9.21. General Remarks. Hoodoos have been described in Sec. 1.92 and it has been shown that their most characteristic feature is the overhanging hat. They occur in badland areas together with pyramidal structures (mesas and buttes) which are quite naturally explained as erosional features due to the action of rain.
If one tries to assume an origin of the hoodoos which would be analogous to that of the pyramidal structures referred to above, one is at once faced with the problem as to how the water gets around the "rim" of the "hat" of the hoodoos so as to wash out their "neck". One might think that the causes for the water turning the corner are surface forces. However, it is the writer's contention that the phenomenon is analogous to that encountered when tea being poured from a teapot runs down the underside of the spout rather than straight on into the cup. This phenomenon has been called teapot effect; it is not due to surface forces, interfacial tensions or such like, but is a consequence of the prevailing flow potentials.
The writer has advanced the above theory in an earlier publication1; we shall reproduce the argument here. In order to do this, we shall first give a description of the teapot effect and then analyse its bearing upon the genesis of hoodoos.
9.22. The Teapot Effect. When tea is being poured from a teapot, it often runs down the underside of the spout rather than straight on into the cup. This has been called the "teapot effect". It is not due to surface forces, but is a consequence of the prevailing flow potentials. It has been closely studied by REINER2 and by KELLER3.
If we neglect gravity forces for one moment, then it can be shown that there are various possible flows when a jet of fluid leaves a nozzle with parallel walls. KELLER3 made a study of this and came up with a variety of flows. He calculated the flow potentials for the planar case where a jet is confined between two parallel plates. The plates end, say, at x = 0 and the jet moves on. There are four possibilities. One is that the jet moves straight on, another that it turns around the upper as well as around the lower plate, filling the whole space. The remaining two possibilities are where the jet turns either around the upper or around the lower plate. This is almost the teapot effect. In the course of his investigations, KELLER found an additional flow which has a direct bearing upon the problem of the hoodoos. We shall discuss it below.
Assume that there is a plate extending along the x-axis (in cross section) towards minus infinity ending at x = 0. Then it can be shown that a free surface flow is possible whose surface has (at minus infinity) the distance ±h from the plate. The geometry of this flow is then as shown in Fig. 203.
The complex potential for this flow is calculated by making a series of conformal mappings until the boundaries are of such a form that the potential can be written down easily. In order to do this, one must assume that the stream function (the imaginary part of the complex potential) is zero on both sides of the plate (since this represents one streamline) and that on the free surface, it is equal to a constant Q representing the total flux. The equation between the complex potential w and the complex variable z turns out to be
It has thus been demonstrated that, provided gravity is neglected, there exists a possible solution to the flow equations where the flow turns a corner (9.22-1). The above solution is valid only for a thin plate.
One still has to investigate the effect of gravity. This effect is presumably small near the corner since the hydrodynamic pressure variations are small there. However, far from the edge, the flow will be parallel to the plate on the underside. An exact solution for such a flow is:
u = constant|
v = 0
h = constant
where u is the horizontal velocity, v the vertical velocity, po the atmospheric pressure, "rho" the density of the fluid and y the distance above the plate (so that the free surface is at y = -h). It is immediately obvious that Eq.(9.22-2) represents an exact solution of the flow equations, which is possible if
It turns out, thus, that the atmospheric pressure can indeed support flow on the underside of a plate.
Eq. (9.22-2) is not sufficient to "explain" hoodoos, nor the teapot effect. For, although a horizontal flow on the underside of a plate can indeed exist, such a flow is obviously an unstable flow: eventually, it will detach itself and drop off downward. One must therefore investigate how long the flow can follow the plate before the always-present disturbances grow sufficiently to make it detach itself. The procedure for doing this is a standard one for investigating hydrodynamic instability: small perturbations are introduced into the flow equations and their growth is analyzed. The above considerations have been applied to precisely our problem by KELLER.
Introducing a perturbation at the end of the plate, one can calculate the distance L at which it will have gown by the factor e. This distance depends on the interfacial tension T between air and water. Furthermore, it depends on the form of the original perturbation. Of interest is that distance L which is the smallest in all the modes of instability that can occur. The expression for this minimum distance cannot be written down in closed form, but in two limit cases, this is possible. KELLER found:
In hydrodynamic stability calculations it is, then, usually assumed that the flow will actually become unstable (i.e. detach itself) after it has travelled a distance of 10 L.
9.23. Bearing of Teapot Effect on Hoodoos. Let us now investigate the significance that the above-mentioned discussion might have with regard to the formation of hoodoos.
In the case of hoodoos, the eroding agent is water. In the case of water, one has T = 80 dynes/cm, p"rho" = 1 g/cm3, g = 980 cm/sec2; thus
where all units are in the c.g.s.-system.
The distance L, as has been explained above, is that distance in which the most significant disturbance grows by the factor e as stated above. In hydrodynamic stability theory, it is usually assumed that the instability will become predominant (i. e. the flow will detached itself) in a distance equal to ten times L. It turns out that the case (b) applies if h is greater than about 1/3 cm. Then
irrespective of the thickness h of the flow. It is difficult to estimate the velocity u in the flow. In a good cloudburst it will probably reach about 1 - 2 m/sec at the edge of the overhang. This means that the flow can continue on the underside for about 28 - 56 cm before detaching itself. According to earlier remarks about the mechanism of erosion, this distance of 28 - 56 cm is the distance by which the "hat" of the hoodoos can overhang, for, in order to erode the soft material below, the water must obviously first reach it.
It thus appears that the values postulated above from a discussion of the teapot effect are in good agreement with those actually found in the hoodoos measured. This would serve to substantiate the theory proposed here.
1. SCHEIDEGGER, A.E.: Geofis. Pura e Appl. 41, 101 (1958).
2. REINER, M.: Physics Today, No. 9, 16 (1956).
3. KELLER, J.B.: J. Appl. Phys. 28, 859 (1957).