Showing papers by "Yuriko Renardy published in 1999"

Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids

Abstract: We provide a mathematical analysis of the spectrum of the linear stability problem for one and two layer channel flows of the upper-convected Maxwell (UCM) and Oldroyd-B fluids at zero Reynolds number. For plane Couette flow of the UCM fluid, it has long been known (Gorodstov and Leonov, J. Appl. Math. Mech. (PMM) 31 (1967) 310) that, for any given streamwise wave number, there are two eigenvalues in addition to a continuous spectrum. In the presence of an interface, there are seven discrete eigenvalues. In this paper, we investigate how this structure of the spectrum changes when the flow is changed to include a Poiseuille component, and as the model is changed from the UCM to the more general Oldroyd-B. For a single layer UCM fluid, we find that the number of discrete eigenvalues changes from two in Couette flow to six in Poiseuille flow. The six modes are given in closed form in the long wave limit. For plane Couette flow of the Oldroyd-B fluid, we solve the differential equations in closed form. There is an additional continuous spectrum and a family of discrete modes. The number of these discrete modes increases indefinitely as the retardation time approaches zero. We analyze the behavior of the eigenvalues in this limit.

Direct simulation of unsteady axisymmetric core-annular flow with high viscosity ratio

Abstract: Axisymmetric pipeline transportation of oil and water is simulated numerically as an initial value problem. The simulations succeed in predicting the spatially periodic Stokes-like waves called bamboo waves, which have been documented in experiments of Bai, Chen & Joseph (1992) for up-flow. The numerical scheme is validated against linearized stability theory for perfect core–annular flow, and weakly nonlinear saturation to travelling waves. Far from onset conditions, the fully nonlinear saturation to steady bamboo waves is achieved. As the speed is increased, the bamboo waves shorten, and peaks become more pointed. A new time-dependent bamboo wave is discovered, in which the interfacial waveform is steady, but the accompanying velocity and pressure fields are time-dependent. The appearance of vortices and the locations of the extremal values of pressure are investigated for both up- and down-flows.

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

Abstract: 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.

comment on "a numerical study of periodic disturbances on two-layer couette flow" phys. fluids 10, 3056 (1998)

Abstract: nt ysiThe flow of fluids with different viscosities, subjected an interfacial perturbation, can lead to fingering a migration. Both Refs. 2 and 3 refer to fingers in two dimensional studies, which would correspond to sheets three dimensions. Figure 19 of Ref. 3 shows a sequenc interface positions for initial amplitude 0.05, wave numb a5p/2, Reynolds number R15500, viscosity ratiom50.5, undisturbed interface height l 150.372, and interfacial ten sion parameterT50.01, at times 0, 5, 10, 15, and 20 carried out on a 256 3256 mesh. A region of high curvatur forms, followed by pinching into a series of horizontal drop However, the drops are at the scale of the numerical mes spatial convergence test is reported here, showing that w refined mesh, the drops are replaced by an elongated fin

Time-Dependent Pattern Formation for Two-Layer Convection

Abstract: This article is a review of double-layer convection in which the pattern formation arises due to a competition between the bulk motions in each fluid (see Figure 1). An instability takes place when the temperature difference between the upper and lower walls reaches a threshold value, and the response of the two-layer system depends on the properties of the constituent fluids. Our motivation is the search for patterns formed in non-equilibrium fluid dynamical systems which exhibit time-dependence at or near the onset of a pattern. Such a time-dependent state is predicted for the two-layer Rayleigh-Benard system and is accessible experimentally as well as theoretically. We expect to see oscillatory and spatio-temporal chaotic behavior. The presence of the interface and the coupling between the fluids in this model problem may provide an understanding of generic behaviors in related applications, such as the modeling of the earth’s mantle as a two-layer convecting system [3, 11], and liquid encapsulated crystal growth [19].

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

Abstract: 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.