Linear stability of a thin non-isothermal droplet spreading on a rotating disk

Previous works show that the linear stability of the contact line of an isothermal spreading droplet depends on the base state curvature of the free interface at the position of the unperturbed contact line [1]. Here the linear stability of a thin liquid droplet on a rotating disk is investigated, where the disk has a certain radial temperature profile and the ambient passive gas a constant temperature. It is shown that two different Marangoni effects influence the spreading beside centrifugal, gravitational, and capillary effects. The stability analysis prevails, that temperature gradients in the disk amplify some and damp other modes, while cooling the entire disk (or heating the gas) damps the growth rate of all modes and therefore seems to be an appropriate way to suppress the instability. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radial spreading and stability of a thin rotating liquid droplet

Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and homogeneous fluid layers. In the present investigations, we treat droplets, spreading on rotating solid substrates. Micro-scale effects appear, firstly, at the wetting front, where the film height tends to zero. Secondly, micro-scale effects may appear at other locations, where the free liquid/gas-interface approaches the solid substrate, as e.g. at film rupture. For such situations, molecular effects need to be considered, e.g. in form of the disjoining pressure (DJP), to get physically-correct solutions. Otherwise, the spreading can be modeled within the frame of continuum mechanics, augmented by the (empirical) law of Tanner to capture the contact-line dynamics. We present, on the one hand, an overview of several interesting issues, as (i) spreading with and without considering the DJP, (ii) spreading after central rupture, including hysteresis effects, and (iii) non-isothermal spreading, including temperature-dependent surface tension (Marangoni effect) and temperature-dependent density (Rayleigh-Bénard effect). On the other hand, we present results for the instability of the contact line, for which the contact line gets corrugated (under isothermal conditions). This instability goes along with a transition from (rotationally-symmetric) two-dimensional to three-dimensional behavior. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thin film dynamics on a vertically rotating disk partially immersed in a liquid bath

The axisymmetric flow of a thin liquid film is considered for the problem of a vertically rotating disk that is partially immersed in a liquid bath. A model for the fully three-dimensional free boundary problem of the rotating disk, that drags a thin film out of the bath is set up. From this, a dimension-reduced extended lubrication approximation that includes the meniscus region is derived. This problem constitutes a generalization of the classic drag-out and drag-in problem to the case of axisymmetric flow. The resulting nonlinear fourth-order partial differential equation for the film profile is solved numerically using a finite element scheme. For a range of parameters steady states are found and compared to asymptotic solutions. Patterns of the film profile, as a function of immersion depth and angular velocity are discussed.     

The boundary layer on a rotating ellipse

Summary The boundary layer at the surface of an elliptic cylinder is considered, when the cylinder is set into uniform rotation about an axis through its centre, in a fluid that is otherwise at rest. It is shown, by numerical solution of the governing equations that, except for the limiting case of a circular cylinder, the solution develops a singularity at a finite time, which is interpreted as the onset of flow separation.

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Zusammenfassung Wir betrachten die Grenzschicht an der Oberfläche eines elliptischen Zylinders welcher in gleichmäige Drehung um sein Zentrum in einer sonst ruhenden Flüssigkeit versetzt wird. Es wirel durch rechnerische Lösung der bestimmenen Gleichunglen gezeigt, daß mit Ausnahme des Grenzfalles eines kreisförmigen Zylinders die Lösung eine Singularität nach endlicher Zeit annimrut welche als Begin der Flussseparation interpreties wired.

     

The boundary layer on a rotating sphere

Zusammenfassung Zur Berechnung der laminaren Grenzschicht auf einer rotierenden Kugel ist ein Differenzenverfahren entwickelt worden. Man kann sich dieses Verfahrens bedienen, um die Lösung schrittweise vom Pol zum Äquator zu erhalten. Die hier durchgeführten Berechnungen werden mit anderen Resultaten verglichen, welche man durch Reihenentwicklungen erhält.     

Note on the boundary layer on a rotating

Ohne Zusammenfassung

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Miscellaneous

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Divers

     

Two-dimensional liquid column and liquid droplet impact on a solid wedge

The hydrodynamic impact due to a two-dimensional (2D) liquidcolumn or a 2D liquid droplet hitting on a solid wedge is analysed.The problem is solved using the complex velocity potential togetherwith the boundary element method. A stretched coordinate systemis used, which is defined through the ratio of the normal Cartesiancoordinate system to an appropriately chosen time-varying lengthscale. Numerical simulations are first made for impact by aliquid wedge. The results from the time-domain method are foundto be in a good agreement with the similarity solution. Simulationsare also made for impact by an elliptic droplet. A conditionfor bisection of the droplet is introduced, which is found toprovide stable and converged results.     

Note on the boundary layer on a rotating sphere

Zusammenfassung Die gleichmässige Strömung in der Grenzschicht bei einer gleichmässig rotierenden Kugel wird behandelt. Die Grenzschicht entsteht an den Polen und entwickelt sich in beiden Hemisphären nach dem Äquator zu. An den Polen verhält sich die Kugel wie eine rotierende Scheibe, und Flüssigkeit strömt hier ein. Am Äquator verhält sich die Kugel wie ein rotierender Zylinder, und die Flüssigkeit strömt in dessen Bereich aus.     

Boundary layer on rotating spheroids

Zusammenfassung Die Grenzschichtgleichungen für dreidimensionale Strömung an der Aussenseite eines rotierenden Sphäroids werden in konfokalen Koordinaten abgeleitet und numerisch mittels der Methode vonPohlhausen für verlängerte und verkürzte Sphäroide gelöst. Die Lösungen zeigen einen Zustrom an den Polen und einen Abfluss an den Äquator. Der Ausfluss hängt vom Verhältnisr/h ab, wobeir der Radius des Äquators undh die Höhe des Körpers in der Rotationsachse ist.     

The thermal laminar boundary layer on a rotating sphere

Résumé Dans cet article, on étudie par un procédé numérique la couche limite laminaire thermique sur une sphère en rotation. En supposant que la sphère est maintenue à une température constante, on obtient la distribution des températures, ainsi que le transport de chaleur. On y discute aussi le problème de la sphère en rotation dans un cas plus général où la répartition de la température sur la surface de la sphère n'est plus uniforme.     

Thermal convection in a rotating porous layer

Summary It is found in this note that basic results concerning thermal convection in a rotating porous layer may be obtained from known analysis of thermal convection in an (non-rotating) anisotropic porous medium.

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Résumé On peut obtenir des résultats de base sur la convection thermique dans une couche poreuse en rotation à partir de l'analyse connue de la convection thermique dans une couche poreuse anisotrope pas en rotation.

     

The stability of the boundary layer on a compliant rotating disc

The boundary layer over a infinite rotating disc is 3D and offinite depth. The breakdown and eventual transition of flowover the surface is preceded by the emergence of crossflow vorticesthat are stationary with respect to the disc. These result froman inviscid instability mechanism associated with an inflexionpoint within the boundary layer's velocity profile or a mechanisminduced by the balance between viscous and Coriolis forces.It has been seen in past studies that compliance can substantiallypostpone the onset of transition, therefore the aim of thisresearch is to investigate whether compliance can be used asa useful tool to do so here. We use numerical and asymptoticmethods to predict possible behaviour by calculating growthrates and producing neutral solutions for the wave number andorientation of both inviscid and viscous modes. The resultsobtained suggest that the inviscid mode of instability willbe stabilized by compliance but the viscous mode will be greatlydestabilized.     

Convective instability of a rotating layer of ferromagnetic fluid

The instability of a hot horizontal layer of ferromagnetic fluid rotating about a vertical axis has been investigated when the Prandtl numberP < 1. Earlier it was shown that forP > 1 the overstability cannot occur. In this paper the convective and overstable marginal states have been investigated separately forP < 1 and it is found that though convective marginal state is possible for alla, the non-dimensional wave number, and N the Taylor number, the overstability is possible only ifN > (1 +P)π ⁴/(1 −P) and in case the condition is satisfied, overstability is possible for all those values ofa which satisfya ² < [N(1 −P)π ²/(1 +P)] ^1/3 − π². IfR _c ^(con) andR _c ^(o.s) are the critical values of the convective and the overstable marginal states respectively, then it is also found thatR _c ^(con) <R _c ^(o.s) providedN is not sufficiently large.     

Using resonances to control chaotic mixing within a translating and rotating droplet

Enhancing and controlling chaotic advection or chaotic mixing within liquid droplets is crucial for a variety of applications including digital microfluidic devices which use microscopic “discrete” fluid volumes (droplets) as microreactors. In this work, we consider the Stokes flow of a translating spherical liquid droplet which we perturb by imposing a time-periodic rigid-body rotation. Using the tools of dynamical systems, we have shown in previous work that the rotation not only leads to one or more three-dimensional chaotic mixing regions, in which mixing occurs through the stretching and folding of material lines, but also offers the possibility of controlling both the size and the location of chaotic mixing within the drop. Such a control was achieved through appropriate tuning of the amplitude and frequency of the rotation in order to use resonances between the natural frequencies of the system and those of the external forcing. In this paper, we study the influence of the orientation of the rotation axis on the chaotic mixing zones as a third parameter, as well as propose an experimental set up to implement the techniques discussed.     

The calculation of the thermal laminar boundary layer on a rotating sphere

Sommaire La communication expose une méthode permettant de déterminer la couche limite thermique d'une sphère en rotation dans un fluide incompressible à écoulement uniforme. La méthode se base sur l'idée deFrössling, selon laquelle la répartition des températures s'exprime en tant que développement en série de puissances, et utilise les résultats déjà obtenus parHoskin pour la répartition des vitesses dans la couche limite d'une sphère en rotation. L'auteur donne des résultats numériques obtenus sur une machine à calculer IBM 704.

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Nomenclature x distance along surface of body in a meridian plane from forward stagnation point - y distance measured normal to surface - U velocity of main flow - U velocity of undisturbed stream - u component of velocity inx-direction - v component of velocity iny-direction - w transverse component of velocity due to spin - p pressure - density - T temperature - R distance of surface from axis of symmetry measured normal to axis of symmetry - coefficient of viscosity - kinematic viscosity - a radius of sphere - k thermal conductivity - angular velocity about axis of symmetry - Re Reynolds number - Pr Prandtl number - Nu Nusselt number - a constant - S area - Q quantity of heat transferred in unit time across areaS - d a characteristic length     

Convection in a rotating layer with nearly insulating boundaries

Summary Finite amplitude steady convection in a horizontal layer of fluid heated from below with nearly insulating boundaries and rotating about a vertical axis is investigated. The main result is that the only steady stable convective motion in the form of square pattern becomes unstable once the rotation parameter exceeds the value 7.45. The disturbances which have the highest growth rates are in the form of rolls inclined at an angle of 45° to the basic wave vectors of the steady motion.

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Zusammenfassung Das Problem der stationären Konvektion endlicher Amplitude wird untersucht für eine horizontale von unten erhitzten Flüssigkeitsschicht, die um eine vertikale Achse rotiert. Das Hauptresultat ist, daß die einzige stabile stationäre Konvektionsströmung in der Form von Quadratzellen instabil wird, wenn der Rotationsparameter den kritischen Wert 7.45 übersteigt. Die Strömungen höchster Anwachsrate haben die Form von Rollen, die einen Winkel von 45° mit den Wellenzahlvektoren der stationären Lösung einschließen.

     

Spreading of thin liquid droplets on rotating disks

In many industrial processes solids are coated to obtain specific surface properties, as e.g. corrosion resistance, mechanical, optical, or electrical properties. Even today many of such coating processes are not fully understood and the choice of parameters is mainly based on experience. Hence, a prediction of the complete hydrodynamic process in its dependency on the parameters appears highly desirable. This would e.g. allow for a precise prediction of the (liquid) layer thickness and shape and help to optimize the quality of the coating. A common coating technique is the so–called spin coating. The coating agent is dissolved or suspended in a liquid, brought onto the solid, spread by rotation, and the carrier liquid is finally removed by evaporation or by chemical reactions. In this article an evolution equation is derived from lubrication theory, valid for thin liquid layers. The model involves a dynamic contact angle, centrifugal, capillary, gravitational, and molecular (London–van–der–Waals) forces. The evolution equation without molecular forces can even be solved analytically, provided the capillary number is small. Otherwise a numerical integration of the governing equations is engaged. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)