Mathematics

The onset of Soret-driven convection in a horizontal layer of a colloidal suspension is investigated by considering a particulate medium model. We consider a dilute suspension of spherical solid particles being subjected to convection in a Rayleigh-Bénard geometry setup. The mathematical model takes into account the effects of thermophoresis, particle sedimentation, and Brownian diffusion. The equations governing the convective motion consist of the momentum equation which includes an extra body force term to account for the thermophoretic force effect, the conservation of particles equation whose mass-flux term couples the Soret and particle diffusion effects and whose advective term includes the sedimentation force, and the heat and mass balance equations. The horizontal boundaries are assumed rigid, perfectly thermally conducting, and impervious to mass flow. Furthermore, the model makes use of the effective viscosity of the suspension whose dependence on the particle concentration is through Einstein's formula. Moreover, we take into account the decrease of both the coefficient of Brownian diffusion and the mixture thermal diffusion with particle concentration due to the particles hindrance effect. The nondimensionalization leads to the emergence of an experimental parameter, β, which depicts the competition between the effects of thermophoresis, sedimentation, and particle diffusion. The parameter β is a function of the particles radius, the shape of which is an inverted parabola having two zeros. A combination of asymptotic and numerical computations is used to determine the threshold for the onset of the mass dominated convection, which corresponds to 0 < β 1. Our findings shed light on the role of particle sedimentation and particle size, as well as the influence of other processing variables on the fluid instability.

We present a dual-reciprocity boundary element method (DRBEM) to investigate bulk surfactant transport dynamics in a free-surface flow system under steady-state conditions. This free-surface flow system consists of semi-infinite bubble progression in a rigid axisymmetric capillary tube. Once adsorbed to the air-liquid interface with a surface concentration , surfactant alters the interfacial surface tension γ . As the interfacial stress balance, which governs the fluid mechanics, is a function of γ , a strong coupling exists between surfactant transport dynamics and the fluid mechanics (physicochemical hydrodynamics). To model this problem over a range of bulk concentrations, C, the bulk convective/diffusive transport of surfactant to the interface must be calculated. In this paper, DRBEM is used to simulate the bulk convection-diffusion relationship while the boundary element method (BEM) is used to solve Stokes flow, and a finite-difference method is used to solve the surface transport equation under steady-state conditions. A nonlinear Langmuir adsorption model is used to determine the surfactant equation of state γ = f ( ). The validity of the DRBEM is first demonstrated by comparing computational and analytical solutions for a test problem. Next, the computational algorithm is used to calculate the bulk concentration field surrounding the bubble as a function of the far-downstream quantity of surfactant, C o , and its influence on interfacial dynamics. These profiles clearly demonstrate the importance of accurately calculating the bulk concentration field under moderate C o conditions. In addition, the variation of mechanical properties of this system as a function of C o indicates that the interfacial pressure jump can be significantly larger when the bulk transport of surfactant to the interface is limited.

Chemical and colloidal transport processes in partially saturated porous and fractured media are dependent on liquid flow along a solid surface, which may be chaotic even for small Reynolds numbers. This paper presents the derivation of a one-dimensional evolution equation describing the slow motion (small Reynolds numbers, R B B1) of a very thin liquid film flowing down an inclined impermeable plane. In this equation, gravitational, capillary, and molecular forces are taken into account. The addition of the molecular force term leads to a highly nonlinear equation governing the spatial and temporal evolution of film thickness. In a weakly nonlinear limit, this evolution equation is rescaled to a canonical form. The latter predicts a chaotic hydrodynamic instability for the film surface. This chaotic behavior is illustrated using the 3D projections of pseudo-phase space attractors for the spatial and temporal variations of the dimensionless film thickness.