Discontinuous Galerkin finite element method applied to the coupled Navier-Stokes/Cahn-Hilliard equations

International audienceTwo-phase flows driven by the interfacial dynamics is studied with a phase-field model to tract implicitly interfaces. The phase field obeys the Cahn-Hilliard equation. The fluid dynamics is described with the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a non-linear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal-order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. Finally, numerical simulations describing the capillary rising in a tube is presented

Intermittent flow in yield-stress fluids slows down chaotic mixing

In this article, we present experimental results of chaotic mixing of Newtonian uids and yield stress fluids using rod-stirring protocol with rotating vessel. We show how the mixing of yield stress fluids by chaotic advection is reduced compared to the mixing of Newtonian fluids and explain our results bringing to light the relevant mechanisms: the presence of fluid that only flows intermittently, a phenomenon enhanced by the yield stress, and the importance of the peripheral region. This finding is confirmed via numerical simulations. Anomalously slow mixing is observed when the synchronization of different stirring elements leads to the repetition of slow stretching for the same fluid particles.Comment: 5 page

Thermoconvective instabilities of a non-uniform Joule-heated high Prandtl number liquid

The Joule heating is commonly used to melt materials in various industries. This is a particular case in glass industry for which electric melting is mainly employed for production of potentially volatile, polluting glasses and also for wool insulation products. In such processes, electrodes are introduced in the bath and the heating is due to the Joule dissipation. The raw materials are introduced from above floating at the surface of the bath forming a cohesive batch leading to a high level of thermal insulation. The maximum of temperature is observed inside the bath which can be a source of thermal instabilities. In order to study these instabilities, we develop a numerical tool coupling the thermoconvection problem written in the framework of Boussinesq approximation with electric potential equation. The numerical method is based on the discontinuous Galerkin finite element approach. This choice is motivated by the fact that the molten silicate giving a glass presents a high Prandtl number. Consequently, a strong impact of the convective process in the thermal transfer is expected. The investigated domain is a 2D-enclosure with two electrodes corresponding to a portion of the two vertical walls. The flow and thermal fields are characterized by the aspect ratio of the enclosure a set at 2 in our numerical simulations, the electrode length Le and the Rayleigh Ra and the Prandtl Pr numbers. We first study the case of the electrodes corresponding to the entire of the vertical walls. In a such case, the flow appears only when the Rayleigh number is larger than a critical value equal to Racr=1700 when a=2. The flow intensity measured by a Péclet number behaves like a square root of the difference between Ra-Racr as expected in supercritical bifurcation when Ra>Racr. The thermal efficiency can be quantified by computing the average temperature over the horizontal direction of the cavity. The inverse of the average temperature corresponds to the Nusselt number. Numerically, we observe that the Nusselt number which is constant below the critical Rayleigh number increases linearly with Ra-Racr. We perform numerical simulations when electrodes are equal to 2/3 of the top of the vertical walls. In this situation, the flow appears without threshold. Two regimes emerge: at low Rayleigh number (below to 1e3), the Péclet number is proportional to the Rayleigh number and above Ra=1e3 the Péclet number is a function of the square root of Ra. These behaviors are completely independent of the Prandtl number. The flow and thermal solution becomes periodic in time above a critical Rayleigh number which is function of the Prandtl number. The occurrence of this instability is mainly due to the creation of two counter-rotating cells forming a jet in the middle of the cavity which can not stay stable. At low Prandtl number (equal to one), the critical Rayleigh above which the flow becomes periodic is larger to 1.6e5 while when the Prandtl number is larger than 10 the critical Rayleigh number decreases to 2.5e4. For a larger Prandtl number, the critical Rayleigh number does not change significantly. Finally, the periodic regime is analyzed by determining the frequency and amplitude of flow oscillations in order to characterize the nature of the instability

Mass-Transfer Enhancement by a Reversible Chemical Reaction Across the Interface of a Bubble Rising Under Stokes Flow

Mass transfer around a bubble rising in a liquid under Stokes regime is investigated when a reversible chemical reaction, A !B, is taken into account. Four dimensionless parameters control the interfacial transfer rate: the Peclet and Damkohler numbers, the ratio of the diffusion coefficient of both species, and the reaction equilibrium constant. The € mass-transfer equations are solved numerically with a finite element technique. A boundary layer approach is also proposed and solved with a coupled technique of finite difference and Chebyshev-spectral method. The equilibrium constant and the ratio of diffusion coefficients have a strong influence on the coupling between the chemical reaction and mass transfer leading to an increase of the Sherwood number. The interaction between the chemical reaction and advection is clearly established by the simulations. Conditions corresponding to Peclet number larger than the Damkohler number reduces the effect of the chemical reaction

Stability of vertical films of molten glass due to evaporation

International audienceFirst, we report observations achieved on a gravitationally-driven film drainage with molten glass pointing out a stabilizing effect when temperature is larger than 1250 C. A model to describe the change of surface tension with the film thickness due to the evaporation of oxide species is proposed. A lubrication model is derived taking into account the gradient of surface tension. The final system of equations describing the mass and the momentum conservations is numerically solved by an implicit time solver using a finite difference method at a second order scheme in time and space. The numerical procedure is applied to study a film drainage of molten soda-lime-silica glass. The effect of the surface tension gradient is investigated pointing out that with an increase of 0.5 % of the surface tension over the spread of the film which is order of few centimeters, the liquid film reaches an equilibrium thickness in agreement with previous experimental work

Low-Reynolds-number rising of a bubble near a free surface at vanishing Bond number

International audienceThis work considers a nearly-spherical bubble and a nearly-flat free surface interacting under buoyancy at vanishing Bond number Bo. For each perturbed surface, the deviation from the unperturbed shape is asymptotically obtained at leading order on Bo. The task appeals to the normal traction exerted on the unperturbed surface by the Stokes flow due to a spherical bubble translating toward a flat free surface. The free surface problem is then found to be well-posed and to admit a solution in closed form when gravity is still present in the linear differential equation governing the perturbed profile through a term proportional to Bo. In contrast, the bubble problem amazingly turns out to be over-determined. It however becomes well-posed if the requirement of horizontal tangent planes at the perturbed bubble north and south poles is discarded or if the term proportional to \Bo is omitted. Both previous approaches turn out to predict for small Bond number quite close solutions except in the very vicinity of the bubble poles. The numerical solution of the proposed asymptotic analysis shows, in the overlapping range Bo=O(0.1) and for both the bubble and the free surface perturbed shapes, a good agreement with a quite different boundary element approach developed in [Phys. Fluids 23, 092102 (2011)]. It also provides approximated bubble and free surface shapes whose sensitivity to the bubble location is examined

An implicit high order discontinuous Galerkin level set method for two-phase flow problems

International audienceAn implicit high order time (BDF) and polynomial degree discontinuous Galerkin (DG) level set method is presented in this talk. The major advantage of this new approach is an accurate mass conservation during the convection of the level set function, thanks to the implicit method. Numerical experiments are presented for the Zalesak and the Leveque test cases. The convergence rates versus time and space are investigated for both BDF and DG high orders. The capture of the zero level set interface is then improved by using an auto-adaptive mesh procedure. The problem is approximated by using the discontinuous Galerkin method for both the level set function, the velocity and the pressure fields

Soft Matter Drainage in a rising foam

International audienceRising foams created by continuously blowing gas into a surfactant solution are widely used in many technical processes, such as flotation. The prediction of the liquid fraction profile in such flowing foams is of particular importance since this parameter controls the stability and the rheol-ogy of the final product. Using drift flux analysis and recently developed semi-empirical expressions for foam permeability and osmotic pressure, we build a model predicting the liquid fraction profile as a function of height. The theoretical profiles are very different if the interfaces are considered as mobile or rigid, but all of our experimental profiles are described by the model with mobile interfaces. Even the systems with dodecanol, which are well known to behave as rigid in forced drainage experiments. This is because in rising foams the liquid fraction profile is fixed by the flux at the bottom of the foam. Here the foam is wet with higher permeability and the interfaces are not in equilibrium. These results demonstrate once again that it is not only the surfactant system that controls the mobility of the interface, but also the hydrodynamic problem under consideration. For example liquid flow through the foam during generation or in forced drainage is intrinsically different

The Sparse Cardinal Sine Decomposition applied to Stokes integral equations

International audienceNumerical simulations of two-phase flows driven by viscosity (e.g. for bubble motions in glass melting process) rely on the ability to efficiently compute the solutions to discretized Stokes equations. When using boundary element methods to track fluid interfaces, one usually faces the problem of solving linear systems with a dense matrix with a size proportional to the system number of degrees of freedom. Acceleration techniques, based on the compression of the underlying matrix and efficient matrix vector products are known (Fast Multipole Method, H-matrices, etc.) but are usually rather cumbersome to develop. More recently, a new method was proposed, called the " Sparse Cardinal Sine Decomposition " , in the context of acoustic problems to tackle this kind of problem in some generality (in particular with respect to the Green kernel of the problem). The proposed contribution aims at showing the potential applicability of the method in the context of viscous flows governed by Stokes equations