Steady wave regimes in a film of viscous liquid flowing down a vertical wall

A study is made of the steady wave solutions of the nonlinear third-order differential equation [1] that describes the behavior of the wave boundary of a thin film of viscous liquid flowing down a vertical wall. It is shown that for long waves of small amplitude the general equation can be reduced [2] to a form containing a unique dimensionless parameter. A qualitative investigation is made of the behavior of the integral curves and the types of the singular points in the phase space. It is shown that a solitary wave exists for discrete values of the dimensionless parameter. A numerical solution is obtained. The structure of the jump in the thickness of the film is investigated qualitatively. Numerical solutions of nonmonotonic structure are obtained for different parameters.     

Stability of a viscous liquid film flowing down a periodic surface

The paper is devoted to a theoretical analysis of linear stability of the viscous liquid film flowing down a wavy surface. The study is based on the Navier–Stokes equations in their full statement. The developed numerical algorithm allows us to obtain pioneer results in the stability of the film flow down a corrugated surface without asymptotic approximations in a wide range over Reynolds and Kapitsa’s numbers. It is shown that in the case of moderate Reynolds numbers there is a region of the corrugation parameters (amplitude and period) where all disturbances decay in time and the wall corrugation demonstrates a stabilizing effect. At the same time, there exist corrugation parameters at which the steady-state solution is unstable with respect to perturbations of the same period as the period of corrugation. In this case the waveless solution cannot be observed in reality and the wall corrugation demonstrates a destabilizing effect.     

Numerical study of boundary layer transition in flowing film evaporation on horizontal elliptical cylinder

A numerical study of the onset of longitudinal transition between turbulent and laminar regimes during the evaporation of a water film is presented. These water film streams along a horizontal elliptical tube under the simultaneous effects of gravity, pressure gradients, caused by the vapor flow and curvature, and viscous forces. At the interface of water vapor, the shear stress is supposed to be negligible. Outside the boundary layer, the vapor phase velocity is obtained from potential flow. In the analysis Von Karmans turbulence model is used and the inertia and convection terms are retained. Transfers equations are discretised by using the implicit Keller method. The effects of an initial liquid flow rate per unit of length, Froude number, temperature difference between the wall and the liquid–vapor interface and ellipticity on the transition position have been evaluated. The transition criterion has been given in term of the critical film Reynolds number (Re)_C.     

Nonisothermal steady flow of a viscoplastic liquid between two coaxial rotating cylinders

Rotational viscosimeters are widely used to determine liquid viscosity. The technique for processing the experimental data is based fundamentally on the analytic solution of the problem of isothermal flow of a viscous liquid between two rotating cylinders.If in the course of the experiment the heat released due to the internal friction leads to significant heating, then the processing of the experimental results using the equations obtained on the assumption of isothermocity of the flow may lead to large errors. The dissipative heating may be reduced by reducing the angular velocity of rotation of the cylinder; however extensive reduction of the angular velocity is not desirable, since this leads to an increase of the measurement relative error. In addition, there is the possibility of conducting the experiments with a wide variation of the angular velocities in order to identify the structural-rheological peculiarities of the liquid. In the latter case we must be able to separate the purely thermal effects from the influence of the rheological factors. All these questions are discussed in detail in [1]. The authors of [1] obtained the solution of the problem of nonisothermal flow of a Newtonian fluid between two rotating cylinders and gave a technique for processing the experimental data which takes account of the dissipative heating of the fluid. The present paper pursues the same objective for a visco-plastic fluid.An attempt to solve the problem of nonisothermal flow of a viscoplastic fluid was made by Dzhafarov in [2], where the problem was solved in two versions. In the first version it was considered that the viscosity varies hyperbolically with the temperature and the gap between the cylinders is small in comparison with the radius of the inner cylinder. As a result of the linearization of the equations of motion and heat balance, it turned out that in fact the problem of Couette flow of a viscoplastic fluid was solved rather than the original problem. In this case, naturally, such a characteristic of the flow of a viscoplastic fluid in an annular gap as the possibility of the formation of an elastic zone was not covered. In the second version the problem was solved under the assumption that the viscosity is independent of the temperature and the angular velocity is small.In the present study the problem is solved without the limitations discussed above on the angular velocity, the fluid visoosity, and the gap between the cylinders. In this case we consider two types of temperature boundary conditions: a) constant temperatures are specified on the surfaces of the cylinders, which in the general case may be different; and b) a constant temperature is given on the surface of the outer cylinder and the inner cylinder is thermally insulated.     

Solitary waves in a viscous liquid film flowing down a thin vertical cylinder

Several equations to describe the flow of a viscous liquid film on a thin cylinder are derived. The solitary-wave solutions to these equations are studied. The families of solutions are constructed for the first two eigenvalues that correspond to single-humped and double-humped waves. It is found that these families become similar as the similarity parameter increases. The dependencies of phase velocities and wave amplitudes on the free parameters of the problem are analyzed. The resulting solutions are compared with solitary waves in films on a flat surface.     

Numerical study of the strong evaporation of a binary mixture

The problem of the one-dimensional evaporation of a binary mixture is investigated by numerically solving a system of two coupled Boltzmann equations. The numerical method is based on the direct discretization of the Boltzmann equation and the Monte Carlo evaluation of the collision integrals. It is assumed that the fluid flows between an evaporating plate and a totally absorbing plate. The spatial profiles of macroscopic quantities as well as evaporation rates have been calculated for values of the Knudsen number between 1/2 and 1/20.     

Perfect gas convection in a porous medium between two coaxial horizontal cylinders of large length

Natural convection of a perfect gas in a porous medium between two coaxial horizontal cylinders of large length located in heat-conducting space is considered. The two-dimensional problem (thin porous ring) is investigated in a plane orthogonal of the axis of the cylinders. The dependence of the criterion of the onset of convection on the non-Boussinesq parameters is studied. In the steady-state case an analytic solution of the nonlinear problem is obtained and its asymptotic behavior is considered for large Rayleigh numbers and when the compressibility criterion tends to zero. The gas flow rate in the ring and other characteristics of convection are studied as functions of the gas compressibility criterion and a constant temperature gradient given far away from the contour.     

Wave regimes on a nonisothermal film of a viscous liquid flowing down a vertical plane

A thin film flow of a viscous liquid flowing down a vertical wall in the field of the gravity force is studied. The values of temperatures on the solid wall and on the free surface are constant. The viscosity and thermal diffusivity are functions of temperature. An equation that describes the evolution of surface disturbances is derived for small flow rates in the long-wave approximation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 89–97, March–April, 2008.     

Flow of a plane film of viscous incompressible liquid down a dry wall

Fluid Dynamics - A study is made of the flow regimes of a plane film of viscous liquid whose leading edge moves down a dry vertical wall with constant velocity, The study is based on a nonlinear...     

Mixed convection film condensation from downward flowing vapors onto a sphere with variable wall temperature

A model is developed for the study of mixed- convection film condensation from downward flowing vapors onto a sphere with variable wall temperature. The model combined natural convection dominated and forced convection dominated film condensation, concerning effects of pressure gradient (P), interfacial vapor shear drag and non-uniform wall temperature variation (A), has been investigated and solved numerically. The effect of pressure gradient on the dimensionless mean heat transfer, NuˉRe^−1/2 will remain almost uniform with increasing P until for various corresponding available values of F. Meanwhile, the dimensionless mean heat transfer, NuˉRe^−1/2 is increasing significantly with F for its corresponding available values of P. Although the non-uniform wall temperature variation has an appreciable influence on the local film flow and heat transfer; however, the dependence of mean heat transfer on A can be almost negligible. Received on 10 October 1996     

Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder

PIV technique is applied for measurements of instant velocity distributions in a liquid film flowing down an inclined tube in the form of a wavy rivulet. An application of special optical calibration is applied to correct distortion effects caused by the curvature of the interface. A vortex flow of liquid is observed inside a wave hump in the reference system moving with wave phase velocity. Conditionally averaged profiles of longitudinal and transverse components of liquid velocity are obtained for different cross-sections of developed non-linear waves. It is shown that the increase in wave amplitude slightly changes the location of the vortex center. The analysis of modification of vortex motion character due to wavy flow conditions, such as tube inclination angle, film Reynolds number, wave excitation frequency, is fulfilled.     

Set of steady-state traveling waves on the surface of a liquid film flowing down an inclined plane

The nonlinear evolution equation often encountered in modeling the behavior of perturbations in various nonconservative media, for example, in problems of the hydrodynamics of film flow, is examined. Steady-state traveling periodic solutions of this equation are found numerically. The stability of the solutions is investigated and a bifurcation analysis is carried out. It is shown how as the wave number decreases ever new families of steady-state traveling solutions are generated. In the limit as the wave number tends to zero a denumerable set of these solutions is formed. It is noted that solutions which also oscillate in time may be generated from the steadystate solutions as a result of a bifurcation of the Landau-Hopf type.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1989.     

Mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux

A model is developed for the study of mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux. The model combined natural convection dominated and forced convection dominated film condensation, including effects of pressure gradient and interfacial vapor shear drag has been investigated and solved numerically. The separation angle of the condensate film layer, φ _s is also obtained for various pressure gradient parameters, P ^* and their corresponding dimensionless Grashof?'s parameters, Gr ^*. Besides, the effect of P ^* on the dimensionless mean heat transfer, will remain almost uniform with increasing P ^* until for various corresponding available values of Gr ^*. Meanwhile, the dimensionless mean heat transfer, is increasing significantly with Gr ^* for its corresponding available values of P ^*. For pure natural-convection film condensation, is obtained.     

Numerical investigation of mixed convection heat transfer past an in-line bundle of cylinders

Two dimensional unsteady Navier-Stokes and the energy equations are solved using finite element method for the case of flow past five row deep in-line bundle of circular cylinders with pitch to diameter ratios (PDR) of 1.5 and 2.0. Numerical solutions of governing equations have been obtained using Euler's explicit algorithm. Analysis have been made for Reynolds number of 100 and Prandtl number of 0.71. The effect of Richardson number (Ri=Gr/Re ²) on the flow and heat transfer have been investigated forRi=?1.0, ?0.5, 0.0, +0.5 and +1.0. Streamlines, isovorticity lines, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distribution around the cylinders are presented. Results obtained for forced convection (Ri=0.0) agree well with the available experimental and numerical results. There is considerable effect of buoyancy over tube bundles both in buoyancy aiding and opposing flows.     

Wavy wall influence on the hydrodynamic instability of a liquid film flowing along an inclined plane

The film dynamic of a thin liquid along an inclined and wavy wall was numerically depicted in a weighted-residual integral boundary layer equation. A qualitative and quantitative analysis was initially carried out and accurate comparisons were obtained from experimental data on film instability along a flat and inclined as well as a wavy wall. To pinpoint the effect of waviness on film instability, 20 wavy wall periods in the computational CFD domain were considered. Several waviness parameters were studied and shown to have taken on a major role in the film instability process. Finally, a wide range of main wall inclination angles was taken into account, and consequent numerical data permitted identification of a threshold angle value. For wall angles higher than the threshold angle, the film behaved as though no corrugations were present. For lower angles, the film was repeatedly altered during the acceleration and deceleration phases.