U2 f

The term containing the vorticity w has disappeared because of the identity u (u A w) 0 (the cross product of u with anything is perpendicular to u and hence the dot product of this vector with u is necessarily zero). Therefore for a steady barotropic flow, the quantity is constant along streamlines, and is referred to as Bernoulli's constant. For a physical interpretation of the constancy of H along streamlines we first consider the p 0 case. Then the constancy of H just implies that the...

Sound waves in a uniform medium

We start with the continuity and momentum equations in Eulerian form In the absence of external forces, the unperturbed state of the fluid, which is in equilibrium (i.e. 0), is one of uniform density p0, pressure p0, and zero velocity (u 0). We then consider perturbations to this equilibrium, i.e. Before substituting these perturbed expressions into the fluid equations, we need to be careful because these perturbations are Lagrangian (i.e. they apply to individual fluid elements) whereas it is...

Exercises

These exercises are a mix of short questions, designed to reinforce concepts developed in the text, and longer ones, some from Cambridge examination papers, which test the reader's knowledge of the subject. The order in which they appear follows the development in the text, and not the degree of difficulty of the questions. 1 Determine the equation of a general streamline of the flow m a, mr b, 0 in cylindrical polar coordinates, and sketch the flow. Repeat for the flow m aR2, mr bR2, 0. If the...

The fluid equations

The equations which describe the motion of fluid elements are based on concepts which are familiar from Newtonian mechanics, namely the conservation of mass, momentum and energy. So we demand that for any region the rate of change of its mass is the net flow of mass into it, that the rate of change of momentum is balanced by momentum flow and net force, and the rate of change of total energy is determined by energy gains minus losses from outside. In this chapter we use these principles to...

Hydrostatic equilibrium

We are now ready to solve the fluid equations and will start with the simplest case, that of hydrostatic equilibrium. Hydrostatic equilibrium implies that u 0 everywhere ('static') and that 3 di 0 ('equilibrium'). The continuity equation is trivially satisfied and in this chapter we will consider barotropic equations of state (see Section 4.2) so that we can dispense with the energy equation. The only equation to be solved is therefore the momentum equation in which the only non-zero terms are...

Viscous flows

The equation of continuity is valid for any fluid, since it expresses the conservation of mass. However, in a viscous fluid the old momentum equation has to be modified to take account of the transfer of momentum between fluid cells due to viscous processes. In Chapter 2 we derived the fluid equations, and expressed them in Cartesian form as where g, dtW in Cartesian coordinates. This equation was based on the premise that in the frame of a moving fluid element, its momentum does not change as...

Gravitation

The gravitational force is of course of central importance in astrophysics. Therefore whereas the force of gravity plays only a minor role in most textbooks on terrestrial fluid mechanics, with such fluids being only subject to the uniform gravitational acceleration of the Earth, the astronomical situation is quite different. We will need routinely to be able to solve for the gravitational field produced by the fluid itself before we can calculate the gravitational term in the momentum equation...

Introduction to concepts

Stated most simply, fluids are 'things that flow'. This definition distinguishes between liquids and gases (both fluids) and solids, where the atoms are held more or less rigidly in some form of lattice. Of course, it is always possible to think of substances whose status is ambiguous in this regard, such as those, normally regarded as solids, which exhibit 'creep' over sufficiently long timescales (glass would fall into this category). Such borderline cases do not undermine the fact that the...

Shocks

Disturbances always propagate at a speed cs relative to the fluid. Consider an observer situated at the source of a spherical disturbance watching the fluid flow past it at a speed v. The velocity of the disturbance relative to the observer is just the vector sum of v and the velocity vectors (of magnitude Cs) of the disturbance relative to the fluid. As may be seen from Figure 7.1, for subsonic flow the resultant vector always sweeps out 4 steradians as seen from the point of view of the...

Equations in curvilinear coordinates

When it comes to applying the fluid equations we generally wish to use a coordinate system in which we minimize the number of equations by using symmetries inherent in the problem to set some derivatives in suitable coordinates to zero. The obvious such coordinate systems are spherical polars, which will apply in the simplest stellar situations for example, and cylindrical polars in those cases where there is some axial symmetry, usually for rotating systems. Here we provide expressions for the...

Structure of the blast wave

What we have done so far is to have developed the description of a blast wave using some approximations, which we have justified with varying levels of conviction, and then written down a self-similar prescription and shown we obtained the same time dependence for the blast wave radius. Now we take the self-similar nature of the blast wave, and investigate the nature of this self-similar solution more rigorously. Among other things, this will show us how good are the approximations we made when...

Fluid instabilities

Consider a fluid in a steady state, i.e. one which satisfies the hydro-dynamic equations with d dt 0 everywhere. If we find that small perturbations to this configuration grow with time, then our chances of finding the initial configuration in nature are very small, and the configuration is said to be unstable with respect to those perturbations. A stable configuration is one where either the perturbations diminish, or there is the possibility of oscillations or waves about the equilibrium...

Solutions for the Lane Emden equation

The Lane-Emden equation for n 0 is d 3 where C is a constant. Therefore This solution has a singularity at the origin, and 0 C as 0. However, the boundary conditions at 0 are 0 1 and j 0, and the only way to satisfy these is for C 0 and D 1. So the solution for n 0 is The surface of the polytrope is where the function 0 is zero, i.e. at V6. (The subscript 0 is just to highlight that this is the solution only for n 0.) This is more tractable if we set 0 - and solve for -. Then the Lane-Emden...

Scaling relations

There are various conditions under which stars behave like poly-tropes amongst which is the case of fully convective stars where the pressure-density relation is very close to the adiabatic one. (The reason for this is explained in Chapter 10 where we examine convection in more detail.) For monatomic gases, y 5 3 (see Section 4.1) and thus p Kp 3, implying n 3 2. For such cases we have to solve the equations numerically, but we can make some progress with the general properties of such stars...

Isothermal shocks

In general, Q will not be zero so eventually the shocked gas will cool, in some cases back to close to its original temperature. So a temperature profile through a shock could look like Provided the flow downstream is in a steady state the pm and p + pm2 are constant in the shocked flow. Therefore, the isothermal portion of the flow also obeys the first two Rankine-Hugoniot relations (7.3) and (7.6), despite the fact that there is an adiabatic portion sandwiched between it and the shock...

Astrophysical Fluid Dynamics

Principles of Astrophysical Fluid Dynamics Fluid dynamical forces drive most of the fundamental processes in the Universe and so play a crucial role in our understanding of astrophysics. This comprehensive textbook introduces the fluid dynamics necessary to understand a wide range of astronomical phenomena, from stellar structures to supernovae blast waves, to accretion discs. The authors' approach is to introduce and derive the fundamental equations, supplemented by text that conveys a more...

The energy equation

So far, we have derived two Eulerian equations which describe the motion of a fluid, To close these we need relations which give us and p in terms of the other variables p and u. We know in principle how to get , since we have V2 4 Gp see Chapter 3 , but we still have to find some relationship which allows us to determine p. The relationship between p and other thermodynamic properties of the system is called the equation of state. As emphasised in Chapter 1, one can only talk about an equation...