The Equations of Fluid Motion
Upon the word, accoutered as I was, I plunged in and bade him follow. So indeed he did. Shakespeare, Julius Caesar We live in an incredibly dynamical universe. Any chance observation bears witness to the enormous diversity of motions and interactions that dominate the structure of matter in astrophysical environments. It is the aim of this book to put someof this in context. We see that gravitation, the ultimate structuring force on the scales typical in astrophysical problems, plays a key role in understanding how matter evolves. But we also must reckon with magnetic ﬁelds, turbulence, and many of the same processes that dominate laboratory studies.
1.1 Introductory Remarks
When you think of a ﬂuid, the idea of a structureless deformablecontinuous medium comes to mind. Other properties, ingrained in childhood, are also likely to occur to you; things like incompressibility, viscosity (if you were really precocious), and perhaps even the statement that a ﬂuid will not support shear and will yield freely in the presence of an applied force. So what have these to do with an astrophysical context? More than these deﬁnitions and propertiesmight lead you to suspect. Of course, you will likely think, since stars are gaseous (something we have known since the 1860s), that we should be dealing with kinetic theory and gas dynamics. The processes of rareﬁed media are likely to dominate our understanding of cosmic bodies. This is not quite true. Since a star is composed of gas which is homogeneous (in most cases) and which actscollectively to create its own gravitational ﬁeld, it mimics rather well the behavior of a ﬂuid moving (or sitting) under gravity. The collision times are so short (or, put another way, the mean free paths are so short compared with any scale lengths in the medium) in the interior of the star that any disturbances can be washed out and the structure can be described as continAstrophysical Hydrodynamics.Second Edition. Steven N. Shore Copyright c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40669-2
1 The Equations of Fluid Motion
uous. Naturally, this is, for the moment, only an assertion. We shall prove it in due time. The most crucial point is that stars and all other cosmic matter can be treated as an ensemble object or system when we have carefully chosen somescales of length and time. In a nutshell, the reason for this book is that we can very often, at some magniﬁcation of scale or some rate of clock ticking, apply a ﬂuid approximation to the problems at hand. This book is meant to provide the machinery, both computational and conceptual, with which to begin treating dynamical and static problems posed by ﬂuids in a nonterrestrial environment.
1.2Equations of Motion 1.2.1 Distribution Functions
We start with a homogeneous medium of identical particles, forgetting for the moment that this may be too restrictive an assumption. Imagine that this group of particles is characterized by a global velocity distribution. Also, assume that we can know this distribution function and that the positional information can eventually be derived for theparticles as well. Let us start with a gas that consists of a collection of myriads of particles, all of identical mass. If we assume that these particles execute collective motions, we will be able to take ensemble averages and treat them as if they were a continuous medium. This is what we mean by a ﬂuid in an astrophysical context. But before we can reach the stage of describing the matter as aclassical substance, we need to consider the microscale phenomena and how to incorporate them into a macroscopic description of the motion and thermal properties. To do this, we begin with a statistical mechanics treatment and then generalize from there. Let us say that there exists a possibly time-dependent distribution function f , which is a function of x, the particle positions, and v, their...