Slavtcho Slavtchev

Thermocapillary flow in a liquid layer at minimum in surface tension

SummaryIn the present paper a class of similarity solutions for the two-dimensional Navier-Stokes and energy equations describing thermocapillary flows in a liquid layer of constant width and infinite extent is presented. The layer is bounded by a horizontal rigid plate from one side and opened to the ambient gas from the other one. The physical properties of the liquid are assumed to be constant except the surface tension which varies as a quadratic function with temperature. It is supposed that a constant temperature gradient exists along either the liquid free surface (case I) or the rigid boundary (case II).In both cases, by means of a similarity transformation, the equations of motion and energy are reduced to a system of three ordinary differential equations, one for the velocity and two for the temperature. The equation for the velocity can be solved separately from the other equations and its solution, found numerically, exists only for the Marangoni number less than a certain finite value. The solution of the whole system depends also on the Prandtl number. The solution of one of the temperature equations is presented in an analytical form and the other equation is solved numerically. Asymptotic formulas of the functions are also obtained for small and large Marangoni numbers. Flow pattern and temperature fields are presented. One convective roll exists in every semi-infinite layer. Fluid velocities at different points of the free surface are evaluated for an aqueous solution of n-heptanol and compared with those measured in the experiments.

Unsteady film flow of power-law liquids on a rotating disk

Abstract A boundary-layer-type approximation of the equations of motion for the flow in a film or power-law liquid on a rotating disk is derived. For a given non-steady angular elocity an exact, similarity, solution to the obtained equations is presented. Flow velocity and the evolution of the initally non-uniform thickness of the film are studied for relatively small times after the start of rotation.

Weakly Nonlinear Marangoni Convection under an External Magnetic Field

Nonlinear Marangoni instability in a thin layer of conducting fluid in the presence of a magnetic field is studied. The layer is bounded from below by a rigid, thermally and electrically perfect conductor and from above by the planar free surface. The ambient gas phase is assumed to be either insulating or conducting. The analysis is carried out by a successive approximation method, which allows for finite amplitude disturbances. Two amplitude equations are derived with coefficients depending on the Chandrasekhar and magnetic Prandtl numbers representing the magnetic field strength and the electromagnetic properties of the fluid. It is shown that the magnetic field does not change the flow pattern established in the case of thermocapillary convection without a magnetic field. But, the relevant parameters determining the conditions for emerging convective cells are influenced by its strength. A domain of subcritical instability is also displayed.

Thermal Marangoni Instability and Magnetic Pressure for a Thin Ferrofluid Layer

We study the linear coupling between the Marangoni and Cowley–Rosensweig instabilities for a thin layer of ferrofluid subjected to a temperature gradient and a magnetic field. Both are perpendicular to the reference horizontal boundaries, one of which is a rigid plate, while the other is a free surface remaining flat as long as the magnetic field is smaller than the critical value of the onset of the static isothermal Cowley–Rosensweig instability. Our study considers at first a ferrofluid layer resting on the rigid border. In the stationary case, when heating is directed from the rigid side, a magnetic field, smaller than the Cowley–Rosensweig critical one, can induce a new pattern: the critical Marangoni number is much lower than in the nonmagnetic undeformable case, for a dimensionless wavenumber of less than 1.992, its Newtonian classical value. When heating from the gaseous phase, an oscillatory marginal case exists theoretically, but for unphysical conditions. We consider also the case when the ferrofluid is hanging down from the rigid side. Only the wavelength critical value of the Rayleigh–Taylor instability that separates a stable region from an unstable one changes.

Weakly nonlinear Marangoni instability in the presence of a magnetic field: effect of the boundary conditions and magnetic Prandtl number

The onset of Marangoni convection in a horizontal layer of conducting liquid in the presence of a vertical magnetic field is considered. The layer is placed on a heated rigid wall and is open to an ambient gas above. The wall and gas phase are assumed to be both thermally and electrically either conducting or insulating. The influence of the boundary conditions and magnetic Prandtl number on the Marangoni instability is studied using a weakly nonlinear analysis. It is shown that the stability conditions are influenced by the electromagnetic properties of the wall when the magnetic field strength is small or moderate, while the properties of the gas are more important under strong magnetic fields. For both conducting and insulating wall, the effect of the magnetic Prandtl number is found to be strong for moderate magnetic fields when the ambient gas is nonconducting, and weak for a conducting gas.

Contribution to the dynamics of a deformable interface between two immiscible electromagnetically controllable fluids

We consider the case of a deformable material interface between two immiscible moving media, both of them being magnetizable, stressing the time dependence of the metric at the interface. This introduces a nonlinear term, proportional to the mean curvature, in the surface dynamical equations of mass momentum and angular momentum. That term intervenes also in the singular magnetic and electric fields inside the interface which lead to the influence of currents and charge densities at the interface. Also, we give the expression for the entropy production and of the different thermodynamical fluxes.

ON FARADAY INSTABILITY IN MAGNETIC LIQUIDS: INCE-ERDELYI APPROACH APPLIED TO THE HILL EQUATION DESCRIBING OSCILLATIONS OF A FERROFLUID FREE SURFACE

Abstract When an isothermal ferrofluid is submitted to an oscillating magnetic field, the initially motionless liquid free surface can start to oscillate. This physical phenomenon is similar to the Faraday instability for usual Newtonian liquids subjected to a mechanical oscillation. In the present paper, we consider the magnetic field as a sum of a constant part and a time periodic part. Two different cases for the constant part of the field, being vertical in the first one or horizontal in the second one are studied. Assuming both ferrofluid magnetization and magnetic field to be collinear, we develop the linear stability analysis of the motionless reference state taking into account the Kelvin magnetic forces. The Laplace law describing the free surface deformation reduces to Hill’s equation, which is studied using the classical method of Ince and Erdelyi. Inside this framework, we obtain the transition conditions leading to the free surface oscillations.

Evolution Equation for Nonlinear Long-Wavelength Monotonic Marangoni Instability in a Binary Liquid Layer with Nonlinear Soret Effect

Abstract The Soret effect in binary systems is called nonlinear when the thermo-diffusive flux is proportional to the temperature gradient with a coefficient being linear function of the concentration of one of the solute components. This effect is significant in highly dilute solutions. The long- wavelength Marangoni instability in a thin layer of binary liquid, in the presence of the nonlinear Soret effect, is considered. The nonlinear dynamic behaviour of the liquid system is studied in the case of monotonic instability. The solution of the dimensionless equations of mass and momentum balances, heat transfer and mass diffusion is searched near the linear instability threshold, in the form of series in a small parameter that measures the supercriticality. An equation for spatiotemporal evolution of the liquid system is derived based on the first two approximations.

Linear Marangoni and Rayleigh-Taylor instabilities of a Ferrofluid thick Layer, in a Vertical Magnetic Field

Abstract This note introduces the linear stability of a ferrofluid thick layer submitted to gravity and to a normal magnetic field. The compatibility condition is derived taking into account the Rayleigh–Taylor instability and the Marangoni one. The pure normal deformation is analytically solved and these results can be extended to explain the heat conducting case.