0.3 The kinematics of fluid motion  (Page 6/9)

the components of the vector potential $\mathbf{A}$ referred to cylindrical co-ordinate lines here being independent of the azimuthal angle $\phi$ .

The relation between and the volume flux 'between two points' may be used to obtain expressions for the velocity components referred to other orthogonal systems of co-ordinates in terms of $\psi$ . For instance, for flow with axial symmetry referred to spherical polar co-ordinates $(r,\theta ,\phi )$ , we find, by evaluating the flux between pairs of neighboring points on the r- and $\theta$ -coordinate lines and equating it to $2\pi$ times the corresponding increments in $\psi$ (allowance being made for the signs in the manner required by (2.2.12)), that

With this co-ordinate system, the vector potential for the velocity has the azimuthal component

Exercise

At time to the position of a material element of fluid has Cartesian coordinates ( $a$ , $b$ , $c$ ) and the density of the fluid is ${\rho }_o$ . At a subsequent time $t$ the position coordinates and density of the element are ( $X$ , $Y$ , $Z$ ) and $\rho$ . Show that with this Lagrangian specification of the flow field the equation of mass conservation is

Reynolds' transport theorem

An important kinematical theorem can be derived from the expression for the material derivative of the Jacobian. It is due to Reynolds and concerns the rate of change not of an infinitesimal element of volume but any volume integral. Let $\Im (\mathbf{x},t)$ be any function and $V\left(t\right)$ be a closed volume moving with the fluid, that is consisting of the same fluid particles. Then

is a function of $t$ that can be calculated. We are interested in its material derivative $DF/Dt$ . Now the integral is over the varying volume $V\left(t\right)$ so we cannot take the differentiation through the integral sign. If, however, the integration were with respect to a volume in $\xi$ -space it would be possible to interchange differentiation and integration since $D/Dt$ is differentiation with respect to $t$ keeping $\xi$ constant. The transformation $\mathbf{x}=\mathbf{x}(\xi ,t)$ , $dV=Jd{V}_o$ allows us to do just this, for $V\left(t\right)$ has been defined as a moving material volume and so come from some fixed volume $V_o$ at $t=0$ . Thus

Since $D/Dt=(\partial /\partial t)+\mathbf{v}•\nabla$ we can express this formula into a number of different forms. Substituting for the material derivative and collecting the gradient terms gives

Now applying Green's theorem to the second integral we have

where $S\left(t\right)$ is the bounding surface of $V\left(t\right)$ . This admits of an immediate physical picture for it says that the rate of change of the integral of $\Im$ within the moving volume is the integral of the rate of change of $\Im$ at a point plus the net flow of over the bounding surface. $\Im$ can be any scalar or tensor component, so that this is a kinematical result of wide application. It is going to be the basis for the conservation of mass, momentum, energy, and species. This approach to the conservation equations differs from the approach taken by Bird, Stewart, and Lightfoot. They perform a balance on a fixed volume of space and explicitly account for the convective flux across the boundaries.