Dynamics of the axisymmetric core-annular flow. II. The less viscous fluid in the core, saw tooth waves

Abstract

The nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube is studied when the viscosity ratio of the fluid in the annulus to that in the core of the tube, μ, is larger than or equal to unity. For these values of the viscosity ratio the perfect core-annular flow (CAF) is linearly unstable and it is necessary to keep the ratio of the thickness of the annulus to the radius of the tube small so that the solutions remain uniformly bounded. The simulations are based on a pseudospectral numerical method while special care has been taken in order to minimize as far as possible the effect of the boundary conditions imposed in the axial direction allowing for multiple waves of different lengths to develop and interact. The time integration originates with the analytical solution for the pressure driven, perfect CAF or the perfect CAF seeded with either the most unstable mode or random disturbances. Quite regular wave patterns are predicted in the first two cases, whereas multiple unstable modes grow and remain even after saturation of the instability in the last case. The resulting waves generally travel in the same direction and faster than the undisturbed interface, except for the case with μ=1 for which they are stationary with respect to it. Depending on parameter values, waves move with the same velocity or interact with each other exchanging their amplitudes or merge and split giving rise to either chaotic or organized solutions. For fluids of equal viscosities and densities (μ=ρ=1) and for a Reynolds number, Re(≡Λρ̂1R̂2Ŵ0/μ̂1)=0.0275 and an inverse Weber number, W(≡T̂/(ρ̂1Ŵ02R̂2))=145.4, both based on the properties of the inner fluid, the tube radius, R̂2, and the average flow velocity, Ŵ0, small amplitude waves are predicted. The increase of μ by almost two orders of magnitude does not affect their amplitudes, but increases their temporal period linearly. Varying W by more than three orders of magnitude increases their amplitudes proportionately, while their period increases with the logarithm of W. Similar to that is the effect of increasing Re. The present analysis confirms and extends results based on long wave expansions, which lead to the Kuramoto–Sivashinsky equation and modifications of it.

Double-diffusive two-fluid flow in a slippery channel: A linear stability analysis

“…Sahu et al 37 showed that the above system becomes absolutely unstable for a certain range of parameters and have indicated the region of absolute and convective instabilities in the Reynolds number and viscosity ratio space. There are also several investigations 29,30,[38][39][40][41][42][43][44][45][46][47][48] not relevant to the present study (but worth mentioning in the present context) that deals with stability characteristics of viscosity stratified flows in rigid channels/pipes, involving the displacement of one fluid by another. The interesting features and the type of instabilities displayed by these flow systems with boundaries as either rigid walls or rigid circular pipes suggest that it is worth analyzing the analogous flow systems in configurations with velocity slip at the boundaries.…”

A study of pressure-driven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach

“…3,[9][10][11][12] This problem was also studied by many researchers experimentally 13,14 and numerically. [15][16][17][18] In miscible core-annular flows, the thickness of the more viscous fluid layer left on the pipe walls and the speed of the propagating "finger" were experimentally investigated by many authors [19][20][21][22][23] and the axisymmetric and "corkscrew" patterns were found. [24][25][26][27][28] In neutrallybuoyant core-annular pipe flows, d'Olce et al 29 observed axisymmetric "pearl" and "mushroom" patterns at high Schmidt number.…”

Linear stability analysis and numerical simulation of miscible two-layer channel flow

“…These have included linear stability analyses for horizontal [6,7,[17][18][19] and vertical pipes [5,20], accounting for viscosity and density contrasts, experiments [21,22] and numerical simulations in straight [9,10,23] and corruagated pipes [24][25][26][27].…”

“…A large number of investigations have focused on the stability of immiscible fluids (see, for instance, [1,[4][5][6][7][8][9][10] and references therein). Starting with the work of Yih [4] and Hickox [5], carried out using long-wave theory in planar channels and cylindrical pipes, respectively, these studies have shown that two-layer flows are destabilized linearly by an 'interfacial' mode at arbitrarily small Reynolds numbers.…”