We report a numerical study of the contact-line effect on the Faraday instability. A momentum balance model is implemented to simulate the hysteresis of the meniscus. By using this model, a stick-slip-motion of the contact line happens on the lateral wall when a vertical vibration is exerted to the container. The numerical result further supports that the contact-line hysteresis can increase the natural frequency by inference. Besides, it can largely delay the timing of the onset because of an extra dissipation provided by the capillary effect. The growth rate can also be affected by the meniscus but the impact of the hysteresis seems limited. The presence of the meniscus is the reason why Faraday instability occurs without any artificial disturbance. Finally, a linear relation between the contact-angle range and the contact-line position is observed from our computation.
This paper describes explicit series solutions for supersonic flat-plate boundary layer flows that are convergent in the whole spatial domain for Mach numbers of up to 50. These series solutions are achieved by means of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Unlike the analytic approximations given by perturbation methods or other approaches, our explicit series solutions are guaranteed to converge with arbitrary physical parameters because of the so-called ``convergence-control parameter' in the HAM framework. Explicit analytic expressions for the local surface skin-friction coefficient and the local heat-transfer coefficient of the supersonic boundary layer flow are also derived. <p>These analytical solutions are found to be in perfect agreement with the corresponding numerical results, allowing the effects of physical parameters on supersonic boundary layer flows to be discussed in detail. The explicit series solutions described in this paper provide a benchmark for supersonic flat-plate boundary layer flows with Mach numbers in the range $0.8\leq M_{a} \leq 50$. To the best of our knowledge, no such explicit series solutions for supersonic flat-plate boundary layer flows have previously been reported. To enable relevant applications, a corresponding Mathematica package is provided to enable convenient access to explicit series solutions for supersonic flat-plate boundary layer flows.
We report a combined method to deal with the contact angle dynamics with hysteresis. The momentum balance model is applied to obtain the transient contact angle by balancing the inertia and the capillary force where the curvatures are estimated by the height function at the contact line. This integrated approach provides a great convenience that no need for a prior knowledge of the contact angle, and possesses the good property of the sharp interface approximation. In order to facilitate the wider use of the present method, we incorporate a dynamic contact line model to estimate the contact angle when the contact line starts to move. This combination plus the hysteresis region will take the ability to solve most of problems related to wetting with more physical sense. This proposed model is finally validated by the droplet equilibrium, spreading, and sliding tests.
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