Fluid dynamics proceeds on the hypothesis that the length scale of the flow is always taken to be large compared with the mean free path of the constitutive particles, so that the fluid may be treated as a continuum. This model makes it possible to treat fluid properties (such as velocity, pressure, density, etc.) at a point in space, with the physical variables being continuous functions of space and time. In other words, we assume that the macroscopic behavior of our systems is the same as if their distribution of matter were perfectly continuous in structure. Accordingly, whenever we speak of the velocity of a "mass element" (or "fluid particle") we always mean the average velocity of a large number of constitutive particles contained within a small volume of finite extent, although this volume must be regarded as a point.
The mathematical description of a fluid motion from the continuum point of view allows two distinct methods of approach. The first one, called the Lagrangian description, identifies each mass element and describes what happens to it over time. Mathematically, we represent the motion by the function r = r (R, t), (2.1)
where R = (X1, X2, X3) is the original position of a fluid particle, at time t = 0 (say), and r = (x1, x2, x3) is the location of the same mass element at the subsequent instant t. The dependent vector r is thus determined as a function of the independent variables R and t. The velocity and acceleration of a fluid particle are d r d 2r v(R, t) = - and a(R, t) = —, (2.2)
where the partial derivative indicates that the differentiation must be carried out for a given mass element (i.e., holding R constant).
The second approach, called the Eulerian description, focuses attention on a particular point in space and describes the flow at that point over time. Mathematically, the state of motion is described by the velocity field v = v(r, t), (2.3)
where the independent variables are location in space, represented by the vector r = (x1, x2, x3), and time. The acceleration of a fluid particle is the material derivative of the velocity. Hence, we let
Similarly, one can define the material derivative
which measures the rate of change of the quantity Q as one follows a fluid particle along its path.
It is not my purpose to demonstrate the basic equations of fluid dynamics, since their derivation can be found in numerous textbooks. In this section I shall list these equations in an inertial frame of reference, making use of the Eulerian specification.
In the absence of sources or sinks of matter within the fluid, the condition of mass conservation is expressed by the continuity equation,
This equation states that the fractional rates of change of density and volume of a mass element are equal in magnitude and opposite in sign. Newton's second law of action can be written in the form
Dt p p where g is the acceleration due to gravity, p is the density, and p is the pressure. The vector f is the viscous force per unit volume, which can be written as the vectorial divergence of the viscous stress tensor t . For Newtonian fluids, the six components of this symmetric tensor are
where the coefficients of shear viscosity ¡x and bulk viscosity ¡x$ both depend on local thermodynamic properties only = 1ifi = j, Sjj = 0ifi = j; summation over repeated indices). Thus, we have f(v) = ¡xV2v + [¡x + 1 ¡x^j grad (div v), (2.9)
where, for the sake of simplicity, we have assumed that the viscosity coefficients remain constant over the field of motion. Equation (2.7) is often known as the Navier-Stokes equation.
If the flow is such that the pressure variations do not produce any significant density variations, the compressibility of the fluid may be neglected. (This occurs in most liquid flows; it also occurs in many of the gas flows for which the speed is everywhere much smaller than the speed of sound.) In compressible flows, however, it is always necessary to augment Eqs. (2.6) and (2.7) with an equation based on the principles of thermodynamics. To be specific, the conservation of thermal energy implies that p DUU + P divv = div(kcgrad T) + + PQ, (2.10)
where U is the thermal energy per unit mass, T is the temperature, kc is the coefficient of thermal conductivity, OV is the rate (per unit volume and unit time) at which heat is generated by viscous friction, and Q is the net heat addition per unit mass by internal heat sources. For all situations to be discussed in this book, the dissipation function OV is utterly negligible. Since the function Q is of particular relevance to stellar interiors, it will be discussed further in Section 3.2.
Now, assuming quasi-static changes at every point of the fluid, we can write
where S is the entropy per unit mass. By virtue of Eq. (2.6), a comparison between Eqs. (2.10) and (2.11) leads to the result
expressing the change of specific entropy as we follow a mass element along its motion.
To complete the system of equations, further thermodynamic relations are required. For example, for an ideal gas, one has
ff where f is the mean molecular weight. One also has R/ f = cp — cV, where cp and cV are the specific heats, at constant pressure and constant volume. Inserting these relations into Eq. (2.11), one obtains
where p0 is a constant reference pressure and y is the adiabatic exponent, is known as the potential temperature. For isentropic motions (i.e., motions for which the right-hand side of Eq. [2.12] identically vanishes) the potential temperature of each fluid particle remains a constant along its path.
In order to solve the partial differential equations that govern the fluid motion, it is necessary to prescribe initial conditions specified over all space and boundary conditions specified over all time. Whereas initial conditions are always peculiar to the problem at hand, the appropriate boundary conditions are of a rather general nature.
On a fixed solid boundary, there can be no fluid motion across the boundary. This condition implies that n ■ v = 0, (2.17)
where n is the outer normal to the surface. A second condition is provided by the no-slip condition that there should be no relative tangential velocity between a rigid wall and the viscous fluid next to it. Hence, we must also prescribe that n x v = 0, (2.18)
on a fixed solid wall.
At an interfacial boundary (such as the top of an ocean or the outer surface of a star), one must prescribe that no mass element cross this boundary so that fluid particles on the boundary will remain on the boundary. Thus, if % (r, t) defines the surface elevation above an equilibrium level, this kinematic boundary condition on the velocity is
Dt v y at the material boundary. Ifthis boundary is fixed (i.e., % = 0), condition (2.19) reduces to n ■ v = 0, (2.20)
expressing that matter is always flowing along the prescribed material boundary.
Beyond this kinematic boundary condition, it is also clear that we must ensure the balance of forces at any nonsolid boundary. For example, the gravitational attraction must be continuous across the free surface of a star. Similarly, the components of the stress vector acting on a nonsolid boundary must be continuous across that boundary (see Eq. [2.8]). Thus, we let
where brackets designate the jump that the quantity experiences on a nonsolid boundary (i = 1, 2, 3). In particular, at the free surface of a stellar model embedded into a vacuum, these three components must identically vanish.
In some applications, it is convenient to describe the motions as they appear to an observer at rest in a frame rotating with the constant angular velocity O. We can write v(r, t) = u(r, t) + O x r, (2.22)
where the velocity u refers to the moving axes. Similarly, the material acceleration (2.4) has the form
is the acceleration relative to the rotating frame. The quantities 2O x u and O x (O x r) represent, respectively, the Coriolis acceleration and the centrifugal acceleration. Since the tensor (2.8) is invariant with respect to a uniform rotation, Eq. (2.7) then becomes
Du + 2O x u = g - O x (O x r) - - grad p + - f(u). (2.25)
It is a simple matter to show that
Because the vector g is derivable also from a scalar potential, V (say), we can thus rewrite the momentum equation (2.25) in the form
is the effective gravity. Comparing Eq. (2.27) with Eq. (2.7), one readily sees that the Coriolis acceleration is the only structural change of Newton's second law for motion relative to a rotating frame of reference. *
* As far back as 1835, the French engineer Gaspard Coriolis (1792-1843) made a detailed mathematical study of the absolute acceleration of moving solids in a rotating frame of reference. His work had little
For steady flows, the relative importance of the acceleration measured in the rotating frame and the Coriolis acceleration can be estimated as
\n x u| QU QL' v ' y where U and L are the characteristic velocity and length of the flow. This ratio is a nondimensional number called the Rossby number, and it is designated by
Similarly, by making use of Eq. (2.9), one can easily estimate the ratio of the viscous force to the Coriolis force. One obtains
\n X u\ QU QL2 v y where v = ¡x/p is the coefficient of kinematic viscosity. The nondimensional number
is known as the Ekman number.
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