Stability of surfactant-laden core–annular flow and rod–annular flow to non-axisymmetric modes

2013 ◽  
Vol 716 ◽  
Author(s):  
M. G. Blyth ◽  
Andrew P. Bassom
Keyword(s):  
Linear Stability ◽  
Annular Flow ◽  
Convective Motion ◽  
Straight Pipe ◽  
Pipe Axis ◽  
Insoluble Surfactant ◽  
Fluid Layers ◽  
Axisymmetric Disturbances ◽  
Quiescent Fluid ◽  
Fluid Configuration

AbstractThe linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but under certain conditions the flow may be linearly stable to axisymmetric disturbances but linearly unstable to non-axisymmetric disturbances.

Linear Stability of Flow in a Concentric Annulus

1977 ◽  
Vol 44 (2) ◽  
pp. 338-340
Author(s):  
A. K. Agarwal ◽  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Annular Flow ◽  
Parallel Flow ◽  
Concentric Annulus ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fully developed, parallel flow in a rigid, concentric annulus to infinitesimal disturbances is analyzed. It is found that the annular flow is spatially stable to axisymmetric disturbances upto a modified Reynolds number of 10,000.

scholarly journals Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes

2019 ◽  
Vol 863 ◽  
pp. 185-214 ◽  
Author(s):  
Alexander L. Frenkel ◽  
David Halpern ◽  
Adam J. Schweiger
Keyword(s):  
Normal Modes ◽  
Bond Number ◽  
Base Shear ◽  
System Parameters ◽  
Critical Curves ◽  
Extremal Points ◽  
Insoluble Surfactant ◽  
Fluid Layers ◽  
Instability Regions ◽  
Unstable Branch

The joint effects of an insoluble surfactant and gravity on the linear stability of a two-layer Couette flow in a horizontal channel are investigated. The inertialess instability regimes are studied for arbitrary wavelengths and with no simplifying requirements on the system parameters: the ratio of thicknesses of the two fluid layers; the viscosity ratio; the base shear rate; the Marangoni number $Ma$; and the Bond number $Bo$. As was established in the first part of this investigation (Frenkel, Halpern & Schweiger, J. Fluid Mech., vol. 863, 2019, pp. 150–184), a quadratic dispersion equation for the complex growth rate yields two, largely continuous, branches of the normal modes, which are responsible for the flow stability properties. This is consistent with the surfactant instability case of zero gravity studied in Halpern & Frenkel (J. Fluid Mech., vol. 485, 2003, pp. 191–220). The present paper focuses on the mid-wave regimes of instability, defined as those having a finite interval of unstable wavenumbers bounded away from zero. In particular, the location of the mid-wave instability regions in the ($Ma$, $Bo$)-plane, bounded by their critical curves, depending on the other system parameters, is considered. The changes of the extremal points of these critical curves with the variation of external parameters are investigated, including the bifurcation points at which new extrema emerge. Also, it is found that for the less unstable branch of normal modes, a mid-wave interval of unstable wavenumbers may sometimes coexist with a long-wave one, defined as an interval having a zero-wavenumber endpoint.

Thermal Instability of a Fluid-Saturated Porous Medium Bounded by Thin Fluid Layers

1987 ◽  
Vol 109 (3) ◽  
pp. 677-682 ◽  
Author(s):  
G. Pillatsis ◽  
M. E. Taslim ◽  
U. Narusawa
Keyword(s):  
Porous Medium ◽  
Porous Layer ◽  
Thermal Instability ◽  
Numerical Solutions ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
Wave Number ◽  
Thermal Conductivity Ratio ◽  
Fluid Layers

A linear stability analysis is performed for a horizontal Darcy porous layer of depth 2dm sandwiched between two fluid layers of depth d (each) with the top and bottom boundaries being dynamically free and kept at fixed temperatures. The Beavers–Joseph condition is employed as one of the interfacial boundary conditions between the fluid and the porous layer. The critical Rayleigh number and the horizontal wave number for the onset of convective motion depend on the following four nondimensional parameters: dˆ ( = dm/d, the depth ratio), δ ( = K/dm with K being the permeability of the porous medium), α (the proportionality constant in the Beavers–Joseph condition), and k/km (the thermal conductivity ratio). In order to analyze the effect of these parameters on the stability condition, a set of numerical solutions is obtained in terms of a convergent series for the respective layers, for the case in which the thickness of the porous layer is much greater than that of the fluid layer. A comparison of this study with the previously obtained exact solution for the case of constant heat flux boundaries is made to illustrate quantitative effects of the interfacial and the top/bottom boundaries on the thermal instability of a combined system of porous and fluid layers.

The linear stability of a core–annular flow in an asymptotically corrugated tube

2002 ◽  
Vol 466 ◽  
pp. 113-147 ◽  
Author(s):  
HSIEN-HUNG WEI ◽  
DAVID S. RUMSCHITZKI
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Stability ◽  
Annular Flow ◽  
Immiscible Fluids ◽  
Base Flow ◽  
Leading Order ◽  
Partial Differential ◽  
The Mean ◽  
Liquid Displacement

This paper examines the core–annular flow of two immiscible fluids in a straight circular tube with a small corrugation, in the limit where the ratio ε of the mean undisturbed annulus thickness to the mean core radius and the corrugation (characterized by the parameter σ) are both asymptotically small and where the surface tension is small. It is motivated by the problems of liquid–liquid displacement in irregular rock pores such as occur in secondary oil recovery and in the evolution of the liquid film lining the bronchii in the lungs whose diameters vary over different generations of branching. We investigate the asymptotic base flow in this limit and consider the linear stability of its leading order (in the corrugation parameter) solution. For the chosen scalings of the non-dimensional parameters the core's base flow slaves that of the annulus. The equation governing the leading-order interfacial position for a given wall corrugation function shows a competition between shear and capillarity. The former tends to align the interface shape with that of the wall and the latter tends to introduce a phase shift, which can be of either sign depending on whether the circumferential or the longitudinal component of capillarity dominates. The asymptotic linear stability of this leading-order base flow reduces to a single partial differential equation with non-constant coefficients deriving from the non-uniform base flow for the time evolution of an interfacial disturbance. Examination of a single mode k wall function allows the use of Floquet theory to analyse this equation. Direct numerical solutions of the above partial differential equation agree with the predictions of the Floquet analysis. The resulting spectrum is periodic in α- space, α being the disturbance wavenumber space. The presence of a small corrugation not only modifies (at order σ2) the primary eigenvalue of the system. In addition, short-wave order-one disturbances that would be stabilized flowing to capillarity in the absence of corrugation can, in the presence of corrugation and over time scales of order ln(1/σ), excite higher wall harmonics (α±nk) leading to the growth of unstable long waves. Similar results obtain for more complicated wall shape functions. The main result is that a small corrugation makes a core–annular flow unstable to far more disturbances than would destabilize the same uncorrugated flow system. A companion paper examines that competition between this added destabilization due to pore corrugation with the wave steepening and stabilization in the weakly nonlinear regime.

Helical instabilities of slowly divergent jets

1979 ◽  
Vol 92 (2) ◽  
pp. 209-215 ◽  
Author(s):  
Peter Plaschko
Keyword(s):  
Linear Stability ◽  
Strouhal Number ◽  
Rapid Growth ◽  
Short Distance ◽  
Numerical Evaluation ◽  
Multiple Scales ◽  
Stability Study ◽  
Spatial Growth ◽  
Axisymmetric Disturbances ◽  
Comparison With Experiments

The inviscid spatial growth of spiral modes of circular, slowly diverging jets is analysed. A multiple-scales expansion is used to develop a linear stability study for non-axisymmetric disturbances of arbitrary helicity. The numerical evaluation is restricted to axisymmetric modes and to the first two helical modes. It is shown that in the case of comparatively high values of the Strouhal number the modes exhibit a very rapid growth and reach their maximal amplification after a short distance, the axisymmetric instabilities being excited more strongly than their spiral counterparts. Contrary to this, the modes grow comparatively slowly in the case of smaller values of the Strouhal number and exhibit their maximal amplification further downstream. In the latter case the first spiral mode is more unstable than the axisymmetric one. A comparison with experiments seems to support these results.

Analogy between laminar flows in curved pipes and orthogonally rotating pipes

1994 ◽  
Vol 268 ◽  
pp. 133-145 ◽  
Author(s):  
Hiroshi Ishigaki
Keyword(s):  
Laminar Flows ◽  
Computational Studies ◽  
Flow Properties ◽  
Straight Pipe ◽  
Pipe Axis ◽  
Transfer Rates ◽  
Curved Pipe ◽  
Curved Pipes ◽  
Data Similarity ◽  
Friction Factors

The secondary flow of a viscous fluid, caused by the Coriolis force, through a straight pipe rotating about an axis perpendicular to the pipe axis is analogous to that of a fluid, caused by the centrifugal force, through a stationary curved pipe. The quantitative analogy between these two fully developed laminar flows will be demonstrated through similarity arguments, computational studies and the use of experimental data. Similarity considerations result in two analogous governing parameters for each flow, which include a new one for the rotating flow. When one of these analogous pairs of parameters of the two flows is large, it will be demonstrated that there are strong similarities between the two flows regarding friction factors, heat transfer rates, flow patterns and flow properties for the same values of the other pair of parameters.

The effects of insoluble surfactants on the linear stability of a core–annular flow

2005 ◽  
Vol 541 (-1) ◽  
pp. 115 ◽  
Author(s):  
HSIEN-HUNG WEI ◽  
DAVID S. RUMSCHITZKI
Keyword(s):  
Linear Stability ◽  
Annular Flow

Thermal Stability of Horizontally Superposed Porous and Fluid Layers

1989 ◽  
Vol 111 (2) ◽  
pp. 357-362 ◽  
Author(s):  
M. E. Taslim ◽  
U. Narusawa
Keyword(s):  
Porous Layer ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
Darcy Number ◽  
Wave Number ◽  
Thermal Conductivity Ratio ◽  
Porous Layers ◽  
Fluid Layers ◽  
Comprehensive Picture

The results of stability analyses for the onset of convective motion are reported for the following three horizontally superposed systems of porous and fluid layers: (a) a porous layer sandwiched between two fluid layers with rigid top and bottom boundaries, (b) a fluid layer overlying a layer of porous medium, and (c) a fluid layer sandwiched between two porous layers. By changing the depth ratio dˆ from zero to infinity, a set of stability criteria (i.e., the critical Rayleigh number Rac and the critical wave number ac) is obtained, ranging from the case of a fluid layer between two rigid boundaries to the case of a porous layer between two impermeable boundaries. The effects of k/km (the thermal conductivity ratio), δ (the square root of the Darcy number), and α (the nondimensional proportionality constant in the slip condition) on Rac and ac are also examined in detail. The results in this paper, combined with those reported previously for Case (a) (Pillatsis et al., 1987), will provide a comprehensive picture of the interaction between a porous and a fluid layer.

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