Stability of a fluid surface in a microgravity environment

Abstract

We introduce a stochastic model to analyze in quantitative detail the effect of the high-frequency components of the residual accelerations onboard spacecraft (often called g jitter) on the motion of a fluid surface. The resid- ual acceleration field is modeled as a narrow-band noise characterized by three independent parameters: its intensity G2, a dominant frequency Q, and a characteristic spectral width T1. The white noise limit corresponds to QT —> 0, with G2T finite, and the limit of a periodic g jitter (or deterministic limit) can be recovered for QT —> , G2 finite. Analysis of the linear response of a fluid surface subjected to a fluctuating gravitational field leads to the stochastic Mathieu equation driven by both additive and multiplicative noise. We discuss the stability of the solutions of this linear equation in the two limits of white noise and deterministic forcing, and in the general case of narrow-band noise. The results are then applied to typical microgravity conditions. STOCHASTIC model is introduced to describe the high- frequency components of the residual accelerations onboard spacecraft (often called g jitter). The model is incorporated into the equations governing fluid motion, and the stability of a surface of discontinuity between two fluids of different density is analyzed. The linear stability of this surface is governed by the stochastic Mathieu equation, driven by both multiplicative and additive noise. We study the range of parameters in which the solutions of the stochastic Mathieu equation are stable and apply the results to a water-air surface in typical microgravity conditions. We find that the component of g jitter normal to the surface at rest couples non- linearly to the surface displacement and can lead to parametric in- stability. On the other hand, the two parallel components appear additively, and we show that they do not modify the linear stability boundaries associated with the normal component. A certain amount of attention has been paid recently to model- ing g jitter as a periodic function of time,13 but little attention has been paid to the more realistic case in which the effective accelera- tion spectrum contains a band of frequency components and is ran- dom in nature. Early work in this direction by Antar 4 considered the stability of the Rayleigh-Benard configuration under a random gravitational field with a uniform frequency spectrum (white noise). Later, Fichtl and Holland5 used a stochastic description to study the distribution of impulses that would exceed a prescribed threshold. We introduce a general model for g jitter, also based on a sto- chastic description,6 with a spectrum that is similar to residual ac- celeration spectra measured in space missions. The model, which we call narrow-band noise, is characterized by three parameters: the mean intensity of the fluctuations G2; their characteristic angu- lar frequency Q; and the characteristic width of the spectrum (peaked at Q) ir1 (Ref. 7). In the limit ^T -> 0, with D = G2T finite, narrow-band noise reduces to white noise of intensity Z), whereas