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 Approximate Lie Group Investigation into the Spreading of a Liquid Drop on a Slowly Dropping Flat Plane

The approximate Lie group method is used to investigate the evolutionof the free surface of a thin liquid drop on a slowly dropping flat plane. Surfacetension effects are ignored. A group classification is performed to determine the rateat which the plane drops. An approximate group invariant solution is then calculatedfor the free surface of an evolving liquid drop on the slowly dropping flat plane. Animportant parameter in the solution is the initial angle of the plane. For small anglesthere is no significant difference in the drop profile. For larger angles, changes in thedrop profile and rate of spreading are significant.     

Gravity-induced spreading of a drop of a viscous fluid over a surface

The unsteady-state nonlinear problem of spreading of a drop of a viscous fluid on the horizontal surface of a solid under the action of gravity and capillary forces is considered for small Reynolds numbers. The method of asymptotic matching is applied to solve the axisymmetrical problem of spreading when the gravity exerts a significant effect on the dynamics of the drop. The flow structure in the drop is determined at large times in the neighborhood of a self-similar solution. The ranges of applicability of the quasiequilibrium model of drop spreading with a dynamic edge angle and a self-similar solution are found. It is shown that the transition from one flow model to another occurs at very large Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen’ 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 59–67, May–June, 1999.     

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.     

Spreading of a liquid over a plane rigid substrate in an electric field

The influence of an electric field on spreading of a thin conducting liquid layer over a plane rigid substrate is investigated theoretically. The conductivity of the liquid is assumed to be so low that the effect of the magnetic field of the currents generated in the liquid under the action of the electric field can be neglected. The spreading is assumed to be so slow that the quasi-steady approximation can be used to calculate the electric field strength which can be considered to be equal to zero inside the liquid. Equations that describe variations in the layer shape are obtained in the lubrication theory approximation. The general formulation of the problem is considered. The solution of the problem is obtained in parametric form when the effect of the gravity force and the surface tension can be neglected. Variations in the layer thickness along the substrate are so smooth that the charge distribution over its surface can be assumed to be the same as that over the substrate surface in the absence of the liquid.     

A new solution for the rotation-driven spreading of a thin fluid film

Lie groups are used to solve the equation governing the flow of a thin liquid film subject to centrifugal spreading and viscous resistance. A new implicit solution is found. It is shown how this relates to the previous known solutions for the spreading of an initially flat film, the steady state and a separable solution. New permissible forms for the film evolution are also studied, including solutions exhibiting finite time blow-up. Near the contact line, where the film height tends to zero, an approximate explicit solution is obtained which may be used to describe a film with any size contact angle.     

Spreading and fingering in a yield-stress fluid during spin coating

We study the deformation, spreading, and fingering of small droplets of a yield-stress fluid subjected to a centrifugal force on a rotating substrate. At low rotation rates and for small enough droplets, the droplets deform elastically but retain their essentially circular contact line. For large enough droplet volumes and rotation speeds, however, one or more fingers eventually form and grow at the edge of the drop. This fingering is qualitatively different from the contact line instability observed in other fluids, and appears to be a localized phenomenon that occurs when the stress at some point on the perimeter of the drop exceeds the yield stress.     

On the dynamics of the fluid balancer

This paper is concerned with the dynamics of a so-called fluid balancer; a hula hoop ring-like structure containing a small amount of liquid which, during rotation, is spun out to form a thin liquid layer on the outermost inner surface of the ring. The liquid is able to counteract unbalanced mass in an elastically mounted rotor. The paper derives the equations of motion for the coupled fluid–structure system, with the fluid equations based on shallow water theory. An approximate analytical solution is obtained via the method of multiple scales. For a rotor with an unbalance mass, and without fluid, it is well known that the unbalance mass is in the direction of the rotor deflection at sub-critical rotation speeds, and opposite to the direction of the rotor deflection at super-critical rotation speeds (when seen from a rotating coordinate system, attached to the rotor). The perturbation analysis of the problem involving fluid shows that the mass center of the fluid layer is in the direction of the rotor deflection for any rotation speed. In this way a surface wave on the fluid layer can counterbalance an unbalanced mass.     

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.     

Note on buble motion in a rotating liquid under residual gravity

The motion of a small gas bubble, presumed to retain its geometrical shape and contained in a rotating liquid, has been investigated. The fluid system liquid-gas is subject to the influence of a reduced gravitational field. It is demonstrated that under certain conditions (spin axis and direction of gravity are perpendicular to each other) the bubble travels on a circular path about the axis of rotation, as seen from an observer moving with the bulk of the liquid.     

Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity

This paper is concerned with the propagation of Rayleigh waves in an incompressible isotropic elastic half-space overlaid with a layer of non-viscous incompressible water under the effect of gravity. The authors have derived the exact secular equation of the wave which did not appear in the literature. Based on it the existence of Rayleigh waves is considered. It is shown that a Rayleigh wave can be possible or not, and when a Rayleigh wave exists it is not necessary unique. From the exact secular equation the authors arrive immediately at the first-order approximate secular equation derived by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. When the layer is assumed to be thin, a fourth-order approximate secular equation is derived and of which the first-order approximate secular equation obtained by Bromwich is a special case. Some approximate formulas for the velocity of Rayleigh waves are established. In particular, when the layer being thin and the effect of gravity being small, a second-order approximate formula for the velocity is created which recovers the first-order approximate formula obtained by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. For the case of thin layer, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate second-order polynomials of the third- and fourth-powers.     

Temperature distribution in a liquid spherical layer around a sphere falling under gravity

An examination is made of the stationary problem of temperature distribution in a liquid of known mass covering a solid sphere falling under gravity, the temperatures of both media with which the liquid is in contact being given. A solution has been obtained under the assumption of constant coefficient of thermal expansion of the liquid. A number of particular cases is examined, the question of stability of the liquid under the temperature distribution obtained is solved, and the total heat content of the liquid is found.     

Dynamic edge angles of wetting upon spreading of a drop over a solid surface

The hydrodynamic free-boundary problem of the axisymmetric spreading of a viscous-fluid drop over the smooth surface of a solid under the action of capillary forces and under the conditions of weak gravitation is considered. For finite inclination angles of the free surface and small capillary numbers, the problem is reduced to the simpler hydrodynamic problem in a region with known boundary by the asymptotic method. An expression for the dynamic edge angle of the drop is obtained. It is shown that in addition to the local inclination angle of the boundary near the contact line of three phases, one drop has several dynamic edge angles. These angles are calculated for small Reynolds and Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 101–107, January–February, 1999.     

Stability of a vertical liquid film with consideration of the marangoni effect and heat exchange with the environment

The stability of a free vertical liquid film under the combined action of gravity and thermocapillary forces has been studied. An exact solution of the Navier-Stokes and thermal conductivity equations is obtained for the case of plane steady flow with constant film thickness. It is shown that if the free surfaces of the film are perfectly heat insulated, the liquid flow rate through the cross section of the layer is zero. It is found that to close the model with consideration of the heat exchange with the environment, it is necessary to specify the liquid flow rate and the derivative of the temperature with respect to the longitudinal coordinate or the flow rate and the film thickness. The stability of the solution with constant film thickness at small wave numbers is studied. A solution of the spectral problem for perturbations in the form of damped oscillations is obtained.     

On the motion of a heavy dynamically symmetric rigid body with vibrating suspension point

The motion of a dynamically symmetric rigid body in a homogeneous field of gravity is studied. One point lying on the symmetry axis of the body (the suspension point) performs high-frequency periodic or conditionally periodic vibrations of small amplitude. In the framework of approximate equations of motion obtained earlier, we find necessary and sufficient conditions for the stability of the body rotation about the vertical symmetry axis and study the existence and stability of regular precessions of the body in the coordinate system translationally moving together with the suspension point.     

The meridian curve of a wetting drop: a boundary value problem and its elliptic integrals solution

In this article we study the shape of free surfaces of a static fluid under gravity. We consider the meridian curve of a heavy liquid drop standing on a horizontal base: the main assumption concerns the liquid wetting capability, namely its contact angle well below \(\pi /2\) . The nonlinear differential boundary problem is solved through the shooting method. Our treatment is self-consistent as holding all demonstrations of existence, uniqueness, and computability. We conclude providing the eigenvalues set to the radius and the meridian curve of the drop through elliptic integrals: such a new exact solution—see (3.9) and (3.10) —is enriching the literature on capillarity.     

Thermocapillary drift of a droplet of viscous liquid

The article discusses the thermocapillary drift of a drop of a viscous liquid, filling the whole space, in another liquid, with a constant temperature gradient at infinity, in the absence of the force of gravity. The distribution of the velocities, temperatures, and pressures in the liquid are obtained in the Ozeenov approximation, and the rate of drift of a drop and its form are found.     

Inertia dominated flow and heat transfer in liquid drop spreading on a hot substrate

The present work deals with computational modeling of the fluid flow and heat transfer taking place in the process of impact of a cold liquid drop (T_d = 20-25 °C) onto a dry heated substrate characterized by different thermophysical properties. The computational model, based on the volume-of-fluid method for the free-surface capturing, is validated by simulating the configurations accounting for the conjugate heat transfer. The simulations were performed in a range of impact Reynolds numbers (Re = 2000-4500), Weber numbers (We = 27-110) and substrate temperatures (T_s = 100-120 °C). The considered temperature range of the drop-surface, i.e. liquid-solid system does not account for the phase change, that is boiling and evaporation. The model performances are assessed by contrasting the results to the reference database originating from the experimental and complementary numerical investigations by Pasandideh-Fard et al. [Pasandideh-Fard, M., Aziz, S., Chandra, S., Mostaghimi, J., 2001. Cooling effectiveness of a water drop impinging on a hot surface. International Journal of Heat and Fluid Flow, 22, 201-210] and Healy et al. [Healy, W., Hartley, J., Abdel-Khalik, S., 2001. On the validity of the adiabatic spreading assumption in droplet impact cooling. International Journal of Heat and Mass Transfer, 44, 3869-3881]. In addition, the thermal field obtained is analyzed along with the corresponding asymptotic analytical solution proposed by Roisman [Roisman, I.V., 2010. Fast forced liquid film spreading on a substrate: flow, heat transfer and phase transition. Journal of Fluid Mechanics, 656, 189-204]. Contrary to some previous numerical studies, the present computational model accounts for the air flow surrounding the liquid drop. This model feature enables a small air bubble to be resolved in the region of the impact point. The reported results agree reasonably well with experimental and theoretical findings with respect to the drop spreading pattern and associated heat flux and temperature distribution.     

Axisymmetric spreading of a thin power-law fluid under gravity on a horizontal plane

Separable solutions admitted by a nonlinear partial differential equation modeling the axisymmetric spreading under gravity of a thin power-law fluid on a horizontal plane are investigated. The model equation is reduced to a highly nonlinear second-order ordinary differential equation for the spatial variable. Using the techniques of Lie group analysis, the nonlinear ordinary differential equation is linearized and solved. As a consequence of this linearization, new results are obtained.     

Two-dimensional problem of the impact of a vertical wall on a layer of a partially aerated liquid

The two-dimensional unsteady problem of the impact of a vertical wall on a layer of a liquid which is mixed with air near the wall and does not contain air bubbles away from the wall is solved in a linear approximation. The gas-liquid mixture is modeled by a homogeneous, ideal, and weakly compressible medium with a reduced sound velocity dependent on the air concentration in the gas-liquid mixture. Outside the gas-liquid layer, the liquid is considered ideal and incompressible. During the initial stage of the impact, the liquid flow and the hydrodynamic pressure are determined using the linear theory of the potential motion of an inhomogeneous liquid. The dependence of the amplitude of the impact pressure along the wall on the air concentration in the gas-liquid layer and on the thickness of this layer is investigated. For a small relative thickness of the layer, the thin-layer approximation is used. It is shown that the solution of the original problem tends to the approximate solution as the thickness of the layer decreases. It is shown that the presence of the gas-liquid layer leads to wall pressure oscillations. Estimates are obtained for the pressure amplitude and the oscillation period. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 34–46, September–October, 2006.