The stability of uniformly accelerated flows with application to convection driven by surface tension

Abstract

The method of energy is used to study the stability of uniformly accelerated flows, i.e. those flows characterized by an impulsive change in boundary temperature or velocity. Two stability criteria are considered: strong stability, in which there is exponential decay of the disturbance energy, and marginal stability, in which the disturbance energy is less than or equal to its initial value. For the important case in which the critical stability parameter (measured by the Marangoni, Rayleigh or Reynolds number) decreases with time, it is proved that an onset time exists. Furthermore, it is shown that the experimental onset time is bounded below by the marginal stability limit, which in turn is bounded below by the strong stability limit.The method is then applied to the problem of an impulsively cooled liquid layer susceptible to instabilities driven by interfacial-tension gradients. The strong stability and marginal stability boundaries are calculated and bounds on the onset time are given. These results represent the first rigorous bounds for convective instability problems of this class. Comparison with the limited available experimental data shows the calculated results to be lower bounds on the experimental onset times, and hence the theory is in agreement with available experimental results.

Onset of marangoni convection in an initially quiescent spherical droplet subjected to the transient heat conduction

“…This means that the amplitude function of temperature disturbances follows the behavior of Θ 0 for small τ. Furthermore, the relation of Ma * =constant is shown even in theoretical results from the frozen-time model [9] and the energy method [12].…”

“…Even though their analyses are indeed appropriate for describing convection with time-independent temperature profiles, its applicability to nonlinear temperature systems of rapid cooling is clouded due to the inherent complexity of time-varying temperature profiles. For the thin liquid layer, the related instability analysis has been conducted by using the quasi-static approximation [9], propagation theory [10,11], energy method [12] and maximum-transientMarangoni-number criterion [13,14]. And, for the spherical droplet, the quasi-static approximation [15] and energy method [16,17] have been employed.…”

The Onset of Tayler-Görtler Vortices in Impulsively Decelerating Circular Flow

“…As discussed by Gummerman and Homsy [12] and Neitzel [4] the growth rate σ(τ) cannot be obtained explicitly and therefore, the calculation of τ m suffers from serious computational burden. Owing to this kind of difficulties Gummerman and Homsy [12] and Neitzel [4] obtained the stability limit for the limited domain.…”

“…Owing to this kind of difficulties Gummerman and Homsy [12] and Neitzel [4] obtained the stability limit for the limited domain. Here, we relax the above stability limits by introducing the relative stability concept: the temporal growth rate of the kinetic energy of the disturbance velocity should exceed that of the base velocity at the onset condition of secondary motion.…”

Görtler vortices in the Rayleigh layer on an impulsively started cylinder

“…12 It is one of the aims of this paper to show that these conflicting conditions can lead to results which have a rigorous mathematical justification. When quasi-steady theory is inapplicable, energy-stability theory provides an alternative approach and this technique has been employed in a range of other papers [13][14][15] concerned with both centrifugal and thermal instabilities.…”