Patrice Laure

Linear pulse structure and signalling in a film flow on an inclined plane

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k -roots of D ( k , omega ) = 0, where k is a wavenumber, omega is a frequency and D ( k , omega ) is the dispersion relation function. The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability—as predicted by some model equations—is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem. The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V . The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.

Large departures from Boussinesq approximation in the Rayleigh–Bénard problem

Flows involving natural convection are generally studied employing the Boussinesq approximation of the Navier–Stokes equations. This paper deals with large departures from this limit by considering the two‐dimensional, periodic Rayleigh–Benard problem as an example. The main effect investigated here is a strong variation in the density considered primarily as a function of temperature as it is accounted for by the low Mach number equations. Additionally, the effect of temperature‐dependent viscosity and heat conductivity is considered. Both types of departure are measured by the nondimensional temperature gradient. They destroy the midplane symmetry of the problem and lead to quantitative and qualitative changes of the flow, in particular in the bifurcation from the conduction state. The present study illustrates how three different approaches—by linear stability analysis, weakly nonlinear analysis, and direct simulation—permit the investigation of complementary aspects of the problem. The first is used t...

Linear stability of multilayer plane Poiseuille flows of Oldroyd B fluids

Abstract The linear stability of plane Poiseuille flows of two and three-symmetrical layers is studied by using both longwave and moderate wavelength analysis. The considered fluids follow Oldroyd-B constitutive equations and hence the stability is controlled by the viscous and elastic stratifications and the layer thicknesses. For the three symmetrical-layer Poiseuille flow, competition between varicose (symmetrical) and sinuous (antisymmetrical) mode is considered. In both cases (two and three symmetrical layers), the additive character of the longwave formula with respect to viscous and elastic terms is largely used to determine stable arrangements at vanishing Reynolds number. It is found that if the stability of such arrangements is due simultaneously to viscous and elastic stratification (the flow is stable for longwave disturbance and the Poiseuille velocity profile is convex), then the Poiseuille flow is also stable with respect to moderate wavelength disturbances and the critical thickness ratio around which the configurations becomes unstable is given by longwave analysis. Note that a convex velocity profile means a positive jump of shear rate at the interface. Finally, the destabilization due to a moderate increase in the Reynolds number is considered and two distinct behaviors are pointed according to the convexity of the Poiseuille velocity profile. Moreover, an important influence of the thickness ratio on the critical wavenumber is found for three symmetrical layer case (for two layer case, the critical wave number is of order one and depends weakly on the thickness ratio).

Numerical simulation of oscillatory convection in semiconductor melts

Abstract This paper concerns the onset of oscillatory flows in horizontal layer subject to horizontal temperature gradient. It summarizes the main difficulties encountered, and typical results presented during a recent GAMM Workshop devoted to numerical simulation of oscillatory convection in low-Prandtl-number fluids. Hydrodynamics stability and bifurcation analysis are shown to be useful complementary tools for a better understanding of the onset of oscillations in metallic melts.

Nonlinear study of the flow in a long rectangular cavity subjected to thermocapillary effect

This paper is devoted to the theoretical study of motions of viscous fluids driven by a constant stress acting on an upper surface of a long rectangular cavity. This problem was originally addressed to the flow behavior of molten metals in open boats driven by thermocapillarity (Bridgman technique solidification). Computations of two‐dimensional Navier–Stokes equations show two totally different end circulations. Considering the spatial disturbances of the core flow (e.g., Couette flow), and, using the local theory of bifurcations (center manifold, normal forms), the appearance of two kinds of disturbances corresponding to these end circulations is explained. Moreover, on one hand a condition for the observability of the fully developed Couette flow in terms of aspect ratio (length/height) and Reynolds–Marangoni number is given, and on the other hand the analytical expression of the space periodic flow occurring in the direction of the cold wall is given.

Marangoni induced force on a drop in a Hele Shaw cell

We analyse the force balance on a cylindrical drop in a Hele-Shaw cell, subjected to a Marangoni flow caused by a surface tension gradient. Depth-averaged Stokes equations, called Brinkman equations, are introduced and a general closed form solution is obtained. The validity of the averaging procedure is ascertained by considering a linear surface tension gradient acting on a cylindrical flattened drop. The Marangoni-driven flow field and resulting force predicted by the Brinkman model are seen to match well a full three-dimensional direct numerical simulation. A closed form expression of the force acting on the drop is obtained, calculated from contributions due to the normal viscous stress, tangential viscous stress, and pressure fields, integrated on the drop perimeter. This expression is used to predict the force balance when a stationary droplet is submitted to both a carrier flow and a Marangoni flow. We show that previous results in the literature had underestimated by a factor two the Marangoni-induced force

Numerical methods for solid particles in particulate flow simulations

The flow motion of solid particle suspensions is a fundamental issue in many problems of practical interest. The velocity field of a such system is computed by a finite element method with a multi-domain approach of two phases (namely a viscous fluid and rigid bodies), whereas the particle displacement is made by a particulate method. We focus our paper on a simple shear flow of Newtonian fluid.

Nonlinear analysis of instability modes in the Taylor–Dean system

The linear and weakly nonlinear stability of flow in the Taylor–Dean system is investigated. The base flow far from the boundaries, is a superposition of circular Couette and curved channel Poiseuille flows. The computations provide for a finite gap system, critical values of Taylor numbers, wave numbers and wave speeds for the primary transitions. Moreover, comparisons are made with results obtained in the small gap approximation. It is shown that the occurrence of oscillatory nonaxisymmetric modes depends on the ‘‘anisotropy’’ coefficient in the dispersion relation, and that the critical Taylor number changes slightly with the azimuthal wave number for large absolute values of rotation ratio. The weakly nonlinear analysis is made in the framework of the Ginzburg–Landau equations for anisotropic systems. The primary bifurcation towards stationary or traveling rolls is supercritical when Poiseuille component of the base flow is produced by a partial filling. An external pumping can induce a subcritical bi...

Experimental Investigation of the Development of Interfacial Instabilities in Two Layer Coextrusion Dies

Abstract The stability of two-layer flow of polyethylene and polystyrene is experimentally studied in different flow geometries and for various flow rate ratios. A first coextrusion device allows to stop the coextrusion flow in a very long slit channel, to cool down the polymer sample and to dismantle the die in order to extract extrudate which is then carefully analyzed. A second device allows to observe the whole slit flow through transparent lateral walls and to record the interfacial waves in both spontaneous and controlled unstable conditions. Both devices point clearly out that the interfacial defect begins to grow after a specific flow distance and is then quickly amplified. This demonstrates the convective character of interfacial wave. Controlled unstable processing conditions in transparent die allow to measure accurately growth rate of defect in the linear regime and show quick occurrence of non linear regime.

Investigation of the Interfacial Instabilities in the Coextrusion Flow of PE and PS

Abstract The interface instability of the coextrusion flow of a polyethylene and a polystyrene is experimentally studied with industrial and laboratory equipments for various flow rates and temperature. Stable and unstable coextrusion conditions are identified as a function of flow rate ratio, shear rate and temperature. It is found that temperature and flow rate ratio are the most relevant parameters. Experimental results are then compared with stability analysis assuming White-Metzner constitutive equations. Agreement is fair at temperatures of 180°C and 200°C. However the longwave stability analysis is not sufficient to predict all experimental data.