Oleg Kabov

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

The Benney equation including thermocapillary effects is considered to study a liquid film flowing down a homogeneously heated inclined wall. The link between the finitetime blow-up of the Benney equation and the absence of the one-hump travelling-wave solution of the associated dynamical system is accurately demonstrated in the whole range of linearly unstable wavenumbers. Then the blow-up boundary is tracked in the whole space of parameters accounting for flow rate, surface tension, inclination and thermocapillarity. In particular, the latter two effects can strongly reduce the validity range of the Benney equation. It is also shown that the subcritical bifurcation found for falling films with the Benney equation is related to the blow-up of solutions and is unphysical in all cases, even with the thermocapillary effect though in contrast to horizontally heated films. The accuracy of bounded solutions of the Benney equation is determined by comparison with a reference weighted integral boundary layer model. A distinction is made between closed and open flow conditions, when calculating travelling-wave solutions; the former corresponds to the conservation of mass and the latter to the conservation of flow rate. The open flow condition matches experimental conditions more closely and is explored for the first time through the associated dynamical system. It yields bounded solutions for larger Reynolds numbers than the closed flow condition. Finally, solutions that are conditionally bounded are found to be unstable to disturbances of larger periodicity. In this case, coalescence is the pathway yielding finite-time blow-up.

Thermocapillary structure formation in a falling film: Experiment and calculations

The present experimental and numerical study is focused on regular structure formation in a film falling down a vertical plate with a built-in rectangular heater. The data on critical Marangoni number and on the wavelength of the most unstable spanwise mode depending on Reynolds and Weber numbers are obtained. The influence of heating intensity on structure geometry is also investigated. The critical Marangoni number is shown to increase nonlinearly with the increase of Reynolds number. The wavelength of the most unstable spanwise mode appears to depend weakly on Reynolds number and much stronger on Weber number. We obtained that the height of the critical two-dimensional bump is not invariant, but depends almost inversely on Reynolds number. The appearance of a reverse flow in the bump is not a criterium for instability onset. For the different fixed Reynolds number values, the structure width significantly grows with the increase of Marangoni number. Numerical results and experimental data are compared ...

Heat transfer and rivulet structures formation in a falling thin liquid film locally heated

An experimental investigation of the heat transfer from a local heat source to a liquid film falling down a vertical plate is performed. The thermocapillary counterflow, induced by non-uniform heating, causes a deformation of the film surface having a horizontal bump-shape. This shape becomes unstable above a critical value of the imposed heat flux and deforms into vertical downstream rivulets. This variation of patterns is expected to modify significantly the heat transfer through the film. Experiments are carried out at atmospheric pressure with three varying parameters: the streamwise heater length, the Reynolds number and the imposed heat flux density. Velocimetry, shadowgraphy and infrared thermography are used to study the behavior of the interface and the heat transfer. We put in evidence the presence of a thermocapillary counter flow producing a stagnation line at the upper edges of the horseshoe structures, beyond the instability threshold, and observe a decrease of the heat transfer with the Reynolds number.

Jet formation in gravitational flow of a heated wavy liquid film

Jet formation was studied in the region of two-dimensional and three-dimensional waves in a heated liquid film flowing down a vertical surface. Jet-to-jet spacings were measured versus the film Reynolds number and the heat flow density. Three-dimensional waves on the film surface were formed naturally or by artificial perturbations. In addition to the thermocapillary mechanism of jet formation, a thermocapillary–wavy mechanism was found to exist.

Critical heat flux in a locally heated liquid film driven by gas flow in a minichannel

The rupture of a liquid film driven by friction with a gas flow in a horizontal minichannel and the heat-exchange crisis in this film locally heated by a 1 × 1 cm source in the channel wall has been experimentally studied. A heat flux of 250 W/cm2 is achieved, which is greater by an order of magnitude than the limiting heat flux for a vertically falling liquid film with the same Reynolds number (Rel = 21). These experiments confirmed good prospects for using gas-flow-driven liquid films in cooling systems of devices with intense local heat evolution.

Deformation of the Free Surface in a Moving Locally-Heated Thin Liquid Layer

Liquid film flow on a vertical surface is studied experimentally and theoretically under the determining influence of the thermocapillary forces. In the two-dimensional steady-state case the shape of the film surface is calculated numerically within the thin layer approximation with allowance for the temperature dependence of the viscosity of the liquid and redistribution of the heat flux in the heating element. A local heat source was used in the experiments to produce temperature gradients up to 10 K/mm on the liquid surface. The film thickness was determined by means of the schlieren method with reflection. The relative thickness of the roller in the upper heater edge zone, characteristic of the formation of regular structures, is measured. The thickness is h/h0=1.32 ±0.07, which agrees satisfactorily with the results of numerical calculations.

Thermocapillary Deformation of a Locally Heated Liquid Film Moving under the Action of a Gas Flow

We propose a two-dimensional model of a steady laminar flow of a liquid film in a channel in the presence of a cocurrent gas flow. An analytical solution for the problem of temperature distribution is obtained for a linear flow velocity profile. The linearized problem of thermocapillary deformation of the film surface caused by local heating at a constant heat flux is solved. It is established that a thermocapillary bump is formed in the region where a thermal boundary layer emerges on the film surface. Additional perturbations, decaying in the upstream direction, can be present on the free surface in front of the bump. A criterion determining this effect is found.

The effect of substrate wettability on the breakdown of a locally heated fluid film

The effect of the equilibrium contact angle of wetting on the dynamics of the dry patch propagation and on the critical heat flux upon the breakdown of a water film that is heated locally from the substrate side is studied experimentally. The equilibrium contact angle is varied from 27° ± 6° to 74° ± 9° (with no changes in the thermophysical properties of the system) through the use of different types of surface grinding. The studies are performed for three flow modes: (a) a fluid film that freely flows down along a substrate with an inclination of 5° to the horizon, (b) a film that moves along a horizontal substrate under the influence of hydrostatic pressure, and (c) a static film on a horizontal substrate. It is found that the substrate wettability has a significant effect on the dry patch propagation rate and its final size in all these cases, but has almost no effect on the threshold heat flux at which the breakdown of a film occurs.

An experimental modeling of gravity effect on rupture of a locally heated liquid film

This study deals with the formation of dry patches in a subcooled liquid film flowing over a locally heated plate at small positive and negative plate inclination angles with respect to the horizon. Prior to film rupture appreciable thermocapillary deformations of the film surface appear, growing with the heat flux. Upon reaching a threshold heat flux the film rupture occurs. By means of high speed imaging it is found that the process of rupture involves two stages: 1) abrupt film thinning down to a thin residual film on the heater; 2) rupture and dryout of the residual film. As the plate inclination angle is reduced the threshold heat flux required for film rupture weakly decreases, however when the angle becomes negative the threshold heat flux begins to rise dramatically, which is associated with an increase of the stabilizing hydrostatic effect due to the growth of the film thickness. The characteristic time of rupture decreases as the threshold heat flux increases. At nucleation of the dry patch the speed of contact line can be as high as 220 mm/s. The results obtained, apart from their intrinsic importance for ground-based applications, can also be of interest for microgravity research as a film flow with different relative contribution of inertia, hydrostatic and thermocapillary forces is considered.

Study of Thermocapillary Film Rupture Using a Fiber Optical Thickness Probe

Rupture of a subcooled water film flowing down an inclined plate with a 150×150 mm heater is studied using a fiber optical thickness probe. The main governing parameters of the experiment and their respective values are: Reynolds number (3.2–30.2), plate inclination angle from the horizon (3–90 deg), heat flux (0–1.53 W/cm2). The effect of the heat flux on the film flow leads to the formation of periodically flowing rivulets and thin film between them. As the heat flux grows the film thickness between rivulets gradually decreases, but, upon reaching a certain critical thickness, the film spontaneously ruptures. The critical film thickness is practically independent on the film Reynolds number as well as on the plate inclination angle and lies in the neighborhood of 60 µm (initial film thickness varies from 93 to 368 µm). The heater surface temperature prior to rupture is also independent of Re and Θ, and is about 45°C (initial film temperature is 24°C). The process of rupture involves two stages: 1) abrupt film thinning down to a very thin residual film remaining on the heater; 2) rupture and dryout of the residual film. The threshold heat flux required for film rupture is scarcely affected by the plate inclination angle but grows with the Reynolds number.