A Taylor drop rising in a liquid co-current flow

The present work presents a numerical study on the behavior of isolated liquid Taylor drops rising in vertical tubes with co-current heavier continuous phase. Numerical simulations were performed with a previously validated model, implementing Volume of Fluid method in an axisymmetric geometry. Detailed flow patterns and drop shapes are provided and discussed for several conditions. The balance between gravity effect and velocity of the continuous phase flow was found to have a great influence in the flow patterns observed. The increase of inertial effects, due to the increase of Eo number and the co-current velocity, leads to the occurrence of closed recirculations below the drops. Furthermore, the continuous phase stabilization distance below the drop is a function of the drop Reynolds number. Drop and continuous phase velocities relationship was studied. A viscosity ratio related term was appended to a pre-existing correlation. The flow in the absence of gravity was also studied. Results demonstrate that micro-scale flow is a lower limit to the cases studied in the present work and suggest that the viscosity ratio affects mainly the inertial part of the drop velocity.     

Wave flow of a liquid film together with a flow of gas

The wave flow of a thin layer of viscous liquid in conjunction with a flow of gas was considered in a linear formulation earlier [1, 2]. In this paper the problem of the wave flow of a liquid film together with a gas flow is solved in a nonlinear setting. On this basis relationships are derived for calculating the parameters of the film and the hydrodynamic quantities.Ivanovo. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 12–18, January–February, 1972.     

Dynamics of a liquid film sheared by a co-flowing turbulent gas

Consider the dynamics of a thin laminar liquid film flowing over an inclined wall in the presence of a co-flowing turbulent gas. The solution to the full two-phase flow problem poses substantial technical difficulties. However, by making appropriate assumptions, the solution process can be simplified and can provide valuable insights. The assumptions allow us to solve the gas and liquid problems independently. Solving for the gas flow reduces to finding perturbations to pressure and tangential stresses at the interface, influencing the liquid problem through the boundary conditions. We analyze the effect of gas flow on the liquid problem by developing an integral-boundary-layer model, which is valid up to moderate liquid Reynolds numbers. We seek solitary-wave solutions of this model under the influence of gas flow via a pseudo-arclength continuation method. Our computations demonstrate that as a general trend, the wave speed increases with increasing the gas shear and the liquid flow rate. Further insight into the problem is provided via time-dependent computations of the integral-boundary-layer model.     

Linear stability in the flow of a liquid film concurrent with a gas flow

The two-phase flow of liquid films are often encountered in practice, but the number of theoretical papers devoted to this problem is limited. The problem of the linear stability of a viscous liquid film subjected to a gas flow has been formulated in [1] and, in somewhat different form, in [2]. The linear stability of plane-parallel motion in films has been studied analytically in [1–8] for some limiting cases. The range of validity of the analytic approaches remains an open question. Therefore, an exact numerical analysis of flow stability over a fairly broad range is required. In the present paper a separate solution of the problem for the gas and the liquid is shown to be possible. The Orr-Sommerfeld equation has been integrated numerically, and the results are compared to the results of analytic calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–146, January–February, 1976.The author is grateful to É. É. Markovich for directing the work and to V. Ya. Shkadov for his interest in the work and many useful comments.     

Wavy flow of a liquid film in the presence of a cocurrent turbulent gas flow

Wavy downflow of viscous liquid films in the presence of a cocurrent turbulent gas flow is analyzed theoretically. The parameters of two-dimensional steady-state traveling waves are calculated for wide ranges of liquid Reynolds number and gas flow velocity. The hydrodynamic characteristics of the liquid flow are computed using the full Navier-Stokes equations. The wavy interface is regarded as a small perturbation, and the equations for the gas are linearized in the vicinity of the main turbulent flow. Various optimal film flow regimes are obtained for the calculated nonlinear waves branching from the plane-parallel flow. It is shown that for high velocities of the cocurrent gas flow, the calculated wave characteristics correspond to those of ripple waves observed in experiments.     

Advective flow in a rotating liquid film

This paper presents a new exact solution of the Navier–Stokes equations in the Boussinesq approximation that describes thermocapillary advective flow in a slowly rotating horizontal layer of incompressible fluid with free boundaries. Such flow occurs in the case of linear temperature distribution over horizontal coordinates or in the case of heat flux distribution at the layer boundaries. The influence of the Taylor, Marangoni, Grashof, and Biot numbers on the flow and temperature velocity profiles is studied.     

Dynamics of nanofibres conveyed by low Reynolds number flow in a microchannel

In this paper we aim to create an experimental and numerical model of nano and micro filaments suspended in a confined Poiseuille flow. The experimental data obtained for short nanofibres will help to elucidate fundamental questions concerning mobility and deformation of biological macromolecules due to hydrodynamic stresses from the surrounding fluid motion. Nanofibres used in the experiments are obtained by electrospinning polymer solutions. Their typical dimensions are 100–1000 μm (length) and 0.1–1 μm (diameter). The nanofibre dynamics is followed experimentally under a fluorescence microscope. A precise multipole expansion method of solving the Stokes equations, and its numerical implementation are used to construct a bead-spring model of a filament moving in a Poiseuille flow between two infinite parallel walls. Simulations show typical behaviour of elongated macromolecules. Depending on the parameters, folding and unfolding sequences of a flexible filament are observed, or a rotational and translation motion of a shape-preserving filament. An important result of our experiments is that nanofibres do not significantly change their shape while interacting with a micro-flow. It appeared that their rotational motion is better reproduced by the shape-preserving Stokesian bead model with all pairs of beads connected by springs, omitting explicit bending forces.     

Primary instabilities of liquid film flow sheared by turbulent gas stream

Evolution of excited waves on a viscous liquid film has been investigated experimentally for the annular gas–liquid flow in a vertical tube. For the first time the dispersion relations are obtained experimentally for linear waves on liquid film surface in the presence of turbulent gas flow. Both cocurrent and countercurrent flow regimes are investigated. As an example of comparison with theory, the experimental data are compared to the results of calculations based on the Benjamin quasi-laminar model for turbulent gas flow. The calculation results are found to be in good agreement with experiments for moderate values of film Reynolds number.     

Dynamics of a liquid with gas bubbles

The hydrodynamics and diffusion of an admixture near an isolated bubble, which simulates the rise of either a chain of identical bubbles or a system of regularly arranged bubbles of the same volume, are analyzed by solving the Navier-Stokes equations numerically. Data are presented for a specific liquid. It is shown that in both cases the maximum flow velocity on the surface of identical bubbles is practically the same, although in the former case the ascent velocity is considerably higher. The stationary admixture diffusion from a bubble also proves to be nearly the same.In relation to the bubbling of a gas through a liquid layer, it is shown that the total admixture diffusion is maximum for regularly arranged bubbles whose diameter is comparable with the liquids capillary constant. Although the flow past the bubble remains continuous, the values of the hydrodynamic parameters are no longer small.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–88, May–June, 1996.     

Instability of a film of viscous liquid under the influence of an adjacent gas flow

A study is made of the instability of a film of viscous liquid adjacent to a gas flow. Despite a number of investigations, there is no unified theory of this problem capable of explaining the experimental results of different authors. The present paper gives a solution of the problem that is valid for a large class of flows of liquid films in the case of laminar and turbulent flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 28–36, March–April, 1979.     

Linear and nonlinear stability of combined plane-parallel flow of a film of liquid and gas

The problem of the linear stability of a layer of liquid entrained by a gas has been investigated for some special cases in [1–7]. In [8], the linear problem was solved numerically and the solution compared with some analytic solutions for special cases of the flow. In the present paper, the results of linear analysis are presented more comprehensively; the problem of finite-amplitude stability of the film is posed and solved numerically; the results of the linear and nonlinear analysis are compared with data of an experiment performed by the authors and by other experimentalists.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–42, January–February, 1979.We are grateful to V. Ya. Shkadov for supervising the work, to all the participants of G. I. Petrov's seminar for helpful discussion, and also to E. L. Kokon for assistance in evaluating the experimental data.     

Characteristics of solitary waves on a falling liquid film sheared by a turbulent counter-current gas flow

We report an experimental investigation of a falling water film sheared by a turbulent counter-current air flow in an inclined rectangular channel. Film thickness and wave velocity measurements associated with visual observation are conducted to study the influence of the air flow on controlled traveling waves consisting of a large wave hump preceded by capillary ripples. First, we focus on the variation of the shape, amplitude and velocity of the waves as the gas velocity is gradually increased. We demonstrate that the amplitude of the main hump grows substantially even for moderate gas velocities, whereas modification of the wave celerity becomes significant above a specific gas velocity around 4 m/s, associated with an alteration of the capillary region. The influence of the gas flow on 3D secondary instabilities of the solitary waves detected in a previous study Kofman et al. (2014), namely rugged or scallop waves, is also investigated. We show that the capillary mode is damped while the inertial mode is enhanced by the interfacial shear. Next, the gas velocity is increased until the onset of upstream-moving patterns referred to as flooding in our experiments. At moderate inclination angles (typically < 7 ^○), flooding occurs for a gas velocity around 8 m/s and is initiated at the scallop wave crests by a backward wave-breaking phenomenon preceded by the onset of ripples on the flat residual film separating two waves. At high inclination angle, a rapid development of solitons is observed as the air velocity is increased preventing the waves to turn back. Finally, at high liquid Reynolds number, sudden and intermittent events are triggered consisting of very large amplitude waves that go back upwards very fast. These “slugs” either extend over the whole width of the channel or are very localized and can thus potentially evolve towards atomization.     

Wave flow of a liquid film in a horizontal separator

The calculation of the motion of separated moisture in a linear horizontal separator is made on the basis of the analysis of the development of the waves in a flow of a thin layer of liquid along a vertical surface without allowance for the transverse flow of mass [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1985.     

Dynamics of a Taylor bubble in steady and pulsatile co-current flow of Newtonian and shear-thinning liquids in a vertical tube

A computational analysis is carried out to ascertain the effects of steady and pulsatile co-current flow, on the dynamics of an air bubble rising in a vertical tube containing water or a solution of Carboxymethylcellulose (CMC) in water. The mass fraction (m_f) of CMC in the solution is varied in the range 0.1% ⩽ m_f ⩽ 1% to accommodate zero-shear dynamic viscosities in the range 0.009–2.99 Pa-s. It was found that the transient and time-averaged velocities of Taylor bubbles are independent of the bubble size under both steady as well as pulsatile co-current flows. The lengths of the Taylor bubbles under the Newtonian conditions are found to be consistently greater than the corresponding shear-thinning non-Newtonian conditions for any given zero-shear dynamic viscosity of the liquid. In contrast to observations in stagnant liquid columns, an increase in the dynamic viscosity of the liquid (under Newtonian conditions) results in a concomitant increase in the bubble velocity, for any given co-current liquid velocity. In shear-thinning liquids, the change in the bubble velocity with an increase in m_f is found to be relatively greater at higher co-current liquid velocities. During pulsatile shear-thinning flows, distinct ripples are observed to occur on the bubble surface at higher values of m_f, the locations of which remain stationary with reference to the tube for any given pulsatile flow frequency, while the bubble propagated upwards. In such a pulsatile shear-thinning flow, a localised increase in dynamic viscosity is accompanied near each ripple, which results in a localised re-circulation region inside the bubble, unlike a single re-circulation region that occurs in Newtonian liquids, or shear-thinning liquids with low values of m_f. It is also seen that as compared to frequency, the amplitude of pulsatile flow has a greater influence on the oscillating characteristics of the rising Taylor bubble. The amplitude of oscillation in the bubble velocity increases with an increase in the CMC mass fraction, for any given value of pulsatile flow amplitude.     

Electroviscous effects in steady fully developed flow of a power-law liquid through a cylindrical microchannel

Electroviscous effects in steady, fully developed, pressure-driven flow of power-law liquids through a uniform cylindrical microchannel have been investigated numerically by solving the Poisson–Boltzmann and the momentum equations using a finite difference method. The pipe wall is considered to have uniform surface charge density and the liquid is assumed to be a symmetric 1:1 electrolyte solution. Electroviscous resistance reduces the velocity adjacent to the wall, relative to the velocity on the axis. The effect is shown to be greater when the liquid is shear-thinning, and less when it is shear-thickening, than it is for Newtonian flow. For overlapping electrical double layers and elevated surface charge density, the electroviscous reduction in the near-wall velocity can form an almost stationary (zero shear) layer there when the liquid is shear-thinning. In that case, the liquid behaves approximately as if it is flowing through a channel of reduced diameter. The induced axial electrical field shows only a weak dependence on the power-law index with the dependence being greatest for shear-thinning liquids. This field exhibits a local maximum as surface charge density increases from zero, even though the corresponding electrokinetic resistance increases monotonically. The magnitude of the electroviscous effect on the apparent viscosity, as measured by the ratio of the apparent and physical consistency indices, decreases monotonically as the power-law index increases. Thus, overall, the electroviscous effect is stronger in shear-thinning, and weaker in shear-thickening liquids, than it is when the liquid is Newtonian.     

The flow of a power-law liquid in a continuous-flow squeeze film

Consideration is given to the flow of an inelastic ‘power-law’ liquid in a continuous flow squeeze film. This simulates the flow in a conventional squeeze film by continuously injecting fluid into the narrow gap between two plates through the lower plate (Oliver et al. [6]). To zero order in the usual lubrication approximation the results are identical with those for the conventional squeeze film. To first order, useful corrections to the normal force due to the effects of inertia are obtained.