0.3 The kinematics of fluid motion  (Page 5/9)

The reader may find useful the general rule for two-dimensional flow, that differentiation of $\psi$ in a certain direction gives the velocity component 90 degree in the clockwise sense from that direction.

Finally, for this case of two-dimensional flow of incompressible fluid, we should note the possibility that $\psi$ is a many-valued function of position. For suppose that across some closed inner geometrical boundary there is a net volume flux $m$ ; this flux might be due to an effective creation of fluid within the inner boundary (as when a tube discharges fluid into this region) or to change of volume of the part of the enclosed region not occupied by the fluid (as when a gaseous cavity surrounded by water expands or contracts).

If now we choose two different paths joining the two points 0 and $P$ which together make up a closed curve enclosing the inner boundary, the fluxes of volume across the two joining curves differ by an amount $m$ (or, more generally, by $pm$ , where $p$ is the number of times the combined closed curve passes round the inner boundary). The value of $\psi -{\psi }_o$ at the point $P$ thus depends on the choice of path joining it to the reference point 0, and may take any one of a number of values differing by multiples of $m$ . This kind of many-valuedness of a scalar function related to the velocity distribution in a region which is not singly-connected will be described more fully in § 2.8. It is not confined to two-dimensional flow, although that is the context in which it occurs most often.

If now the flow has symmetry about an axis, the mass-conservation equation for an incompressible fluid takes the form

in terms of cylindrical co-ordinates $(x,\sigma ,\phi )$ with corresponding velocity components $(u,v,w)$ , the axis of symmetry being the line $\sigma =0$ . This relation ensures that $\sigma u\delta \sigma -\sigma v\delta x$ is an exact differential, equal to $\delta \psi$ say. Then

and the function $\psi (x,\sigma ,t)$ is defined by

where the line integral is taken along an arbitrary curve in an axial plane joining some reference point 0 to the point $P$ with co-ordinates $(x,\sigma )$ . It will be noticed that the azimuthal component of velocity $w$ does not enter into the mass-conservation equation in a flow field with axial symmetry and cannot be obtained from $\psi$ .

Again it is possible to interpret $\psi$ both as a measure of volume flux and as one component of a vector potential. The flux of fluid volume across the surface formed by rotating an arbitrary curve joining 0 to $P$ in an axial plane, about the axis of symmetry, the flux again being reckoned as positive when it is in the anti-clockwise sense about $P$ , is $2\pi$ -times the right-hand side of (2.2.12). Lines in an axial plane on which $\psi$ is constant are everywhere parallel to the vector ( $u$ , $v$ , $o$ ), and can be described as `streamlines of the flow in an axial plane'. $\psi$ is here termed the Stokes stream function . A sketch of lines on which $\psi$ is constant, with the same increment in $\psi$ between all pairs of neighboring lines (see figure 2.5.2 for an example), does not give quite as direct an impression of the distribution of velocity magnitude here as in two-dimensional flow, owing to the occurrence of the factor $1/\sigma$ in the expressions for $u$ and $v$ in (2.2.11). The relations (2.2. 11) are readily seen to be equivalent to