation. Thisresulting equation is solved using the classical Lie group method. Thegroup invariant solution is found to model the long time behaviour ofthe liquid drop.     

An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk

The axisymmetric spreading of a thin liquid drop under the influence of gravity and rotation is investigated. The effects of the Coriolis force and surface tension are ignored. The Lie group method is used to analyse the non-linear diffusion-convection equation modelling the spreading of the liquid drop under gravity and rotation. A stationary group invariant solution is obtained. The case when rotation is small is considered next. A straightforward perturbation approach is used to determine the effects of the small rotation on the solution given for spreading under gravity only. Over a short period of time no real difference is observed between the approximate solution and the solution for spreading under gravity only. After a long period of time, the approximate solution tends toward a dewetting solution. We find that the approximate solution is valid only in the interval t∈[0,t∗), where t∗ is the time when dewetting takes place. An approximation to t∗ is obtained.     

Equilibrium of a free nonisothermal liquid film

The equilibrium of a free weightless liquid film fixed over a planar contour and acted upon by thermocapillary forces is studied. Trends in the behavior of free liquid films are important for understanding the processes occurring in foams. The equilibrium equations for a nonisothermal weightless free film are derived for the two limiting cases: the temperature of the film is considered a known function of the coordinates; the free surface of the film is thermally insulated. For the plane and axisymmetric cases, the existence conditions for the solutions of the resulting nonlinear boundary-value problems are found and their properties are studied. For the general case, an approximate solution of the equilibrium problem is obtained provided that the analogue of the Marangoni number is small. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 16–29, May–June, 2007.     

Axisymmetric spreading of a thin liquid drop with suction or blowing at the horizontal base

The axisymmetric spreading under gravity of a thin liquid drop on a horizontal plane with suction or blowing of fluid at the base is considered. The thickness of the liquid drop satisfies a non-linear diffusion equation with a source term. For a group invariant solution to exist the normal component of the fluid velocity at the base, v_n, must satisfy a first-order quasi-linear partial differential equation. The general form of the group invariant solution for the thickness of the liquid drop and for v_n is derived. Two particular solutions are considered. Each solution depends essentially on only one parameter which can be varied to yield a range of models. In the first solution, v_n is proportional to the thickness of the liquid drop. The base radius always increases even for suction. In the second solution, v_n is proportional to the gradient of the thickness of the liquid drop. The thickness of the liquid drop always decreases even for blowing. A special case is the solution with no spreading or contraction at the base which may have application in ink-jet printing.     

On the undamped free vibration of a mass interacting with a Hertzian contact stiffness

Three methods are used to determine the natural frequency of undamped free vibration of a mass interacting with a Hertzian contact stiffness. The exact value is determined using the first integral of motion. The harmonic balance method is used on a transformed equation for an approximate solution, and the multiple scales method is used on an approximate equation. The maximum initial displacement avoiding contact loss is also determined, and the corresponding exact natural frequency is also obtained analytically. The methods are evaluated by studying the free vibration of an elastic sphere on a flat rigid surface.     

Surface shape of a magnetic liquid drop near the edge of a magnetic wedge

The two-dimensional problem of the shape of the free surface of a magnetic fluid in a gravity field, a uniform external magnetic field and the nonuniform field of a magnetized metal wedge is considered. The results of numerically calculating the shape of the free surface of a magnetic liquid drop retained on an inclined plane by the field of a magnetizing wedge are presented. The changes in the shape of the free surface of an infinite volume of magnetic liquid near the edge of a wedge with increase in the external field are investigated. It is shown that for a certain critical field some of the magnetic liquid separates and adheres to the edge of the wedge. Experimental data on the determination of the maximum cross-sectional area of a drop retained by the magnetic field of a wedge and the critical rise of the magnetic liquid relative to the level outside the field are presented. The experimental and theoretical results are in agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 113–119, July–August, 1992.The authors wish to thank V. V. Gogosov for useful discussions and his interest in the work.     

Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat stamp profile.     

The cooling of a lava flow spreading over a flat surface

The cooling of a lava flow modeled by a viscous incompressible fluid spreading over a flat surface is considered. In order to model the free surface, a known analytical solution is used in the thin-layer approximation. The thermal boundary layer thickness is determined and the evolution of thermal fields in the lava profile is studied.     

沿垂等温平板的自然对流问题的近似解析解

本文用加权残值得到了沿垂等温平板的自然对流问题的近似解析解。和已有准确解的比较表明，在平板表面附近，近似解有很好的精度，而且速度场的精度比温度场更好一些。文末还对配域法的准收敛问题作了说明。     

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.     

Evolution and detachment of slowly growing drops and bubbles

The deformation of the free surface is considered for a slowly growing drop or bubble with a horizontal solid wall (either continuous or having a circular hole). The volume at the instant of detachment is determined. It is assumed that the liquid is subject to surface tension and gravitational forces.     

Nonaxisymmetric equilibrium shapes of a drop on a plane

A nonuniform temperature distribution, the presence of surface-active substances and impurities, and also other factors lead to a change in the wetting angle along a plane. A study is made of the influence of a small perturbation of the equilibrium contact angle on the shape of the free surface of the liquid. Two cases are considered: a surface of small slope in a gravity field and a nearly spherical shape under conditions of weightlessness. The equilibrium shapes of a liquid drop on an inclined plane under conditions of hysteresis of the wetting are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 164–167, July–August, 1983.I thank I. E. Tarapov and I, I, Ievlev for constant interest in the work and valuable comments.     

Standing waves of finite amplitude on surface of ideal liquid of finite depth

The nonlinear problem of plane gravitational standing waves of finite amplitude on the surface of an ideal incompressible liquid of infinite depth has been solved analytically [1], It was also solved in [2] using a new method, in which the dimensionless velocity potential, the profile of the free surface, and the frequency were expressed as power series in the parameter , equal to the ratio between the the amplitude and the wavelength. The results of these two papers agree. Below, the method of [2] is used to study plane standing surface waves of finite amplitude on the surface of a liquid of finite depth. The frequency, the profile of the free surface, and the velocity potential are expressed as power series in the small parameter . The solution is obtained in the third approximation. An expression for the amplitude dependence of the frequency is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–43, March–April, 1978.     

Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

A weakly nonlocal phase-field model is used to define the surface tension in liquid binary mixtures in terms of the composition gradient in the interfacial region so that, at equilibrium, it depends linearly on the characteristic length that defines the interfacial width. Contrary to previous works suggesting that the surface tension in a phase-field model is fixed, we define the surface tension for a curved interface and far-from-equilibrium conditions as the integral of the free energy excess (i.e., above the thermodynamic component of the free energy) across the interface profile in a direction parallel to the composition gradient. Consequently, the nonequilibrium surface tension can be widely different from its equilibrium value under dynamic conditions, while it reduces to its thermodynamic value for a flat interface at local equilibrium. In nonequilibrium conditions, the surface tension changes with time: during mixing, it decreases as the inverse square root of time, while in the linear regime of spinodal decomposition, it increases exponentially to its equilibrium value, as nonlinear effects saturate the exponential growth. In addition, since temperature gradients modify the steepness of the concentration profile in the interfacial region, they induce gradients in the nonequilibrium surface tension, leading to the Marangoni thermocapillary migration of an isolated drop. Similarly, Marangoni stresses are induced in a composition gradient, leading to diffusiophoresis. We also review results on the nonequilibrium surface tension for a wall-bound pendant drop near detachment, which help to explain a discrepancy between our numerically determined static contact angle dependence of the critical Bond number and its sharp-interface counterpart from a static stability analysis of equilibrium shapes after numerical integration of the Young-Laplace equation. Finally, we present new results from phase-field simulations of the motion of an isolated droplet down an incline in gravity, showing that dynamic contact angle hysteresis can be explained in terms of the nonequilibrium surface tension.     

Reflection of a beam of elastic waves by a periodic surface profile

Reflection of elastic waves from a traction-free solid-air boundary of periodic saw-tooth profile is investigated analytically and experimentally. For an incident plane wave the surface displacements on the profile are computed as the solution of a singular integral equation. The reflected field is subsequently obtained by using an integral representation. Incident beams of finite width are represented by Fourier superpositions of plane waves. The dependence of the reflected signal spectra on the incident beam width is examined closely near the fundamental surface resonance frequency. Experimental spectra which were obtained using two different diameter transducers, are compared to the corresponding theoretical spectra. It is found that the depth of the spectral minima depends on the incident beam width. Both analytical and experimental results exhibit the splitting of an incident beam of elastic waves into two reflected beams. The beam splitting is more pronounced for a narrower incident beam and for frequencies close to a resonance frequency of the profile.     

Brinkman Flow Past a Stretching Sheet

The problem of steady viscous flow of an incompressible fluid over a flat deformable sheet in a porous medium, when the sheet is stretched in its own plane is revisited. An exact solution is recovered for the two-dimensional case and a totally analytic approximate solution is developed for the axisymmetric case. Stretching rate of two-dimensional case is assumed as double the stretching rate of axisymmetric case. The analytical expressions of residual errors, horizontal, vertical velocity distributions, stream lines, vorticity lines, pressure distributions have been obtained and plotted. The values of skin friction, entrainment velocity, boundary layer thickness, momentum thickness and energy thickness have been tabulated. For the first time, two-dimensional and axisymmetric cases are compared by means of a unified scale.     

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 3 September 1997     

On the slow translation of a solid submerged in a fluid with a surfactant surface film—II

In Shail & Gooden (1982) the problem of a solid particle translating in a semi-infinite fluid, whose surface is contaminated with a surfactant film, was examined in the quasi-steady Stokes flow régime. Various linearised models governing the variation of film concentration were considered, but the analysis was approximate in that the fluid motion generated was represented by that due to a Stokeslet situated at the centre of the particle. In this paper we remove the latter restriction and treat two specific solids, namely a rigid flat circular disk and a sphere, which move axisymmetrically perpendicular to the fluid surface. This surface is assumed to remain plane throughout the motion. The velocity field in the translating-disk problem is represented in terms of harmonic functions, and the resulting mixed boundary-value problems are reduced, for each of the film behaviours examined, to the solution of sets of simultaneous Fredholm integral equations of the second kind. These equations are solved both iteratively and numerically, and the drag on the disk is computed. For the sphere a stream-function formulation in bispherical coordinates is used. Application of the boundary conditions at the sphere and film results in infinite sets of simultaneous linear equations for the coefficients in the eigenfunction expansion of the stream function. These equations are solved by the method of truncation, and the drag on the sphere is determined.     

An integral treatment for non-Darcy free convection over a vertical flat plate and cone embedded in a fluid-saturated porous medium

An integral treatment was proposed for analysis of non-Darcy free convection over a vertical flat plate and cone within a fluid-saturated porous medium. A flexible one-parameter family of third order polynomials was employed to cope with vast changes in the velocity and temperature profiles encountered in the Darcy flow limit through to the Forchheimer flow limit. Zero curvature requirement for the temperature profile at the wall was exploited as an auxiliary relation to determine the shape parameter. Comparison of the approximate results with the exact solution reveals a high performance of the present integral procedure for heat transfer rat prediction.     

Scattering of pulsed Rayleigh surface waves by a cylindrical cavity

A pulsed Rayleigh surface wave of prescribed shape is incident on a cylindrical cavity which is parallel to both the plane free surface and the plane wave front. Multiple reflections at the cylindrical and plane free surface are considered and the resulting displacements and stress components are calculated in the surrounding of the cavity by approximately summing infinite double sums. Use is made of the stationary loading case simulated by a periodic train of wave pulses and its time Fourier series representation and of expansions of all incident and reflected waves in terms of cylindrical wave functions. For reflection, the free surface of the half-space is approximated by a fictitious convex (or concave) cylindrical surface of “large” radius. The wave pattern due to a single pulse loading is constructed from the stationary solution by enforcing homogeneous initial conditions in the half-space ahead of the single loading pulse and by prescribing a wide spacing in the periodically set-forth train of pulses. The numerical results for stresses and dynamic stress magnification factors are especially useful for the interpretation of recent measurements in dynamic photoelasticity.