We examine the buoyancy-driven rise of a bubble in a Newtonian or a viscoplastic fluid assuming axial symmetry and steady flow. Bubble pressure and rise velocity are determined, respectively, by requiring that its volume remains constant and its centre of mass remains fixed at the centre of the coordinate system. The continuous constitutive model suggested by Papanastasiou is used to describe the viscoplastic behaviour of the material. The flow equations are solved numerically using the mixed finite-element/Galerkin method. The nodal points of the computational mesh are determined by solving a set of elliptic differential equations to follow the often large deformations of the bubble surface. The accuracy of solutions is ascertained by mesh refinement and predictions are in very good agreement with previous experimental and theoretical results for Newtonian fluids. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham Bn=$\tau_y^{\ast}/\rho^{\ast}g^{\ast} R_b^{\ast}$ Bond Bo =$\rho^{\ast}g^{\ast} R_b^{\ast 2}/\gamma^{\ast}$ and Archimedes Ar=$\rho^{\ast2}g^{\ast} R_b^{\ast3}/\mu_o^{\ast2}$ numbers, where ρ* is the density, μ*o the viscosity, γ* the surface tension and τ*y the yield stress of the material, g* the gravitational acceleration and R*b the radius of a spherical bubble of the same volume. If the fluid is viscoplastic, the material will not be deforming outside a finite region around the bubble and, under certain conditions, it will not be deforming either behind it or around its equatorial plane in contact with the bubble. As Bn increases, the yield surfaces at the bubble equatorial plane and away from the bubble merge and the bubble becomes entrapped. When Bo is small and the bubble cannot deform from the spherical shape the critical Bn is 0.143, i.e. it is a factor of 3/2 higher than the critical Bn for the entrapment of a solid sphere in a Bingham fluid, in direct correspondence with the 3/2 higher terminal velocity of a bubble over that of a sphere under the same buoyancy force in Stokes flow. As Bo increases allowing the bubble to squeeze through the material more easily, the critical Bingham number increases as well, but eventually it reaches an asymptotic value. Ar affects the critical Bn value much less.

The mechanisms driving the surfactant-enhanced spreading of droplets on the surface of solid substrates, and particularly those underlying the superspreading behaviour sometimes observed, are investigated theoretically. Lubrication theory for the droplet motion, together with advection-diffusion equations and chemical kinetic fluxes for the surfactant transport, leads to coupled evolution equations for the drop thickness, interfacial concentrations of surfactant monomers and bulk concentrations of monomers and micellar, or other, aggregates. The surfactant can be adsorbed on the substrate either directly from the bulk monomer concentrations or from the liquid-air interface through the contact line. An important feature of the spreading model developed here is the surfactant behaviour at the contact line; this is modelled using a constitutive relation, which is dependent on the local surfactant concentration. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis for cases of both insoluble and soluble surfactants with concentrations that can, in the latter case, exceed the critical micelle, or aggregate, concentration. The results show that basal adsorption of the surfactant plays a crucial role in the spreading process; the continuous removal of the surfactant that lies upon the liquid-air interface, due to the adsorption at the solid surface, is capable of inducing high Marangoni stresses, close to the droplet edge, driving very fast spreading. The droplet radius grows at a rate proportional to t a with a = 1 or even higher, which is close to the reported experimental values for superspreading. The spreading rates follow a non-monotonic variation with the initial surfactant concentration also in accordance with experimental observations. An accompanying feature is the formation of a rim at the leading edge of the droplet. In some cases, the drop spreads to form a 'pancake' or creates a 'secondary' front separated from the main droplet.

We consider the flow dynamics of a thin evaporating droplet in the presence of an insoluble surfactant and non-interacting particles in the bulk. Based on lubrication theory, we derive a set of evolution equations for the film height, the interfacial surfactant and bulk particle concentrations, taking into account the dependence of the liquid viscosity on the local particle concentration. An important ingredient of our model is that it takes into account the fact that the surfactant adsorbed at the interface hinders evaporation. We perform a parametric study to investigate how the presence of surfactants affects the evaporation process as well as the flow dynamics with and without the presence of particles in the bulk. Our numerical calculations show that the droplet life-time is affected significantly by the balance between the ability of surfactant to * To whom correspondence should be addressed † Department of Chemical Engineering, University of Patras, Patras 26500, Greece ‡ Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 285, Telangana, India ¶ Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK 1 enhance spreading suppressing the effect of thermal Marangoni stresses-induced motion and to hinder the evaporation flux through the reduction of effective interfacial are of evaporation, which tend to accelerate and decelerate the evaporation process, respectively. For particle-laden droplets and in the case of dilute solutions, the droplet life-time is found to be weakly-dependent on the initial particle concentration. We also show that the particle deposition patterns are influenced strongly by the direct effect of surfactant on the evaporative flux; in certain cases, the "coffee stain" effect is enhanced significantly. A discussion of the delicate interplay between the effects of capillary pressure, solutal and thermal Marangoni stresses, which drive the liquid flow inside the evaporating droplet giving rise to the observed results is provided herein.

ABSTRACT:We study the thermocapillary-driven spreading of a droplet on a nonuniformly heated substrate for fluids associated with a non-monotonic dependence of the surface tension on temperature. We use lubrication theory to derive an evolution equation for the interface that accounts for capillarity and thermocapillarity. The contact line singularity is relieved by using a slip model and a Cox-Voinov relation; the latter features equilibrium contact angles that vary depending on the substrate wettability, which, in turn, is linked to the local temperature. We simulate the spreading of droplets of fluids whose surface tension−temperature curves exhibit a turning point. For cases wherein these turning points correspond to minima, and when these minima are located within the droplet, then thermocapillary stresses drive rapid spreading away from the minima. This gives rise to a significant acceleration of the spreading whose characteristics resemble those associated with the "superspreading" of droplets on hydrophobic substrates. No such behavior is observed for cases in which the turning point corresponds to a surface tension maximum. ■ INTRODUCTIONThe motion of sessile droplets over liquids and solids is of central importance to a number of industrial applications such as coating flow technology, inkjet printing, microfluidics and microelectronics, and medical diagnostics. Despite the apparent simplicity of the physical setup involved, this motion is rather complex and some of its aspects remain poorly understood; in particular, the mechanisms underlying the dynamics of the threephase contact line are still the subject of debate. In view of its complexity 1 and its practical importance, droplet motion has received considerable attention in the literature and has been the subject of two major reviews. 2,3 In this work, we consider the motion of sessile droplets on non-isothermal solid walls, driven by thermocapillarity. The walls underlying the droplets are subjected to a temperature gradient which induces surface tension gradient-driven droplet deformation and migration from low to high surface tension regions. Thermocapillary-driven droplet motion was studied by Bouasse 4 who demonstrated the possibility of inducing droplet-climbing on a heated wire, against the action of gravity, by heating its lower end. Studies involving horizontal substrates have shown that, unless the magnitude of the imposed temperature gradient is sufficiently large, no droplet motion is possible due to contact angle hysteresis, while under certain conditions, steady migration of droplets has been shown. 5,6 A number of studies have examined the thermocapillary motion of droplets theoretically. Brochard 7 determined the spreading characteristics of a wedge-shaped drop in the presence of chemical or thermal gradients via local force and energy balances. This work was generalized by Ford and Nadim 8 to arbitrary, two-dimensional droplet shapes and different contact angles at the two contact lines. Lubrication theory was used to describe the...

ABSTRACT:We study the two-dimensional dynamics of a droplet on an inclined, nonisothermal solid substrate. We use lubrication theory to obtain a single evolution equation for the interface, which accounts for gravity, capillarity, and thermocapillarity, brought about by the dependence of the surface tension on temperature. The contact line motion is modeled using a relation that couples the contact line speed to the difference between the dynamic and equilibrium contact angles. The latter are allowed to vary dynamically during the droplet motion through the dependence of the liquid−gas, liquid−solid, and solid−gas surface tensions on the local contact line temperature, thereby altering the local substrate wettability at the two edges of the drop. This is an important feature of our model, which distinguishes it from previous work wherein the contact angle was kept constant. We use finite-elements for the discretization of all spatial derivatives and the implicit Euler method to advance the solution in time. A full parametric study is carried out in order to investigate the interplay between Marangoni stresses, induced by thermo-capillarity, gravity, and contact line dynamics in the presence of local wettability variations. Our results, which are generated for constant substrate temperature gradients, demonstrate that temperature-induced variations of the equilibrium contact angle give rise to complex dynamics. This includes enhanced spreading rates, nonmonotonic dependence of the contact line speed on the applied substrate temperature gradient, as well as "stick−slip" behavior. The mechanisms underlying this dynamics are elucidated herein.