The axisymmetric long-wave interfacial stability of core-annular flow of power-law fluid with surfactant

The long wave stability of core-annular flow of power-law fluids with an axial pressure gradient is investigated at low Reynolds number. The interface between the two fluids is populated with an insoluble surfactant. The analytic solution for the growth rate of perturbation is obtained with long wave approximation. We are mainly concerned with the effects of shear-thinning/thickening property and interfacial surfactant on the flow stability. The results show that the influence of shear-thinning/thickening property accounts to the change of the capillary number. For a clean interface, the shear-thinning property enhances the capillary instability when the interface is close to the pipe wall. The converse is true when the interface is close to the pipe centerline. For shear-thickening fluids, the situation is reversed. When the interface is close to the pipe centerline, the capillary instability can be restrained due to the influence of surfactant. A parameter set can be found under which the flow is linearly stable.

The capillary pressure curves from upscaling interfacial and unsaturated flows in porous layers with vertical heterogeneity

We provide the capillary pressure curves p_(c)(s)as a function of the effective saturation s based on the theoretical framework of upscaling unsaturated flows in vertically heterogeneous porous layers proposed recently(Z.Zheng,Journal of Fluid Mechanics,950,A17,2022).Based on the assumption of vertical gravitational-capillary equilibrium,the saturation distribution and profile shape of the invading fluid can be obtained by solving a nonlinear integral-differential equation.The capillary pressure curves p_(c)(s)can then be constructed by systematically varying the injection rate.Together with the relative permeability curves k_(rn)(s)that are already obtained.One can now provide quick estimates on the overall behaviours of interfacial and unsaturated flows in vertically-heterogeneous porous layers.

Kinetics of cerium(Ⅳ) and fluoride extraction from sulfuric solutions using bifunctional ionic liquid extractant(Bif-ILE)[A336][P204]

The extraction kinetics of Ce（Ⅳ） and Ce（Ⅳ）-F^- mixture systems from sulfuric solutions to n-heptane solution containing Bif-ILE[A336][P204]（[trialkylmethylammonium][di-2-ethylhewanxylphosphinate]） with a constant interfacial area cell with laminar flow were studied,just to elucidate the extraction mechanism and the mass transfer models.The data were analyzed in terms of pseudo-first-order constants.The effects of stirring speed,specific interfacial area and temperature on the extraction rate in both systems were discussed,suggesting that the extractions were mixed bulk phases-interfacial control process.Supported by the experimental data,the corresponding rate equations for Ce（Ⅳ） extraction system and Ce（Ⅳ）-F^- mixture extraction system were obtained.The experimental results indicated the rate-controlling step.The kinetics model was deduced from the rate-controlling step and consistent with the rate equation.

A three-dimensional robust volume-of-fluid solver based on the adaptive mesh refinement

The present study provides a three-dimensional volume-of-fluid method based on the adaptive mesh refinement technique.The projection method on the adaptive mesh is introduced for solving the incompressible Navier-Stokes equations.The octree structure mesh is employed to solve the flow velocities and the pressure.The developed solver is applied to simulate the deformation of the cubic droplet driven by the surface tension without the effect of the gravity.The numerical results well predict the shape evolution of the droplet.

Mass transfer coefficients in metallurgical reactors

An overview on the application and achievements of physico-mathematicalmodeling of metallurgical processes in China is briefly declared. The important role of coefficientsin model formulation is shown from our experience. The mass transfer coefficients of the slag-metalreactions and the gas-metal reactions are discussed referring to the flow conditions near theinterface. The influence of the surface-active species on the mass transfer and the interfacialreaction is also discussed briefly.

Formation of radially expanding liquid sheet by impinging two round jets

A thin circular liquid sheet can be formed by impinging two identical round jets against each other. The liquid sheet expands to a certain critical radial distance and breaks. The unsteady process of the formation and breakup of the liquid sheet in the ambient gas is simulated numerically. Both liquid and gas are treated as incompressible Newtonian fluids. The flow considered is axisymmetric. The liquid-gas interface is modeled with a level set function. A finite difference scheme is used to solve the governing Navier-Stokes equations with physical boundary conditions. The numerical results show how a thin circular sheet can be formed and break at its circular edge in slow motion. The sheet continues to thin as it expands radially. Hence, the Weber number decreases radially. The Weber number is defined as ρu 2 h/σ, where ρ and σ are, respectively, the liquid density and the surface tension, and u and h are, respectively, the average velocity and the half sheet thickness at a local radial location in the liquid sheet. The numerical results show that the sheet indeed terminates at a radial location, where the Weber number reaches one as observed in experiments. The spatio-temporal linear theory predicts that the breakup is initiated by the sinuous mode at the critical Weber number We c =1, below which the absolute instability occurs. The other independent mode called the varicose mode grows more slowly than the sinuous mode according to the linear theory. However, our numerical results show that the varicose mode actually overtakes the sinuous mode during the nonlinear evolution, and is responsible for the final breakup. The linear theory predicts the nature of disturbance waves correctly only at the onset of the instability, but cannot predict the exact consequence of the instability.

Thin film dynamics in coating problems using Onsager principle

A new variational method is proposed to investigate the dynamics of the thin film in a coating flow where a liquid is delivered through a fixed slot gap onto a moving substrate. A simplified ODE system has also been derived for the evolution of the thin film whose thickness hf is asymptotically constant behind the coating front. We calculate the phase diagram as well as the film profiles and approximate the film thickness theoretically, and agreement with the well-known scaling law as Ca2/3 is found.

A finite volume algorithm based on overlapping meshes for simulation of hydrodynamic problems

A finite volume algorithm was established in order to investigate two-dimensional hydrodynamic problems. These include viscous free surface flow interaction with free rigid bodies in the case of large and/or relative motions. Two-phase flow with complex deformations at the interface was simulated using a fractional step-volume of fluid algorithm. In addition, body motions were captured by an overlapping mesh system. Here, flow variables are transferred using a simple fully implicit non-conservative interpolation scheme which maintains the second-order accuracy of implemented spatial discretisation. Code was developed and an appropriate set of problems investigated. Results show good potential for development of a virtual hydrodynamics laboratory.

The impact of interphase forces on the modulation of turbulence in multiphase flows

The modulation of turbulence by particles has been rigorously investigated in the literature yielding either a reduction or an enhancement of the turbulent kinetic energy at different spatial length scales.However,a general description of the turbulence modulation in multiphase flows due to the presence of an interphase force has attracted less attention.In this paper,we investigate the turbulent modulation for interfacial and fluid-particle flows analytically and numerically,where surface tension and drag define the interphase coupling,respectively.It is shown that surface tension and drag appear as additional production/dissipation terms in the transport equations for the turbulent kinetic energies(TKE),which is of particular importance for the turbulence modelling of multiphase flows.Furthermore,we study the modulation of turbulence in decaying homogenous isotropic turbulence(HIT)for both types of multiphase flow.The results clearly unveil that in both cases the energy is reduced at large scales,while the small-scale energy is enhanced compared to single-phase flows.Particularly,at large scales surface tension works against the turbulent eddies and hinders the ejection of droplet from the corrugated interface.In contrast,at the small scales,the surface tension force and the velocity fluctuations are aligned leading to an enhancement of the energy.In the case of fluid-particle flows,particles retain their energy longer than the surrounding fluid increasing the energy at the small scales,while at the large scales the particles do not follow exactly the surrounding fluid reducing its energy.For the latter effect,a considerable dependence on the particle Stokes number is found.

Numerical Simulation of Moving Contact Lines with Surfactant by Immersed Boundary Method

In this paper,we present an immersed boundary method for simulating moving contact lines with surfactant.The governing equations are the incompressible Navier-Stokes equations with the usual mixture of Eulerian fluid variables and Lagrangian interfacial markers.The immersed boundary force has two components:one from the nonhomogeneous surface tension determined by the distribution of surfactant along the fluid interface,and the other from unbalanced Young’s force at the moving contact lines.An artificial tangential velocity has been added to the Lagrangian markers to ensure that the markers are uniformly distributed at all times.The corresponding modified surfactant equation is solved in a way such that the total surfactant mass is conserved.Numerical experiments including convergence analysis are carefully conducted.The effect of the surfactant on the motion of hydrophilic and hydrophobic drops are investigated in detail.