Two-layer Couette Flow Research Articles

Perturbation-Based Analysis to Investigate the Effects of Temperature-Dependent Viscosity in a Two-Layer Plane Couette Flow

Perturbation-Based Analysis to Investigate the Effects of Temperature-Dependent Viscosity in a Two-Layer Plane Couette Flow

Stability of two-layer Newtonian plane Couette flow past a deformable solid layer

The linear stability of two-layer plane Couette flow of Newtonian fluids (designated by labels A and B) of thicknesses (1−β)R and βR, and viscosities μa and μb past a soft, deformable linear viscoelastic solid of thickness HR, shear modulus G, and viscosity ηw is determined using a combination of low wavenumber asymptotic analysis and a numerical method. There are two qualitatively different interfacial modes in this system, viz., the two-fluid interfacial mode due to viscosity stratification [“mode 1;” C. S. Yih “Instability due to viscosity stratification,” J. Fluid Mech. 27, 337 (1967)], and the fluid–solid interfacial mode [“mode 2;” Kumaran, Fredrickson, and Pincus, “Flow induced instability of the interface between a fluid and a gel at low Reynolds number,” J. Phys II 4, 893 (1994)]. The respective effects of solid layer deformability and fluid viscosity stratification on mode 1 and mode 2 are analyzed in detail using both asymptotic and numerical methods. Results of our low wavenumber asymptotic analysis show that the deformability of the solid layer has a dramatic effect on the interfacial instability (mode 1) between the two Newtonian fluids: When the more viscous fluid is of smaller thickness (an unstable configuration for the two-fluid mode 1 instability), the solid layer could completely stabilize the two-fluid interfacial instability, when the nondimensional elasticity parameter Γ=Vμb/(GR) increases beyond a critical value. Here V is the dimensional velocity of the top moving plate. When the more viscous fluid is of larger thickness compared to the less viscous fluid (a stable configuration in rigid channels), it is shown that the solid layer could destabilize or stabilize the two-fluid interfacial mode, depending on the solid layer thickness H. Numerical results at finite values of wavenumber k reveal that the stabilization of the two-fluid interfacial mode predicted by the low wavenumber analysis extends to moderate values of k. For high values of k, the perturbations are localized near the two-fluid interface. Increase in Γ therefore does not have any effect on the high k unstable modes, which are stabilized by the presence of nonzero interfacial tension in the two-fluid interface. When Γ is further increased, the interfacial mode between fluid B and the solid layer becomes unstable. It is demonstrated here that the parameters Γ (representing the shear modulus of the solid), solid layer thickness H, and the solid layer viscosity ηw can be chosen such that both the interfacial modes are stabilized at all wavenumbers, for a fixed top plate velocity.

Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer

The stability of two-layer plane Couette flow of upper-convected Maxwell (UCM) fluids of thicknesses (1− β) R and βR, with matched viscosities η, and relaxation times τ 1 and τ 2 past a soft, deformable solid layer (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity η w is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the two fluids and the solid layer is negligible. Interfaces with and without the interfacial tension between the two fluids are considered, while the interfacial tension between the UCM fluid and the deformable solid is neglected. It is well known that the two-layer Couette flow of UCM fluids with different relaxation times undergoes a purely elastic interfacial instability (referred here as mode 1) in rigid-walled channels, even in the creeping flow limit. It was recently shown [J. Non-Newtonian Fluid Mech 116 (2004) 371] that the plane Couette flow of a single UCM fluid past a deformable solid layer also undergoes an instability (referred here as mode 2) in the creeping flow limit when the nondimensional solid elasticity parameter Γ= Vη/( GR) exceeds a certain critical value, for a given Weissenberg number W= τV/ R. In this study, the respective effects of solid layer deformability and fluid elasticity stratification on mode 1 and mode 2, and the concomitant interaction of these two qualitatively different interfacial modes are analyzed in detail. It is shown that the deformability of the solid layer has a dramatic effect on the purely elastic (mode 1) interfacial instability between the two UCM fluids: When the more elastic UCM fluid is present in between the less elastic UCM fluid and the solid layer, the layer deformability has a stabilizing effect. In this configuration, if the thickness of the more elastic fluid is smaller than the less elastic fluid, it is shown that it is possible to stabilize the mode 1 purely elastic instability (except for very long and very short waves, the latter being stabilized usually by the nonzero interfacial tension between the two fluids) by making the solid layer sufficiently deformable (i.e., Γ exceeding a critical value), while such a configuration is unstable in the case of two-layer flow in rigid channels (i.e., in the absence of the deformable solid layer). Increase in the solid layer deformability, on the other hand, destabilizes the interfacial mode between the UCM fluid and the deformable solid (mode 2). It is demonstrated that it is possible to choose the solid elasticity parameter Γ and viscosity ratio η w/ η such that both mode 1 and mode 2 are completely stable in finite experimental geometries. In marked contrast, when the less elastic fluid is present in between the more elastic fluid and the solid layer, it is shown that solid layer deformability has a destabilizing effect on mode 1. For this configuration, if the thickness of the less elastic fluid is smaller, it is shown that mode 1 can be rendered unstable by increasing solid layer deformability, while it is stable in rigid channels (in the absence of the solid layer). In both of the above configurations, the nondimensional solid layer elasticity required respectively to stabilize or destabilize mode 1 varies as Γ∝( k * R) −1 for ( k * R)≪1, where k * is the wavenumber of perturbations. However, for finite experimental geometries the minimum allowed k * is dictated by the system length, and in such cases, it is argued that complete stabilization or destabilization can be achieved for realistic values of Γ. In both the configurations, our results show that deformable solid layer coatings, by design or default, can completely suppress or induce purely elastic interfacial instabilities in two-layer flow of viscoelastic fluids past a deformable solid surface.

‘Phase diagram’ of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability

A unified view is given of the instabilities that may develop in two-layer Couette flows, as a ‘phase diagram’ in the parameter space. This view is obtained from a preliminary study of the single-fluid Couette flow over a wavy bottom, which reveals three flow regimes for the disturbances created at the bottom, each regime being characterized by a typical penetration depth of the vorticity disturbances and an effective Reynolds number. It appears that the two-layer flow exhibits the same flow regimes for the disturbances induced by the perturbed interface, and that each type of instability can be associated with a flow regime. Typical curves giving the growth rate versus wavenumber are deduced from this analysis, and favourably compared with the existing literature. In the second part of this study, we propose a mechanism for the long wavelength instability, and provide simple estimates of the wave velocity and growth rate, for channel flows and for semi-bounded flows. In particular, an explanation is given for the ‘thin-layer effect’, which is typical of multi-layer flows such as pressure driven flows or gravity driven flows, and according to which the flow is stable if the thinner layer is the less viscous, and unstable otherwise.

Small Reynolds number instabilities in two-layer Couette flow

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.

Problems and Techniques

In this issue of Problems and Techniques, we feature articles on the formation of structure in flows involving immiscible liquids and techniques for solving nonlinear knapsack problems. Free surface and interface problems introduce the added complication of having to determine the position of unknown surfaces into a typically complex scientific computation problem. When surfaces develop intricate patterns and change topology as solutions evolve, numerical approaches must simultaneously maintain accuracy and control computational cost. In the first article, Jie Li and Yuriko Renardy describe a volume of fluid (VOF) technique for addressing instabilities in slow flows involving two immiscible, incompressible, viscous liquids subject to shearing. After describing their variant of the VOF method, Li and Renardy use it to analyze the simple shearing of two fluids in parallel flow (two-layer Couette flow), the flow of a core of fluid surrounded by a second fluid in a pipe (core-annular flow), and droplet breakup. Their results explain how the interface forms "fingers" of one fluid into the other, "bamboo waves" between the fluids, and the rupture of larger drops into trains of smaller ones. Experimental observations nicely confirm and complement the calculations. The second article, by A. Melman and G. Rabinowitz, describes a globally convergent iterative procedure for a class of nonlinear knapsack problems. As I learned, a knapsack problem involves a maximization of a concave function subject to a linear constraint. The optimization presented by Melman and Rabinowitz is motivated chemical processing where periodic servicing is necessary to remove waste by-products and to add raw materials. The operations research problem is to determine the optimal service schedule. Using quadratically convergent Newton-like iterations, the authors construct a fast solution procedure and show its global convergence for a subclass of knapsack problems. Complications arise because the function whose roots they seek is singular and has only an implicit representation. Once again, I'm pleased to thank our authors for their interesting and informative contributions.

Comment on “A numerical study of periodic disturbances on two-layer Couette flow” [Phys. Fluids 10, 3056 (1998)

Comment on “A numerical study of periodic disturbances on two-layer Couette flow” [Phys. Fluids <b>10</b>, 3056 (1998)

Wave dynamics in two-layer Couette flow

We study the longwave dynamics of two-layer Couette flow with high viscosity ratio by rigorously deriving a depth-averaged integral momentum equation for the internal flow dynamics beyond criticality and a modified Kuramoto–Sivashinsky (mKS) equation for near-critical longwave interfacial dynamics. For large viscosity stratification, the primary long-wave instability is triggered by an inertia-delayed lubrication pressure that is generated when the waves introduce a Poiseuille component to the flat-film Couette flow. These primary waves can then undergo secondary modulation, due to a coupling between nonlinear wave dispersion and mean-flow gradient, to form longer waves. The secondary modulation only occurs if the plate speed exceeds a critical value that scales as ( l− l c ) −1/4 near criticality where l is the thickness of the less viscous inner fluid and l c is the critical thickness for the onset of the primary waves. Both primary and secondary wave transition scenarios are favorably compared to experimental data.

Instability due to second normal stress jump in two-layer shear flow of the Giesekus fluid

The two-layer Couette flow of superposed Giesekus liquids is examined. In order to emphasize the effect of a jump in the second normal stress difference, the analysis is focused on flows where the shear rate and first normal stress difference are continuous across the interface. In this case, the flow is neutrally stable to streamwise disturbances, but can be unstable for spanwise disturbances driven by a jump in the second normal stress difference. Whether the long and order one waves are stable or not depends on the sign of this difference. Short waves are unstable. In the case of order one wave instability, the mode of maximum growth rate gives rise to stationary ripples perpendicular to the flow. The eigenvalue problem for purely spanwise wave vectors can in principle be solved analytically, although, in general, the analytical solution is too complicated to obtain. In most cases, however, a simplifying assumption can be made which makes analytical solutions feasible. We present such solutions and compare them with purely numerical solutions.

Secondary instabilities of interfacial waves due to coupling with a long wave mode in a two-layer Couette flow

We report experiments on the stability of interfacial waves in a two-layer Couette flow. As the shear rate is increased, the periodic wave train arising from the primary instability undergoes a secondary instability which results in wave coalescence or nucleation, after a long transient. This secondary instability crucially involves the coupling with a long wave mode, which corresponds to variations of the mean interface level. These observations are favourably compared to stability results on travelling wave solutions for a set of two coupled equations, one for the envelope of a weakly unstable wave packet, and the other for the marginal long wave mode with zero wave number. A physical mechanism for this instability is proposed, as well as an interpretation for the onset of chaos.

A numerical study of periodic disturbances on two-layer Couette flow

The flow of two viscous liquids is investigated numerically with a volume of fluid scheme. The scheme incorporates a semi-implicit Stokes solver to enable computations at low Reynolds numbers, and a second-order velocity interpolation. The code is validated against linear theory for the stability of two-layer Couette flow, and weakly nonlinear theory for a Hopf bifurcation. Examples of long-time wave saturation are shown. The formation of fingers for relatively small initial amplitudes as well as larger amplitudes are presented in two and three dimensions as initial-value problems. Fluids of different viscosity and density are considered, with an emphasis on the effect of the viscosity difference. Results at low Reynolds numbers show elongated fingers in two dimensions that break in three dimensions to form drops, while different topological changes take place at higher Reynolds numbers.

Temporal Evolution of Periodic Disturbances in Two-Layer Couette Flow

The time-dependent motion for a two-layer Couette flow consisting of fluids of different viscosities is simulated numerically by using an algorithm based on the Volume of Fluid (VOF) method. Interfacial tension is included via a continuous surface force (CSF) algorithm. The algorithm is fine-tuned to handle the motion which is driven by a shear-induced interfacial instability due to the viscosity stratification. The code is validated against linear theory. Two prototypical situations are presented, one at a moderately high Reynolds number and the other at a lower Reynolds number. The initial condition is seeded with the eigenmode of largest growth rate, with amplitudes that are varied from those that capture the linear regime to larger values for nonlinear regimes. Issues of free surface advection and viscosity interpolation are discussed. The onset of nonlinearity occurs at the interface and is quadratic, followed by wave steepening.

Experimental study of interfacial long waves in a two-layer shear flow

Interfacial stability of two-layer Couette flow was investigated experimentally in a channel bent into an annular ring. This paper is focused on the supercritical long-wave instability which arises for a broad range of flow parameters. Above the critical upper plate velocity, a slowly growing long wave appears with wavelength equal to the perimeter of the channel. Transients of this wave were studied within the theoretical frame of amplitude equations obtained from the long-wave interface equation. Near the onset of instability, the unstable fundamental harmonic is described by the Landau–Stuart equation, and the nonlinear dynamics of the harmonics closely follows the central and slaved modes analysis. For the higher upper plate velocity, harmonics gain some autonomy but they eventually are enslaved by the fundamental, through remarkable collapses of amplitudes and phase jumps leading to wave velocity and frequency locking. Dispersive effects play a crucial role in the nonlinear dynamics. Far from the threshold, the second harmonic becomes unstable and bistability appears: the saturated wave is dominated either by the fundamental harmonic, or by the even harmonics, after periodic energy exchange.

Elastic instability of the interface in Couette flow of viscoelastic liquids

An error in the literature of two-layer Couette flow of viscoelastic liquids is corrected. It is shown that even in the absence of viscosity stratification, a long wave interfacial instability due to elastic stratification can occur when the more elastic component occupies less than half of the total volume inside the channel.

Stability of the interface in two-layer couette flow of upper convected maxwell liquids

The linear stability of a two-layer Couette flow of upper convected Maxwell liquids is considered. The fluids have different densities, viscosities, and elasticities, with surface tension at the interface. At low speeds, the interfacial mode may become unstable, while other modes stay stable. The shortwave asymptotics of the interfacial mode is analyzed. It is found that an elasticity difference can stabilize or destabilize the flow even in the absence of a viscosity difference. As the viscosity difference increases, the range of elasticities for which there is shortwave stability widens. A linearly stable arrangement can be achieved by placing the less viscous fluid in a thin layer to stabilize longwaves and using elasticities to stabilize shortwaves. Such an arrangement can be stable even when the density stratification is adverse.

The thin-layer effect and interfacial stability in a two-layer Couette flow with similar liquids

The linear stability of Couette flow composed of two layers of immiscible fluids, one lying on top of the other, is considered for the special case when the two fluids have similar mechanical properties. The interfacial eigenvalue is found in closed form by considering the two-fluid problem as a perturbation of the one-fluid problem. The importance of the role played by the viscosity difference, when one of the fluids is in a thin layer, is illustrated.

Effect of fluid microstructure on stability of plane two-layer flows

The stability of plane two-layer Couette and Poiseuille flows, where the lower layer consists of a Grad-model fluid and the upper layer is a viscous Newtonian fluid, is investigated. The disturbances are assumed to be of the long-wave type, and the analysis involves expansion in wave numbers and is limited by two approximations. Numerical calculations are made for some values of the parameters. The calculations indicate that the rotational energy of the fluid in the lower layer has a destabilizing effect on the flow.