FIGURE 6.8. The velocity of a fluid parcel viewed in the rotating frame of reference: vrot = (vq, vr) in polar coordinates (see Appendix A.2.3).
We now consider the balance of forces in the vertical and radial directions, expressed first in terms of the absolute velocity Vq and then in terms of the relative velocity vq.
Vertical force balance We suppose that hydrostatic balance pertains in the vertical, Eq. 3-3. Integrating in the vertical and noting that the pressure vanishes at the free surface (actually p = atmospheric pressure at the surface, which here can be taken as zero), and with p and g assumed constant, we find that
where H(r) is the height of the free surface (where p = 0), and we suppose that z = 0 (increasing upwards) on the base of the tank (see Fig. 6.5, left).
Radial force balance in the non-rotating frame If the pitch of the spiral traced out by fluid particles is tight (i.e., in the limit that VVq << 1, appropriate when Q is sufficiently large) then the centrifugal force directed radially outward acting on a particle of fluid is balanced by the pressure gradient force directed inward associated with the tilt of the free surface. This radial force balance can be written in the nonrotating frame thus:
Using Eq. 6-16, the radial pressure gradient can be directly related to the gradient of free surface height, enabling the force balance to be written5:
Radial force balance in the rotating frame
Using Eq. 6-15, we can express the centrifugal acceleration in Eq. 6-17 in terms of velocities in the rotating frame thus:
Equation 6.19 can be simplified by measuring the height of the free surface relative to that of a reference parabolic surface (see
Then, since dq/dr = dH/dr - Q2r/g, Eq. 6-19 can be written in terms of n thus:
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