The Onset of Fluid Rotation in a Thermogravitational Boundary Layer with Local Cooling of the Free Surface

Axisymmetric regimes of flows of an inhomogeneous fluid in the boundary layer near a free surface are calculated for a nonuniform temperature distribution on this surface. For the fluid motion equations written in the Oberbeck-Boussinesq approximation, the leading terms of asymptotic expansions of solutions of a steady-state problem are constructed. It is shown that in the presence of local cooling of the free surface and a rising outer fluid stream, as a result of a bifurcation, a pair of rotational regimes may develop in a thin boundary layer near the free surface, with no rotation observed outside this layer. No bifurcation of rotation was detected in the case of local heating of the free surface.     

Formation of a cavity in the initial stage of motion of a circular cylinder in a fluid with a constant acceleration

This paper considers the joint motion of an ideal fluid and a circular cylinder completely immersed in it at small times. It is assumed that the cylinder, which was initially at rest, moves in a horizontal direction with a constant acceleration. The dynamics of the internal and external free boundaries of the fluid at small times is studied. An asymptotic analysis of the form of the internal free surface near the separation points is performed. It is shown that at high acceleration of the circular cylinder, a large cavity is formed behind, with a strong perturbation of the external free surface of the fluid over the surface of the cylinder.     

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.     

Nonlinear analysis of the finite-amplitude oscillations of a charged liquid layer on a rigid spherical core for the multimodal initial deformation of the free surface

Within the framework of the analytical asymptotic procedure, in the quadratic approximation in the small dimensionless amplitude of the capillary oscillation of a charged electroconductive fluid layer on the surface of a rigid spherical core, the problem of calculation of the free surface shape at an arbitrary moment is solved for a multimodal initial deformation. The conditions under which the internal nonlinear resonant interaction between the liquid layer oscillation modes and the intermodal energy exchange are realized are analyzed for both the degenerate and the secondary combinational resonances.     

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.     

Initial stage of the circular cylindermotion in a fluid after an impact with the formation of a cavity

The joint motion of an ideal fluid and a submerged circular cylinder is considered in the initial stage after an impact. The dynamics of separation points on the inner free boundary (cavity boundary) and the shapes of the inner and outer free boundaries of the fluid are determined. An asymptotic analysis of the inner free boundary near the separation points is made. The effects of the Froude number, the pressure difference, and the cylinder immersion depth are investigated.     

Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.     

Nonlinear faraday waves in a parametrically excited circular cylindrical container

IntroductionIn 1 83 1 ,Faraday[1]reportedhisexperimentalobservationofsurfacewavesindifferentfluidscoveringahorizontalplatesubjectedtoaverticalvibration ,andheobservedthesurfacestandingwavesoffluidsliketheteethofaveryshortcoarsecomb .Heremarksthatthesesurfacewaveshaveafrequencyequaltoonehalfthatoftheexcitation .ThisisthefamousFaradayexperiment.WedesignatethosefluidsurfacewavesformedbyverticallyexcitationandhaveafrequencyequaltoonehalfthatoftheexcitationasFaradaywaves.FollowingthisproblemMatth…     

An asymptotic theory for a gradually varied flow problem

The steady laminar flow of a heavy liquid around a horizontal and circular cylinder is investigated in this paper. The fluid flows with a constant flow rate from an inclined flat plate and then reaches tangentially the cylinder. An asymptotic method is used to calculate the hydrodynamic characteristics of the flow and special attention is put on the equation of the free surface far from the entry zone.     

Effect of capillary wave radiation on the oscillations of a vibrator

The oscillations of a rigid body on an elastic tie (vibrator) in an ideal incompressible fluid with a free boundary, on which surface tension forces act, are considered. The linearized problem of hydrodynamics is solved approximately in the self-consistent formulation, the reaction forces exerted on the body by the fluid are calculated, and an integrodifferential equation of motion is obtained. Using asymptotic methods, the average characteristics determining the damping coefficient and the frequency shift of the oscillations of the vibrator are obtained with allowance for the effect of the capillary waves radiated by the vibrator. Qualitative effects depending on the parameters of the system are revealed. The authors' numerical simulation of the motion of the vibrator completely confirms the qualitative conclusions concerning the nature of the oscillations of a body in a fluid having surface tension.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–132, March–April, 1995.     

An asymptotic iteration method for the numerical analysis of near-critical free-surface flows

Satisfying the boundary conditions at the free surface may impose severe difficulties to the computation of turbulent open-channel flows with finite-volume or finite-element methods, in particular, when the flow conditions are nearly critical. It is proposed to apply an iteration procedure that is based on an asymptotic expansion for large Reynolds numbers and Froude numbers close to the critical value 1.The iteration procedure starts by prescribing a first approximation for the free surface as it is obtained from solving an ODE that has been derived previously by means of an asymptotic expansion (Grillhofer and Schneider, 2003). The numerical solution of the full equations of motion then gives a surface pressure distribution that differs from the constant value required by the dynamic boundary condition. To determine a correction to the elevation of the free surface we next solve an ODE that is obtained from the asymptotic analysis of the flow with a prescribed pressure disturbance at the free surface. The full equations of motion are then solved for the corrected surface, and the procedure is repeated until criteria of accuracy for surface elevation and surface pressure, respectively, are satisfied.The method is applied to an undular hydraulic jump as a test case.     

Generation of free-surface gravity waves by an unsteady Stokeslet

The interaction of unsteady Stokeslets with the free surface of an initially quiescent incompressible fluid of infinite depth is investigated analytically for two- and three-dimensional cases. The disturbed flows are generated by an unsteady singular force moving perpendicularly downwards away from the surface. The analysis is based on the assumption that the motion satisfies the linearized unsteady Navier–Stokes equations with linear kinematic and dynamic boundary conditions. Firstly, the asymptotic representation for the transient free-surface waves due to an instantaneous Stokeslet is derived for a large time with a fixed distance-to-time ratio. As is well known, the corresponding inviscid waves predicted by the potential theory do not decay to zero as the time goes to infinity. In the present study, the transient waves predicted by the viscous theory eventually vanish due to the presence of viscosity, which is consistent with reality from the physical point of view. Secondly, the asymptotic solutions are obtained for the unsteady free-surface waves due to a harmonically oscillating Stokeslet. It is found that the unsteady waves can be decomposed into steady-state and transient responses. The steady state can be attained as time approaches infinity. It is shown that the viscosity of the fluid plays an important role in the evolution of the singularity-induced waves.     

Symmetry of free-surface flows

Certain geometrical properties of steady two-dimensional potential flows in the presence of gravity are considered. The two main problems are that of a solitary wave in an interface between two immiscible fluids of different densities, and that of steady waves in a free surface of a fluid flowing past an object (such as a ship hull, hydrofoil or surfboard). Our results include statements of positivity (or negativity), monotonicity, symmetry and the asymptotic behavior at infinity of such flows. The methods are extensions of those presented in earlier work on the solitary wave [7].     

Convective wall plume in power-law fluid: Second-order correction for the adiabatic wall

This paper considers the steady-state free convection flow arising from an infinitely long horizontal line source of heat embedded in the base of a vertical adiabatic surface when the ambient fluid is a non-Newtonian fluid for moderately large values of the generalized Grashof numbers by the method of matched asymptotic expansions. In particular, the second-order corrections to account for the non-boundary layer effects have been predicted. A family of numerical solutions for the power-law fluid behavior indexn ranging from 0.4 to 2.0 and for the Prandtl numberPr=10 and 100 are reported.     

Axisymmetric deformation of poroelastic shells of revolution

A linear theory is developed for axisymmetric deformation of thin poroelastic shells of revolution. With fluid solid coupling included through Biot's consolidation theory, results are presented for cylindrical shells with an oscillating internal pressure and various surface boundary conditions on the fluid. First, the effects of fluid flow and shell inertia on the stretching behavior are studied through a separation of variables solution. Then, the bending behavior near a clamped edge is examined through an asymptotic solution of a matrix form of the governing equations. The results show that the asymptotic solution is accurate in the low frequency range, when the loading time is large compared to the consolidation time. In addition, for the examples studied, the fluid flow influences the membrane more than the bending behavior, but damping due to flow resistance is limited near resonance.     

The slow rotation of an axisymmetric solid submerged in a fluid with a surfactant surface layer—II. The rotating solid in a bounded fluid

This paper extends the work in Shail (1979) on the problem of an axisymmetric submerged solid rotating slowly and steadily in a fluid whose surface is covered with a surfactant film. The bulk fluid is of finite extent, and both asymptotic and numerical results (the latter in the case of a thin circular disk) are given for boundary effects on the resistive torque and surface velocity profile when the container is a right circular cylinder and the fluid is of finite depth.     

Separation Impact on a Plate Floating on the Surface of an Ideal Incompressible Fluid in a Bounded Tank

The paper studies the planar problem of separation impact on a plate floating on the surface of an ideal incompressible fluid in a bounded tank. The problem is solved using an asymptotic method under the assumption that the immovable rigid walls of the tank are at a large distance from the plate. It is concluded that the tank walls of arbitrary shape have an ambiguous effect on the fluid particle separation zone formed on the plate surface is revealed. Examples of solutions are given.     

Marangoni convection in weld pools with a free surface

A steady-state two-dimensional model of heat transfer and fluid flow was developed to describe Marangoni convection in the weld pool. Both the pool surface and the fusion boundary were calculated. The validity of the model was verified against an asymptotic solution for Marangoni-convection-induced free surface geometry. Two different cases were studied, i.e. a negative surface tension temperature coefficient ?γ/?T a positive one, and the resultant shapes of the weld pool surface were compared.     

Stability of wave forms of motion of a viscous fluid layer under tangential stress

We study the stability of wave flow of a viscous incompressible fluid layer subjected to tangential stress and an inclined gravity force with respect to long-wave disturbances.An asymptotic solution is constructed for the equations of the disturbed motion and the problem is reduced to the study of a second-order ordinary differential equation. It is shown that after loss of stability by a Poiseuille flow the laminar nature of the flow is not destroyed, but the form of the free surface acquires a wave-like profile. The Poiseuille regime is stable for low Reynolds numbers. The critical Reynolds number for wave flow is found, and the stability and instability regions are determined.     

On thin film flow of a third-grade fluid down an inclined plane

An exact solution for the thin film flow of a third-grade fluid on an inclined plane is presented. This is a corrected version of the solution obtained by Hayat et al. (Chaos Solitons Fractals 38:1336–1341, 2008). An alternative parametric form for the solution is also derived. The variation of the dimensionless velocity and average velocity is given for a wide range of parameter values. An asymptotic solution for large parameter values is obtained giving rise to a boundary-layer structure at the free surface.