S. Tabakova

Freezing of a supercooled spherical droplet with mixed boundary conditions

The freezing of a supercooled droplet occurs in two steps: recalescence, that is, a rapid return to thermodynamic equilibrium at the freezing temperature leading to a liquid–solid mixture and a longer stage of complete freezing. The second freezing step can be modelled by the one-phase Stefan problem for an inward solidification of a sphere, assuming the droplet to be spherical. A convective heat transfer with the ambient immiscible fluid is modelled by a mixed boundary condition on the outer surface of the droplet. This condition depends on the Biot number (ratio of the heat transfer resistances inside the droplet and at its surface). A novel asymptotic solution is developed for a small Stefan number and an arbitrary Biot number. Applying the method of matched asymptotic expansions, uniformly valid solutions are obtained for the temperature profile and freezing front evolution in the whole stage of complete freezing. For an infinite Biot number, that is, for a fixed temperature at the droplet outer boundary, known solutions are recovered. In parallel, numerical results are obtained for an arbitrary Stefan number using a finite-difference scheme based on the enthalpy method. The asymptotic and numerical solutions are in good agreement.

Approximation of the oscillatory blood flow using the Carreau viscosity model

The analysis of non-Newtonian flows in tubes is very important when studying the blood flow in different types of arteries. Usually the blood viscosity is defined by shear-dependent models, for example by the Carreau model, which represents the viscosity as a non-linear function of the shear-rate. In this paper the unsteady (oscillatory) 2D model of the blood flow in a straight tube is discussed theoretically and numerically. The solution of the quasilinear parabolic equation for the velocity is constructed using appropriate analytical functions. Further the corresponding numerical solution is approximated by similar analytical functions.

Numerical modelling of the one-phase stefan problem by finite volume method

The one-phase Stefan problem in enthalpy formulation, describing the freezing of initially supercooled droplets that impact on solid surfaces, is solved numerically by the finite volume method on a non-orthogonal body fitted coordinate system, numerically generated. The general case of third order boundary conditions on the droplet is considered. The numerical results for the simple case of a spherical droplet touching a surface at first order boundary conditions are validated well by the known 1D asymptotic solution. The proposed solution method occurs faster than another method, based on ADI implicit finite-difference scheme in cylindrical coordinates, for the same droplet shapes.

Dynamics and Stability of Free Thin Films

The modeling of the free thin film dynamics and stability is usually performed in the framework of the extended lubrication approach, including inertial, viscous, capillary and intermolecular van‐der‐Waals forces. For films with mobile surfaces, this approach leads to a system of nonlinear PDE for the film thickness and lateral velocities. In the present work the 1D variant of this system is studied numerically in the case of a laterally bounded free film by a frame, with a prescribed wetting angle. A linear and non‐linear stability analysis of this problem is performed and their results generalize the stability results obtained for periodic free films by other authors. It is shown that the film rupture can be caused either by the non‐existence of a static film shape or by the instability of an existing static shape. It is found that the static film shapes are always stable when subjected to symmetrical disturbances, but can be unstable if asymmetrical disturbances are imposed on them. The latter fact depends on the wetting angle in combination with the action of the van‐der‐Waals attraction and of the surface tension.

Freezing of a Suspended Supercooled Droplet with a Heat Transfer Mixed Condition on Its Outer Surface

The problem of water droplets freezing before or after their impact on solid surfaces is of major importance when modeling the aircrafts icing in wind tunnels. The object in this study is to estimate the freezing time of a suspended supercooled droplet in an air flow. It is known that freezing occurs into two steps: a short stage of rapid return to thermodynamic equilibrium, when the droplet becomes a water‐ice mixture and a longer stage of its complete freezing. Mathematically, the second freezing step can be modeled by the one‐phase Stefan problem. A convective heat transfer with ambient air is modeled here by a mixed boundary condition on the droplet outer surface. Assuming a spherical droplet, an asymptotic solution is developed for small Stefan numbers, while for arbitrary Stefan numbers a numerical solution based on the enthalpy method is constructed. Both solutions are compared. The numerical solution is also compared with other authors experimental results.

On the Stability of a Free Viscous Film

The free films have applications in some technological processes including liquid‐liquid or liquid‐gas disperse systems, as well as in the production of semiconductor micro‐materials from melt. The problem of film stability during these processes is of major interest. In the present work a free thin film, attached on a rectangular frame surrounded by an ambient gas, is considered. The film is assumed viscous and under the action of the capillary forces. The linear stability problem is studied, when a small perturbation is imposed on the steady‐state solution. An evolutionary nonlinear system for the longitudinal velocity and film thickness is obtained from the long‐wavelength approach. This system is solved numerically for the perturbed flow by the method of differential Gauss elimination. The results show, that for a large range of Re and wetting angles α, the film steady shapes are stable with respect to antisymmetrical disturbances.

Initial Stages of Viscous Drop Impact on Solid Surface

Drop impact has various applications like supercooled drops freezing when hitting air‐crafts or electrical power lines, spray cooling, soldering, ink jet printing. When a drop hits a solid surface, a circular wall jet appears, which may spread and/or splash afterwards. The modeling of this jet is still an open problem. In the present paper we propose an axisymmetric dynamic model of the jet appearance, as an inviscid fluid, and its evolution in time towards a thin viscous film, as a boundary layer develops from the wall. Typical values of the drop diameter considered here are O(1 mm) and typical values of the impact velocity are O(1 m/s). Numerical results for the jet thickness and lateral velocity are shown for some typical process parameters and are compared with experimental data obtained with a rapid shutter video camera.

Numerical modelling of the free film dynamics and heat transfer under the van der waals forces action

In the present work a numerical model of the heat transfer of a hot free thin viscous film attached on a rectangular colder frame is proposed. If the film is cooled down to its solidification temperature the Stefan boundary condition for the heat flux jump is introduced and a part of the liquid film is transformed into a rigid one. The film is assumed to be under the action of the capillary forces and attractive intermolecular van der Waals forces and to be symmetric to a middle plane. Taking the film thickness as a small parameter, the thermal-dynamic problem in its one-dimensional form is solved numerically by a conservative finite difference scheme on a staggered grid.