Minimal formulation of the linear spatial analysis of capillary jets: Validity of the two-mode approach

A rigorous and complete formulation of the linear evolution of harmonically stimulated capillary jets should include infinitely many spatial modes to account for arbitrary exit conditions [J. Guerrero et al., J. Fluid Mech. 702, 354 (2012)]. However, it is not rare to find works in which only the downstream capillary dominant mode, the sole unstable one, is retained, with amplitude determined by the jet deformation at the exit. This procedure constitutes an oversimplification, unable to handle a flow rate perturbation without jet deformation at the exit (the most usual conditions). In spite of its decaying behavior, the other capillary mode (subdominant) must be included in what can be called a “minimal linear formulation.” Deformation and mean axial velocity amplitudes at the jet exit are the two relevant parameters to simultaneously find the amplitudes of both capillary modes. Only once these amplitudes are found, the calculation of the breakup length may be eventually simplified by disregarding the subdominant mode. Simple recipes are provided for predicting the breakup length, which are checked against our own numerical simulations. The agreement is better than in previous attempts in the literature. Besides, the limits of validity of the linear formulation are explored in terms of the exit velocity amplitude, the wave number, the Weber number, and the Ohnesorge number. Including the subdominant mode extends the range of amplitudes for which the linear model gives accurate predictions, the criterion for keeping this mode being that the breakup time must be shorter than a given formula. It has been generally assumed that the shortest intact length happens for the stimulation frequency with the highest growth rate. However, we show that this correlation is not strict because the amplitude of the dominant mode has a role in the breakup process and it depends on the stimulation frequency.Ministerio de Economía, Industria y Competitividad, Spain, under Contract No. FIS2014-25161Junta de Andalucía under Contract No. P11-FQM-791

Spatial modes in one-dimensional models for capillary jets

One-dimensional (1D) models are widely employed to simplify the analysis of axisymmetric capillary jets. These models postulate that, for slender deformations of the free surface, the radial profile of the axial velocity can be approximated as uniform (viscous slice, averaged, and Cosserat models) or parabolic (parabolic model). In classical works on spatial stability analysis with 1D models, considerable misinterpretation was generated about the modes yielded by each model. The already existing physical analysis of three-dimensional (3D) axisymmetric spatial modes enables us to relate these 1D spatial modes to the exact 3D counterparts. To do so, we address the surface stimulation problem, which can be treated as linear, by considering the effect of normal and tangential stresses to perturb the jet. A Green’s function for a spatially local stimulation having a harmonic time dependence provides the general formalism to describe any time-periodic stimulation. The Green’s function of this signaling problem is known to be a superposition of the spatial modes, but in fact these modes are of fundamental nature, i.e., not restricted to the surface stimulation problem. The smallness of the wave number associated with each mode is the criterion to validate or invalidate the 1D approaches. The proposed axial-velocity profiles (planar or parabolic) also have a remarkable influence on the outcomes of each 1D model.We also compare with the classical 3D results for (i) conditions for absolute instability, and (ii) the amplitude of the unstable mode resulting from both normal and tangential surface stress stimulation. Incidentally, as a previous task, we need to re-deduce 1D models in order to include eventual stresses of various possible origins (electrohydrodynamic, thermocapillary, etc.) applied on the free surface, which were not considered in the previous general formulations.Gobierno de España FIS2011-25161Junta de Andalucía P11-FQM-791

Whipping of electrified liquid jets

Weapply an electric field to amoderately conducting liquid surrounded by another coflowing liquid, all inside a glass-based microfluidic device, to study nonaxisymmetric instabilities.Wefind that the bending of the electrified jet results in a steady-state, helicoidal structure with a constant opening angle. Remarkably, the characteristic phase speed of the helicoidalwave only depends on the charge carried by the jet in the helicoidal region and its stability critically depends on the properties of the coflowing liquid. In fact, the steady-state helical structure becomes chaotic when the longest characteristic time is that of the inner liquid rather than that of the outer coflowing liquid. We also perform a numerical analysis to show that the natural preference of the jet is to adopt the conical helix structure observed experimentally

Aplicación de Campos Eléctricos a Chorros Capilares

Los resultados de este trabajo no se restringen a la mera caracterización de la estimulación EHD, sino que por un lado el formalismo es útil para la descripción de cualquier tipo de estimulación realizada mediante esfuerzos en la superficie del chorro, y por o tro, se han obtenido resultados relativos a la dinámica del chorro que tienen interés por sí mismos. Nos referimos a la descripción de los modos espaciales. Por ello, la organización más natural de la tesis nos parece aquella en la que primero se plantee el problema fluidomecánico de un chorro sometido a esfuerzos superficiales de cualquier origen, y posteriormente se aborde el problema eléctrico propio de la estimulación EHD. Para ello, en el capítulo dos se presentarán las ecuaciones y condiciones de contorno que describirán la dinámica del problema fluido sometido a cualquier tipo de estimulación superficial. Se resolverá el problema fluido sometido a una estimulación puntual con una dependencia temporal armónica (problema de la señal), mediante el formalismo de la función de Green. En esta solución aparecerá una combinación de los modos del sistema. Se mostrará un estudio detallado del análisis espacial del chorro capilar y se comprobará que la función de Green es una herramienta imprescindible para la comprensión de los modos espaciales. Una vez realizado el análisis modal, el objetivo del capítulo tres es realizar un estudio análogo mediante los modelos unidimensionales, lo cual suministra una herramienta de análisis más simple y, como se verá, suficientemente precisa. Se estudiarán los modos que aparecen en cada uno de los modelos y se comprobará su bondad comparando los resultados obtenidos para la función de Green con los del caso 3D. La caracterización de la estimulación EHD se realizará en el capítulo cuatro. Aunque en general están acoplados el problema fluidomecánico y eléctrico, en la práctica estos problemas pueden considerarse desacoplados. Se resolverá el problema eléctrico mediante el método de los elementos finitos usando el software comercial COMSOL Multiphysics y a partir de sus resultados se obtendrán los esfuerzos eléctricos que actúan sobre el chorro. Para comprobar y entender estos resultados numéricos, se propondrá un modelo circuital del problema. A partir del estudio numérico se diseñará un estimulador que actúe de forma eficiente. Por último, para dicho diseño, se estudiará la respuesta del chorro a la estimulación. Para ello se calculará la convolución de la función de Green obtenida en capítulos anteriores con los esfuerzos que se han obtenidos de forma numérica. De estos resultados se podrá obtener mucha información, como por ejemplo, cómo afecta la estimulación a cada uno de los modos espaciales, o a qué tipo de condición de contorno corresponde la deformación y velocidad obtenidas a la salida de la estimulación. También se verá cuál de las dos componentes de la estimulación, normal o tangencial, es más eficiente