3 results
Start-up flow in shallow deformable microchannels
- Alejandro Martínez-Calvo, Alejandro Sevilla, Gunnar G. Peng, Howard A. Stone
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- Journal:
- Journal of Fluid Mechanics / Volume 885 / 25 February 2020
- Published online by Cambridge University Press:
- 27 December 2019, A25
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Microfluidic systems are usually fabricated with soft materials that deform due to the fluid stresses. Recent experimental and theoretical studies on the steady flow in shallow deformable microchannels have shown that the flow rate is a nonlinear function of the pressure drop due to the deformation of the upper soft wall. Here, we extend the steady theory of Christov et al. (J. Fluid Mech., vol. 841, 2018, pp. 267–286) by considering the start-up flow from rest, both in pressure-controlled and in flow-rate-controlled configurations. The characteristic scales and relevant parameters governing the transient flow are first identified, followed by the development of an unsteady lubrication theory assuming that the inertia of the fluid is negligible, and that the upper wall can be modelled as an elastic plate under pure bending satisfying the Kirchhoff–Love equation. The model is governed by two non-geometrical dimensionless numbers: a compliance parameter $\unicode[STIX]{x1D6FD}$, which compares the characteristic displacement of the upper wall with the undeformed channel height, and a parameter $\unicode[STIX]{x1D6FE}$ that compares the inertia of the solid with its flexural rigidity. In the limit of negligible solid inertia, $\unicode[STIX]{x1D6FE}\rightarrow 0$, a quasi-steady model is developed, whereby the fluid pressure satisfies a nonlinear diffusion equation, with $\unicode[STIX]{x1D6FD}$ as the only parameter, which admits a self-similar solution under pressure-controlled conditions. This simplified lubrication description is validated with coupled three-dimensional numerical simulations of the Navier equations for the elastic solid and the Navier–Stokes equations for the fluid. The agreement is very good when the hypotheses behind the model are satisfied. Unexpectedly, we find fair agreement even in cases where the solid and liquid inertia cannot be neglected.
The structure of the absolutely unstable regions in the near field of low-density jets
- Wilfried Coenen, Alejandro Sevilla
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- Journal:
- Journal of Fluid Mechanics / Volume 713 / 25 December 2012
- Published online by Cambridge University Press:
- 17 October 2012, pp. 123-149
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The viscous spatiotemporal stability properties of low-density laminar round jets emerging from circular nozzles or tubes are investigated numerically providing, for the first time, a separate treatment of the two particular cases typically studied in experiments: a hot gas jet discharging into a quiescent cold ambient of the same species, and an isothermal jet consisting of a mixture of two gases with different molecular weight, discharging into a stagnant ambient of the heavier species. To that end, use is made of a realistic representation for the base velocity and density profiles based on boundary-layer theory, with account taken of the effect of variable transport properties. Our results show significant quantitative differences with respect to previous parametric studies, and reveal that hot jets are generically more unstable than light jets, in the sense that they have larger associated critical density ratios for values of the Reynolds number and momentum thickness typically used in experiments. In addition, for several values of the jet-to-ambient density ratio, $S$, the downstream evolution of the local stability properties of the jet is computed as a function of the two main control parameters governing the jet, namely the Reynolds number, $\mathit{Re}$, and the momentum thickness of the initial velocity profile, ${\theta }_{0} / D$. It is shown that, for a given value of $S$, the $(\mathit{Re}, {\theta }_{0} / D)$ parameter plane can be divided in three regions. In the first region, defined by low values of $\mathit{Re}$ or very thick shear layers, the flow is locally convectively unstable everywhere. In the second region, with moderately large values of $\mathit{Re}$ and thin shear layers, the jet exhibits a localized pocket of absolute instability, away from boundaries. Finally, in the third region, that prevails in most of the $(\mathit{Re}, {\theta }_{0} / D)$ parameter plane, the absolutely unstable domain is bounded by the jet outlet. All the experiments available in the literature are shown to lie in the latter region, and the global transition observed in experiments is demonstrated to take place when the absolutely unstable domain becomes sufficiently large. The marginal frequency of the resulting global self-excited oscillations is shown to be fairly well described by the absolute frequency evaluated at the jet outlet, in agreement with the numerical results obtained by Lesshafft et al. (J. Fluid Mech., vol. 554, 2006, pp. 393–409) for synthetic jets.
The effect of viscous relaxation on the spatiotemporal stability of capillary jets
- Alejandro Sevilla
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- Journal:
- Journal of Fluid Mechanics / Volume 684 / 10 October 2011
- Published online by Cambridge University Press:
- 02 September 2011, pp. 204-226
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The linear spatiotemporal stability properties of axisymmetric laminar capillary jets with fully developed initial velocity profiles are studied for large values of both the Reynolds number, , and the Froude number, , where is the injector radius, the volume flow rate, the kinematic viscosity and the gravitational acceleration. The downstream development of the basic flow and its stability are addressed with an approximate formulation that takes advantage of the jet slenderness. The base flow is seen to depend on two parameters, namely a Stokes number, , and a Weber number, , where is the surface tension coefficient, while its linear stability depends also on the Reynolds number. When non-parallel terms are retained in the local stability problem, the analysis predicts a critical value of the Weber number, , below which a pocket of local absolute instability exists within the near field of the jet. The function is computed for the buoyancy-free jet, showing marked differences with the results previously obtained with uniform velocity profiles. It is seen that, in accounting for gravity effects, it is more convenient to express the parametric dependence of the critical Weber number with use made of the Morton and Bond numbers, and , as replacements for and . This alternative formulation is advantageous to describe jets of a given liquid for a known value of , in that the resulting Morton number becomes constant, thereby leaving as the only relevant parameter. The computed function for a water jet under Earth gravity is shown to be consistent with the experimental results of Clanet and Lasheras for the transition from jetting to dripping of water jets discharging into air from long injection needles, which cannot be properly described with a uniform velocity profile assumed at the jet exit.