P. M. J. Trevelyan

Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells

Buoyancy-driven instabilities of a horizontal interface between two miscible solutions in the gravity field are theoretically studied in porous media and Hele-Shaw cells (two glass plates separated by a thin gap). Beyond the classical Rayleigh-Taylor (RT) and double diffusive (DD) instabilities that can affect such two-layer stratifications right at the initial time of contact, diffusive-layer convection (DLC) as well as delayed-double diffusive (DDD) instabilities can set in at a later time when differential diffusion effects act upon the evolving density profile starting from an initial step-function profile between the two miscible solutions. The conditions for these instabilities to occur can therefore be obtained only by considering time evolving base-state profiles. To do so, we perform a linear stability analysis based on a quasi-steady-state approximation (QSSA) as well as nonlinear simulations of a diffusion-convection model to classify and analyse all possible buoyancy-driven instabilities of a stratification of a solution of a given solute A on top of another miscible solution of a species B. Our theoretical model couples Darcys law to evolution equations for the concentration of species A and B ruling the density of the miscible solutions. The parameters of the problem are a buoyancy ratio R quantifying the ratio of the relative contribution of B and A to the density as well as δ, the ratio of diffusion coefficients of these two species. We classify the region of RT, DD, DDD and DLC instabilities in the (R, δ) plane as a function of the elapsed time and show that, asymptotically, the unstable domain is much larger than the one captured on the basis of linear base-state profiles which can only obtain stability thresholds for the RT and DD instabilities. In addition the QSSA allows one to determine the critical time at which an initially stable stratification of A above B can become unstable with regard to a DDD or DLC mechanism when starting from initial step function profiles. Nonlinear dynamics are also analysed by a numerical integration of the full nonlinear model in order to understand the influence of R and δ on the dynamics.

Viscous fingering of a miscible reactive A+B --> C interface: a linear stability analysis

When one solution of reactant A is displacing another miscible solution of reactant B , a miscible product C can be generated in the contact zone if a simple A + B → C chemical reaction takes place. Depending on the relative effect of A , B and C on the viscosity, different viscous fingering (VF) instabilities can be observed. In this context, a linear stability analysis of this reaction–diffusion–convection problem under the quasi-steady-state approximation is performed to classify the various possible instability scenarios. To do so, we determine the criteria for an instability, obtain dispersion curves both at initial contact time using an analytical implicit solution and at later times via numerical stability analysis. Along with recovering known results for non-reactive systems where the displacement of a more viscous fluid by a less viscous one leads to a VF instability, it is found that in the presence of a chemical reaction, injecting a more viscous fluid into a less viscous fluid can also lead to instabilities. This occurs when the chemical reaction leads to the build up of non-monotonic viscosity profiles. Various instability scenarios are classified in a parameter plane spanned by R b and R c representing the log-mobility ratios of the viscosities of the B and C solution respectively with respect to that of the injected solution of A . A parametric study of the influence on stability of the Damkohler number and of the time elapsed after contact of the two reactive solutions is also conducted.

Heated Falling Films

We present new insights and results for the problem of a film falling down a heated wall: (i) treatment of a mixed heat flux boundary condition on the substrate; (ii) development of a long-wave theory for large Peclet numbers; (iii) refined treatment of the energy equation based on a high-order Galerkin projection in terms of polynomial test functions which satisfy all boundary conditions; (iv) time-dependent computations for the free-surface height and interfacial temperature; (v) numerical solution of the full energy equation; (vi) demonstration of the existence of a thermal boundary layer at the front stagnation point of a solitary pulse; (vii) development of models that prevent negative temperatures and are in good agreement with the numerical solution of the full energy equation.

Convective mixing induced by acid-base reactions.

When two miscible solutions, each containing a reactive species, are put in contact in the gravity field, local variations in the density due to the reaction can induce convective motion and mixing. We characterize here both experimentally and theoretically such buoyancy-driven instabilities induced by the neutralization of a strong acid by a strong base in aqueous solutions. The diverse patterns obtained are shown to depend on the type of reactants used and on their relative concentrations. They have their origin in a combination of classical hydrodynamic instabilities including differential diffusion of the solutes involved while temperature effects only play a marginal role.

Dynamics of a horizontal thin liquid film in the presence of reactive surfactants

We investigate the interplay between a stable horizontal thin liquid film on a solid substrate and an excitable or bistable reactive mixture on its free surface. Their coupling is twofold. On the one hand, flow in the film transports the reacting surfactants convectively. On the other hand, gradients in the surfactant concentration exert Marangoni stresses on the free surface of the film. A reduced model is derived based on the long-wave approximation. We analyze the linear stability of the coupled system as well as the nonlinear behavior, including the propagation of solitary waves, fronts, and pulses. We show, for instance, that the coupling of thin film hydrodynamics and surfactant chemistry can either stabilize instabilities occurring in the pure chemical system, or in a regime where the pure hydrodynamic and chemical subsystems are both stable, the coupling can induce instabilities.

Differential diffusion effects on buoyancy-driven instabilities of acid–base fronts: the case of a color indicator

Buoyancy-driven hydrodynamic instabilities of acid-base fronts are studied both experimentally and theoretically in the case where an aqueous solution of a strong acid is put above a denser aqueous solution of a color indicator in the gravity field. The neutralization reaction between the acid and the color indicator as well as their differential diffusion modifies the initially stable density profile in the system and can trigger convective motions both above and below the initial contact line. The type of patterns observed as well as their wavelength and the speed of the reaction front are shown to depend on the value of the initial concentrations of the acid and of the color indicator and on their ratio. A reaction-diffusion model based on charge balances and ion pair mobility explains how the instability scenarios change when the concentration of the reactants are varied.

Dynamics of a reactive falling film at large Péclet numbers. I. Long-wave approximation

We study the dynamics of a vertically falling film in the presence of a first-order (exothermic or endothermic) chemical reaction. We extend the work by Trevelyan et al. [Phys. Fluids 14, 2402 (2002)] on the same problem to large heat/mass transport Peclet numbers and so we take into account the convective terms of the heat/mass transport equations. Our analysis is based on a long-wave expansion of the equations of motion and associated boundary conditions. Previous results by Trevelyan et al. are reviewed and compared with present results. Particular emphasis is given on permanent-form traveling solitary waves. We show that the inclusion of the heat/mass transport convective effects can have a dramatic effect on the evolution of the interface and in fact can make the solitary waves dispersive. The size of dispersion depends on the size of the Prandtl and Schmidt numbers while its sign can change from positive to negative leading to negative-hump solitary waves. We show that for large dispersion and for a...

Dynamics of a reactive falling film at large Péclet numbers. II. Nonlinear waves far from criticality: Integral-boundary-layer approximation

We consider the dynamics of a reactive falling film far from criticality. Our analysis is based on the integral-boundary-layer (IBL) approximation of the equations of motion, energy and concentration, and associated free-surface boundary conditions. We develop a hierarchy of IBL models starting from a simplified Shkadov approach to large IBL systems based on high-order Galerkin projections. We show that these high-order models correct the deficiencies of Shkadov’s approach and predict correctly all relevant quantities including the critical Reynolds number. We also numerically construct nonlinear solutions of the solitary wave type for a simplified Shkadov approximation and we show that unlike the long-wave theory in Paper I which leads to branch multiplicity and limit points as well as points where the solitary wave solution branches terminate, the IBL model predicts the existence of solitary waves for all Reynolds numbers.

Dynamics of a vertically falling film in the presence of a first-order chemical reaction

The evolution of a vertically falling film in the presence of a simple first-order (exothermic or endothermic) chemical reaction is considered. The heat of reaction sets up surface tension gradients that induce thermocapillary stresses on the free-surface, thus affecting the evolution of the film. By using a long-wave expansion of the equations of motion and associated boundary conditions, we derive a nonlinear partial differential equation of the evolution type for the local film thickness. We demonstrate that, when the surface tension is an increasing function of temperature an exothermic reaction has a stabilizing effect on the free surface while an endothermic reaction is destabilizing. We construct bifurcation diagrams for permanent solitary waves and show that, in all cases the solution branches exhibit limit points and multiplicity with two branches, a lower branch and an upper branch. Time-dependent computations of the free-surface evolution equation show that the system always approaches a train of coherent structures that resemble the lower branch solitary waves. We also examine the absorption characteristics through the interface and we demonstrate that an endothermic reaction enhances absorption and mass transport. The opposite is true for an exothermic reaction.

Mass-transport enhancement in regions bounded by rigid walls

The mass transport into a fluid bounded by stationary rigid walls in the limit of large Péclet number, Pe, is examined analytically. Two model systems are considered in detail: a stationary cavity and a model involving two concentric rotating cylinders. A macroscopic gradient is imposed between the top and bottom surfaces. It is demonstrated that mass transport into the fluid is enhanced owing to a recirculation zone which is connected to the solid boundary through a boundary layer of thickness O(Pe−1/3) in which cross-stream molecular diffusion is balanced by convection. The associated enhancement is large and scales as Pe1/3. Our asymptotic analysis is found to be in good agreement with numerical solutions of the full transport equation.