Centrifugal force: an appreciation

doi:10.4236/ns.2011.37086

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Vol.3, No.7, 633-639 (2011) Natural Science

http://dx.doi.org/10.4236/ns.2011.37086

Centrifugal force: an appreciation

Kern E. Kenyon

4632 North Lane, Del Mar, USA; *Corresponding Author: [email protected]

Received 19 April, 2011; revised 2 May, 2011; accepted 8 May, 2011.

ABSTRACT

The centrifugal force is used to increase the

physical understanding of five examples taken

from fluid dynamics, geophysics and the solar

system, as well as four hypothetical orbital

problems. Each example involves a balance of

forces between the centrifugal force and one or

two other forces, such as a pressure gradient

and a component of the force of gravity. Among

the examples chosen for examination are: the

orbital motion of fluid particles in surface grav-

ity waves, the boundary layer character of

steady flow next to a curved rigid surface, the

tornado, the rotating self-gravitating mass and

the three-body problem.

Keywords: Centrifugal Force

1. INTRODUCTION

It may be easier for a trained dynamical oceanogra-

pher (the author) or meteorologist to take the centrifugal

force seriously than for a practitioner of classical physics.

In the field of physics there has existed for a long time a

considerable amount of confusion surrounding the cen-

trifugal force: is it real or fictitious? For example, see the

descriptions of the centrifugal force in the various avail-

able dictionaries of physics, of which there are at least

three. Here are two quotes from one such dictionary [1].

“Centrifugal force is said to be ‘fictitious’ force, and the

use is best avoided.” Also, “It is sometimes said that the

centrifugal force is ‘reaction’ to the centripetal force—

this is not true.” Both quotes illustrate the present state

of confusion with respect to the centrifugal force.

Throughout the discussions that follow support can be

found for the second quote but none for the first one.

Strangely, use of the centrifugal force in physics appears

to have been restricted to purely circular motion. Why

can’t it be applied also to elliptical orbital motion? Ex-

actly when the dissatisfaction with the centrifugal force

began is not known; apparently no single event or person

started it, unless it was Newton. He, on the one hand,

pointed out that the flattening of the poles of the earth is

direct evidence for the working of the centrifugal force,

but, on the other hand, he used “centrifugal force” very

sparingly in the Principia.

However, the physical oceanographer, operating in a

relatively new field and not being hampered by all the

physics history, learns in his first year of graduate work

to balance the Coriolis force against a horizontal pres-

sure gradient in order to help him understand the dy-

namics of the major permanent ocean currents. I believe

the same thing happens when studying the large-scale

wind systems in graduate courses in meteorology. Then

it is only a small step to seek to balance the centrifugal

force against another force, such as a pressure force, in

order to increase understanding of some smaller scale

motions. But there has never been much of a motivation

to consider the centrifugal force in either ocean or atmo-

spheric science. One reason for this is that the spatial

scales of most weather features, even a hurricane, and

ocean circulations are large enough to cause the Coriolis

force to dominate over the centrifugal force.

In about 1982, a question was asked by me of myself

regarding understanding the centrifugal force in surfaced

gravity waves, since the fluid particles have an orbital

motion, as indicated by observing small neutrally buoy-

ant particles. This is discussed further in Section 2. By

attempting to answer the question a chain of similar qu-

estions was triggered resulting in several totally different

papers, all connected by a single thread (or ‘most’ com-

mon denominator): the centrifugal force. This chain of

events was not planned or organized as such and the

time sequence of the related publications did not always

follow the order that is most convenient to put down

here. But it all started with the surface gravity wave, a

special subject of interest to me, before I found out from

the dictionaries that I was not supposed to use the cen-

trifugal force.

2. SURFACE GRAVITY WAVES

Thirty years ago, I began to think about the motion of

the fluid particles in a surface gravity wave. Imagine the

fluid particles in the fixed frame (fixed to the solid bot-

K. E. Kenyon et al. / Natural Science 3 (2011) 633-639

634

tom of the body of water), which go round and round in

circles while the sinusoidal wave shape propagates past

the observer. During circular motion each fluid particle

must experience a centrifugal force acting radially out-

ward from the orbit’s center. For the organized circular

motion to be maintained and not break down into chaos

or turbulence, there has to be an equal but opposite force

to balance the centrifugal force. What is this other force?

From looking into the literature it appears that nobody

had ever asked this question before. The answer is a

pressure force related to the different elevations of the

sea surface and to the vertical fluid accelerations [2].

Immediately popping out of the force balance equation is

the usual theoretical dispersion relation of these waves,

derived on the basis of potential theory. There was a

name already waiting in the oceanographic literature for

the pressure/centrifugal force balance, the cyclostrophic

balance, but it had never been applied to this example in

the past.

At the time I asked the question of myself about how

the centrifugal force on the orbiting fluid particles is

balanced, my mind was not exactly prepared to do an-

swer it. Ever since graduate school (in the 1960s), when

I discovered a little known paper on water waves by Ein-

stein [3], I had been under the spell of his elementary

model that explained both the existence of the waves and

their propagation speed with simple physics. He used a

steady state model by assuming that the observer moves

at the speed of the wave so that the wave shape remains

steady and fluid under the wavy surfaces flows past the

observer. This point of view had a pretty firm grip on my

mind because of its usefulness for understanding some

features of the waves. However, the observed orbital

motion of the particles in the fixed frame does not come

naturally out of the Einstein model. In other words, for

all these years I had not been thinking much at all about

the orbital movements of the particles. In another sense

the question was ripening and just waiting to occur to me

(or to somebody).

An opportunity came up, as it does every four years,

to present a paper at an international meeting (IUGG),

which happened at that time to be in Vienna. The timing

was right to take into the public arena my thoughts on

the centrifugal force as it applies to the surface gravity

wave. My oral presentation (I also presented a poster)

was scheduled for the largest lecture hall I had ever been

in. Only a handful of people were there to hear the talk

including my wife and our two kids. There I was up on a

platform behind a podium far removed from my small

audience in this giant room. One of the many amusing

experiences of my academic career to think back on lat-

er.

After seeing this project through to publication, a re-

lated question came up. According to the standard irrota-

tional wave theory the particle orbits transform from cir-

cles to ellipses when the mean total depth of water be-

comes comparable to or less than a wave length and the

long axes of the ellipses are horizontal, parallel to the

mean free surface. The centers of rotation of the particles

are the centers of the ellipses. So how does the cyclos-

trophic balance work in that case, since in general the

centrifugal force cannot be anti-parallel to the central

attractive (restoring) force? The restoring force, what-

ever its true nature is, must always point to the ellipse’s

center due to the inherent symmetry whereas the cen-

trifugal force is always normal to the orbit. For guidance

I turned to an undergraduate mechanics text I had used at

MIT, which has a brief section on celestial mechanics.

Of special interest to me then were the elliptical paths of

the planets around the sun. I had gotten an “A” in this

physics course many years before, but on revisiting the

book I plunged into a really deep confusion, lasting for

about two years. Sometimes in the book it seemed as if

the author was dancing around the centrifugal force and

other times not. For example, for purely circular motion

the centrifugal force was occasionally mentioned whe-

reas for elliptical orbital paths it was never used. A reso-

lution of this confusion is described in Section 4.

In a way it was not completely logical for me look for

help in astronomy for my problem with surface gravity

waves in the ocean. Elliptical orbits are similar in both

cases but there is a fundamental difference in the con-

figurations with respect to the central attractive forces.

For the planets, the gravitational force is centered at one

focus of the ellipse whereas the restoring force center for

the fluid particles in the surface gravity wave must be at

the center of the ellipse, although the nature of the cen-

tral attractive force was not yet identified at this point.

At any rate, that is the path I chose to take, and I am now

living with some of its consequences.

3. TORNADOES

Wouldn’t a person think that among all observable na-

tural geophysical phenomena the one with the fastest

circular speed and relatively small radius of curvature-

should be a perfect subject for the application of the cen-

trifugal force concept? However, there is little evidence

of this in the literature on tornadoes (of course Newton

never knew about tornadoes). Circular winds parallel to

and near the ground have been measured to sometimes

exceed 200 mph by aiming Doppler radar beams at solid

objects caught up in the tornado. What holds these or-

biting winds together and keeps them from flying apart

due to the outward acting centrifugal force? Evidently

the core pressure must be a lot lower than the ambient

pressure outside. Then how is such low pressure gene-

K. E. Kenyon et al. / Natural Science 3 (2011) 633-639

635

rated?

Very few direct or indirect measurements have ever

been made of the characteristics of a tornado, which is

understandable in view of the high wind velocities and

the dangers of getting too close to them. A man can stand

against a 100 mph wind, at a rather large angle from the

vertical to be sure, but very few things, including mete-

orological instruments, can stand up to a 200 mph wind.

A lack of direct observations may have led to some ra-

ther mysterious ideas about tornadoes that have taken

root in the minds of scientists.

Independent of any of my particular research topics

that were occupying my time, my son took up the tor-

nado as a project in grade school. He built a model tor-

nado and used smoke to make the vortex motion visible.

He got some assistance from a colleague of mine who

knew some meteorology but he got little help from me

(who didn’t know any). Summers spent in Iowa, where

his mother was born and raised and where his grandpar-

ents still lived, may have put this notion into his head. I

could not help noticing from the sidelines, though, that

his model tornado went from the ground up! After parts

of many summers spent in Iowa, I have yet to see a tor-

nado, except in still photographs and news videos. There

is a strong suggestion in these visual aids, though, that

the tornado comes down from the clouds to the ground.

In other words, the vertical motion in the tornado is

downward. For example, the funnel of the tornado is

made visible because some cloud material is brought

down to the ground with it. If one adopts this starting

point, it makes the explanations of a few observed fea-

tures easier, such as the funnel shape, the very large cir-

cular winds at the ground and the extremely low pres-

sure inside.

Some key observations needed for a complete under-

standingof tornadoes are missing since they are difficult

to obtain. Outside the tornado the air is always reported

to be very warm and very humid, but is it cold and dry

inside? Nobody knows for sure, except possibly one

crazy storm chaser who apparently ran and dove into one

and survived, according to a TV news report in May

2008. He came out saying that it was cool inside. Actu-

ally there is another storm chaser who probably knows

the answer to some of these questions because at least

once he successfully placed in the way of a tornado an

instrument which had the capability of measuring tem-

perature and humidity as well as pressure, and the instru-

ment was not destroyed when the tornado went right

over it on a paved road. However, so far only the record

breaking low pressure has been reported to the public for

some reason or other.

Suppose cool dry air, because it is heavier, falls down

through the warm humid air, from the cloud to the

ground. Then acceleration under gravity of the down-

ward flow plus conservation of mass qualitatively ac-

counts for the shape of the tornado’s funnel, narrowing

toward the ground. Suppose also that in or above the

cloud base there has been generated somehow a hori-

zontal circular motion which is comparatively weak. As

it falls down this circulation is magnified by conserva-

tion of angular momentum as the funnel diameter be-

comes smaller. It is not difficult in this way to go from a

relatively small horizontal velocity in the clouds to a 200

mph wind at the ground. What balances the outward

centrifugal force is a pressure gradient in which there is

low pressure at the core and relatively high pressure in

the outside air. Low pressure is consistent with Ber-

noulli’s law (pressure is least where the speed is greatest

along a streamline) as the air falls (accelerates) down-

ward at the center of the tornado. This is the gist of an

elementary model [4] of a tornado in which the centri-

fugal force is strategically embedded.

4. NEWTON’S HYPOTHETICAL ORBITS

In addition to the one “real” Kepler problem, in the

Principia Newton examined mathematically the follow-

ing four hypothetical orbits of a single particle moving

about a force center in two dimensions:

1) Elliptical orbit with the force center in the middle

of the ellipse;

2) Circular orbit with the force center on the circum-

ference;

3) Circular orbit with the force center in the middle of

the circle;

4) Spiral with the force center at the pole of the spiral.

Some years ago five books were published within a

few months of each other that all attempted to explain

Newton’s Principia to the educated public. Why this

sudden and apparently widespread burst of interest in

trying to understand the Principia, and why now after

about 300 years? Earlier I had attempted on my own to

read the Principia, but I couldn’t see the forest for the

trees, being lost in a tangle of theorems, corollaries,

lemmas, etc. The reason for my searching the Principia

was to hopefully relate my derivation of Kepler’s laws to

Newton’s derivation of these laws, which is well-known

to be the first theoretical prediction of the empirical laws.

Perhaps I had found an independent derivation. But I

could not follow Newton’s geometrical and logical de-

ductions, and I still can’t. To this day I have trouble even

locating the universal law of gravitation in the Principia!

However, one of the recent books allowed me to see

better just what it was that Newton was trying to accom-

plish in one small part of the Principia. Now, although I

still do not completely understand all the details of New-

ton’s method, I saw that I can independently check his

K. E. Kenyon et al. / Natural Science 3 (2011) 633-639

636

results on the four hypothetical orbits, plus his result of

the Kepler problem, because my force balance method is

the inverse of his method. He posed his orbital problems

in the following way. Given the shape of the orbit and

the location of the force center, find the functional form

of the central attractive force that will keep a body in

this orbit. My method is [5]: given the functional form of

the central force and the force balance, find the shape of

the orbit and its relation to the force center.

The Kepler problem is: for an elliptical orbit with the

force center at one focus of the ellipse, find the form of

the central attractive force. Newton obtained the inverse

square law for the force in this case, i.e. the force varies

as the inverse square of the distance between the focus

and the body. My force balance method puts the inverse

square law in and comes out with an elliptical orbit with

the force center at one focus.

Four hypothetical orbital problems worked out in the

Principia have not led to any practical applications so far,

with one recent exception: the elliptical orbit with the

force center at the center of the ellipse. Here Newton

proved that the attractive force varies directly as the dis-

tance, like a linear spring, which is a special case of

Hooke’s Law. To the best of my knowledge Newton

never used the term “spring” in connection with this hy-

pothetical problem in the Principia, but it is known that

Hooke and Newton were not the closest of colleagues.

Recently a practical application of this orbital problem to

surface gravity waves has been found, as discussed in

Section 6.

By putting Newton’s results for the central forces, one

at a time, into my force balance equation in polar coor-

dinates, that includes the centrifugal force, I can easily

get the differential equations to be solved for the shapes

of the orbits. In all cases the left side of the equation is

the linear harmonic oscillator equation in polar coordi-

nates where the variable of oscillation is the inverse of

the radius (which may seem a bit unusual). Depending

on the form of the force, the right side of the force bal-

ance equation can be linear or more generally nonlinear.

Then if the corresponding shapes of the orbits that New-

ton began with are inserted into the differential equations,

these equations can be solved and complete consistency

is found between the two independent methods for all

four hypothetical orbital problems in the Principia as

well as for the Kepler problem. One striking offshoot is

that analytic solutions in closed form, in terms of ele-

mentary functions, exist for two different differential

equations that are both fully nonlinear, which one would

never expect to find (one simply would not bother to

search for it). Faced with such nonlinear differential equ-

ations the contemporary graduate student would automa-

tically rush to the computer to solve them by means of

numerical techniques.

This consistency between Newton’s geometric method

and my independent force balance method further

strengthens the position that the centrifugal force is a

real and useful force and not a fictitious one to be

avoided. What is really needed, though, is a comparison

between theory and measurements. That is not likely to

happen with most of the hypothetical orbital problems.

Perhaps the elliptical problem, with the force center at

the center of the ellipse, may lead to such a comparison

someday by means of a mechanical model (i.e. a mass

rotating and vibrating on a spring, see Section 6).

Could one use the same force balance method with the

centrifugal force to attack the famous unsolved 3-body

problem of astronomy? This is a far more ambitious

theoretical project than any that occurred to me before

(or any that came after either).

5. THREE-BODY PROBLEM

At the end of the 1990s, I attempted to solve the cele-

brated 3-body problem in a plane (i.e. in two dimensions)

using my force balance method, including the centrifugal

force, and I got farther than I had expected to get on the

first try. However, the equations started to look a little

messy so I stopped. But the recent progress with New-

ton’s Principia, in which all five of the orbital problems

were found to be consistent with the force balance me-

thod, gave me the extra courage to try again. On a vaca-

tion trip to New Mexico I did not have any notes with

me, but nevertheless I sat down and drew a set of train-

gles connecting the three masses. In doing so I just hap-

pened to define two angles differently than before, and

this produced enough of a simplicity in the trigonometry

that it eventually led to an initial breakthrough, which

allowed me to proceed further than I had done previ-

ously. Without my books and papers I had to rederive

most of the necessary mathematical relations; a few I re-

membered but had to check. Four pages in the notebook

were enough to contain the derivation of the kernel of

the solution of the 3- and 4-body problems, and then the

generalization to the n-body problem was just a rela-

tively simple extension of the same procedure.

As it turns out now my original solution had errors in

it, which were pointed out by one peer reviewer in due

course, but these mistakes were corrected one by one

until the paper was finally accepted after about a two-

year review period. Of course I am very grateful to this

particular reviewer, whoever he or she is. The word “so-

lution” is not quite accurate, since what I have done so

far is to set up the differential equations to be solved for

the orbits of the gravitating masses. I have not actually

solved these equations and I am not the most likely per-

son to do so either. The equations are strongly nonlinear

K. E. Kenyon et al. / Natural Science 3 (2011) 633-639

637

and in the general case almost certainly must be addre-

ssed by numerical methods, about which I am almost

totally ignorant. Only after that work is done can a com-

parison between theory and measurement be carried out,

upon which the theory will either stand or fall.

For 300 years the 3-body problem has teased resear-

chers, mathematicians, physicists, and astronomers. It

gave Newton a headache. It gave Euler a headache. Many

mathematicians and physicists devoted a considerable

amount of time to the problem. A 1967 Encyclopedia of

Science estimates that about 1000 papers had been pub-

lished on the 3-body problem. There must be consider-

ably more papers than that now, and there may even

have been more then.

On the other hand, I never declared in the beginning

that I was working on the 3-body problem, but rather

bumped into it sideways: picked it up for a day, put it

down for three years, and then picked it up again for a

few days and thought I had finished with it (until the re-

viewer got hold of it). If I have really solved this famous

problem, which at the moment I think I have, it will be

something to wonder at briefly before hopefully going

on to write other papers. Then I predict that the physi-

cists will more readily accept the centrifugal force as

being a real useful tool.

The actual working out of the setting up of the solu-

tion of the 3-body problem took a few interesting twists

and turns through the review process. One anonymous

revie- wer thought I had made a fundamental mistake in

setting up the force balance equations for the three

masses in the first place. I was so sure of myself that I

did not come to realize that he was right for a consider-

able time, but kept trying to justify my position. How-

ever, the reviewer was unyielding. Eventually, with the

help of some friends on a particular point of the problem

(and mostly by e-mail), I saw my mistake and stayed up

all night correcting it.

Then the same reviewer had another major objection

to my suggested method of solution of the corrected

force balance equations. Again I was sure of my pro-

posed method, because it was a straight forward exten-

sion, I thought, of an earlier published paper of mine on

the 2-body or Kepler problem. I was not happy with the

result either and could not think of a way around the

obstacle. None of my friends came back with any use-

ful information when I asked them about it. Finally, the

reviewer pushed me hard enough that I found a way to

solve the difficulty which was at the same time more

mathematically correct and not physically inconsistent,

but it confused me because of my published results and

made me think I had made a mistake before. There was a

tug of war going on between my feelings of happiness

for having found a way to solve the 3-body problem and

my confusion over the way I had previously solved the

2-body problem. Gradually the feeling of happiness be-

gan to win out after a few anxious days. Naturally I am

indebted to this reviewer for his patience and persistence

in sticking to his position, and the editor was unbelieva-

bly great throughout; a real test for all three of us actually.

But this was not the end of the story either, because I

had gone out on a limb in offering an opinion about a

way that the equations could be solved numerically. One

more time the reviewer was very critical of my sugges-

tion, and probably rightly so. After taking this section

out of the paper, the editor finally accepted it for publi-

cation [6]. No solutions are contained in the paper, and

no comparisons are made between theory and measure-

ment. These important pieces of work are left for the fu-

ture. About one year and a half after the review process

was finished the paper finally saw the light of day.

6. ROTATING MASS ON A SPRING

Since Newton’s day one of his hypothetical orbital

problems, an elliptical path with the force center in the

middle of the ellipse, has come to be associated with a

practical situation: particle orbits of the surface gravity

wave [8]. In this case Newton discovered that the central

force needed to keep a mass in its orbit varied directly as

the distance the body was from the force center. In other

words the force behaved like a linear spring. Thus the

present exploration of problems incorporating the centri-

fugal force has come full circle, so to speak, starting and

ending with the surface gravity wave. However, the end

of the tale has not yet been reached because two more

sections follow this one.

Another way this particular hypothetical problem

could become a practical one is by simply attaching a

mass to a spring, anchoring the other end of the spring,

giving the mass a push while restricting the motion to

the horizontal plane, for example. But there are two ide-

alizations of the theoretical spring that may take a bit of

creative engineering in order to build such a device: the

spring is supposed to be mass less and have zero equilib-

rium length. Also friction must be eliminated (or greatly

reduced).

A noteworthy property of such a theoretical device,

apparently not noticed by Newton, is that the orbital

period of the mass is independent of the size, shape and

orientation of the elliptical orbit. Such a machine would

be a very good clock in the sense that in spite of several

types of disturbances of the mass, within its plane of

motion, it will keep perfect time!

The period of the mass rotating on a spring is directly

related to the spring constant and to the mass of the ro-

tating body but totally independent of the major and mi-

nor axes of the elliptical trajectory as well as its orienta-

K. E. Kenyon et al. / Natural Science 3 (2011) 633-639

638

tion. This is in stark contrast to Kepler’s third law by

which the elliptical orbital period of a planet around the

sun does depend on the major axis of the planet. [It

seems a bit strange, when one thinks about it, that of the

two length-scales of the elliptical orbit, the period only

depends on one of them (the major axis).]

Identifying the surface gravity wave with Newton’s

hypothetical orbital problem of an elliptical orbit due to

a force emanating at the center of the orbit did not come

easily or by a straight path although it very well could

have. Circles are special cases of ellipses and the fluid

particles in propagating surface gravity waves move in

circles for infinitesimal amplitudes in deep water. Also

in the cyclostrophic balance of forces on each particle

the outward centrifugal force is equal and opposite to the

pressure force, and each of these two forces varies line-

arly with the distance of the particle from the center of

its orbit, the attractive pressure force acting like a spring.

From another point of view, the radii of the circular

particle orbits are known observationally to decrease

with increasing mean depth. Therefore, all radii from

zero to the wave amplitude exist simultaneously in the

water column. And each particle makes one complete

revolution around its orbit in the same length of time, the

wave period. In other words, the orbital period of the

particle is independent of the size of the orbit! There is

only one known orbital problem with this characteristic:

the mass rotating in a plane on a linear spring.

When the mean depth is comparable to or less than the

wavelength the particles orbit in ellipses, theoretically,

with the major axes being parallel to the flat bottom.

Both major and minor axes decrease with increasing

depth but the period of each orbit remains constant, the

wave period. This example comes closer to being an

analogue of Newton’s hypothetical orbital problem.

7. ROTATING SELF-GRAVITATING

MASS

An extension of the simple model of the mass rotating

and vibrating on a spring in two dimensions is as follows:

a rotating self-gravitating mass in three dimensions, or a

model of a planet made of either solid or gaseous mate-

rial. Here, although counter-intuitive for a solid, compre-

ssibility is the key characteristic for solving the problem.

Obviously each particle completes its orbit in the same

period of time, the rotational period of the planet, inde-

pendent of the size of the orbit. This extension might

seem a bit farfetched until it is realized what the con-

necting link is. At an interior point of a spherical mass of

uniform density the gravitational attraction acting on it is

known to be a linear function of the distance of the point

from the center of the body. In other words, the attrac-

tion inside the body acts like an ideal linear spring,

which is both mass less and has zero equilibrium length.

The linear spring behavior is the common feature, along

with the centrifugal force, between the two- and three-

dimensional problems.

One difference between the two and three dimensional

problems is that for the rotating self-gravitating body

there are three instead of two forces that need to be bal-

anced. In addition to the restoring spring force due to

gravitational attraction and the outward centrifugal force

there is also a pressure force inside the body. It needs to

be demonstrated that the pressure force also depends

linearly on the distance from the center of the mass.

Then all three forces are directly proportional to the ra-

dius, since the centrifugal and gravitational forces al-

ready are, and when they are balanced, the radius drops

out of the problem. The result is a formula for the rota-

tional frequency of the body that is independent of the

radius, and also independent of the latitude, as it turns

out [8].

If the mean density and the rotational periods of the

six planets that have nonzero values of their flattenings

are supplied from measurements, the model accounts for

35% - 75% of the observed flattenings. Increasing den-

sity with decreasing radius inside the planets accounts

qualitatively for the discrepancies. Therefore, what ap-

pears to be a rather rough kind of agreement between

data and a theory involving the centrifugal force has

been obtained. Hopefully in the future a closer corre-

spondence can be established.

The balance of forces leads both to the equilibrium

form of the rotating body as well as the stability of cer-

tain perturbations of it. For the equilibrium form the

small excess of the equatorial radius over the polar ra-

dius, due to the stretching of the compressible solid ma-

terial under the centrifugal force, trails off like the

square of the cosine of the latitude. Therefore the equi-

librium form is not strictly an ellipsoid. Superimposed

on this equilibrium form is a perturbation that is taken to

be in planes perpendicular to the rotation axis of the

body. Such a perturbation might be caused, for example,

by the gravitational attraction of a sufficiently distant

object, like a moon or a sun. Before the perturbation the

planes normal to the rotation axis are circles, but after

the perturbation they are more than likely concentric

ellipses with centers on the axis of rotation. Thus there

are two bulges on opposite sides of the mass. Conse-

quently, in one rotational period there are two high and

two low bodily tides. The model in its present form has

nothing to say about the expected amplitude or phase of

the bulges in relation to the given perturbing force. In

any case the seeds have been planted for a possible al-

ternative explanation for why two tides should occur per

K. E. Kenyon et al. / Natural Science 3 (2011) 633-639

639

day, at least in the solid earth.

8. FLOWS ADJACENT TO A CURVED

SOLID BOUNDARY

A relatively new type of boundary layer, called the

curvature boundary layer [9], that involves the centrifu-

gal force but no friction, comes last in this list of topics

that was initiated by wondering about the orbital motion

of fluid particles in a surface gravity wave. Consider

fluid flowing steadily next to a smooth flat rigid bound-

ary with one bump protruding into the flow. Observa-

tions will show that if the fluid speed is not too great and

the slopes of the bump are small, then the fluid will fol-

low the bump’s contours without separation or eddies

being shed off the back face. Also flow visualization

techniques will illustrate that the perturbed velocity

caused by the bump only penetrates a finite distance

above the bump. Within a few lengths of the bump,

marked off in the vertical direction, the curved stream-

lines have degenerated into straight lines. This is an in-

viscid boundary layer due to the working of the centri-

fugal force on the curved flow. When measurements in-

dicate that flow near a curved solid boundary is steady,

there must be an equal but opposite force to counter ba-

lance the centrifugal force wherever it exists. This force

can only be a pressure gradient under the given circum-

stances.

Bernoulli’s law states that where the speed is greatest

the pressure is least along a streamline in steady friction-

less flow. For the streamline just going over the top of

the bump the pressure is least at the top of the bump

because the speed is largest there by conservation of

mass. Through each cross-section equal amounts of mass

must flow in equal times. Above the top of the bump the

cross-section is least due to the fact that the perturbed

velocity dies off upward away from the bump. Conser-

vation of mass only works in this way if the complete

trailing off of the perturbed velocity takes place within a

finite distance of the bump. Assuming incompressibility

of the fluid is fine but not essential.

To see why it is that the disturbed pressure also decays

upward over the bump, look at the force balance in the

cross-stream direction: the pressure gradient equals the

centrifugal force. By convention the radius of curvature

of the flow over the bump is positive and the square of

the flow speed is positive so the centrifugal force is posi-

tive. That means the counterbalancing pressure gradient

must be positive. In other words the pressure, which has

a deficit at the top of the bump, increases upward and

will equal the ambient pressure some finite distance

away from the bump.

One of the consequences of the “vertical” gradients in

pressure, normal to the bump, is that there are “horizon-

tal” variations in pressure parallel to the bump. Since the

bump is assumed rigid, these pressure variations will not

distort its shape. Relatively low pressure at the bump is

the origin of the well-known lift force on the bump

As the pressure increases upward above the bump the

speed of the horizontal flow must decrease in accordance

with Bernoilli’s law. For a smaller flow speed the up-

ward centrifugal force is smaller requiring a smaller pre-

ssure gradient to balance it. So not only does the pres-

sure increase upward but the pressure gradient decreases

upward. Therefore, pressure and velocity perturbations,

caused by the presence of the bump, are confined to the

neighborhood of the bump in a boundary layer configu-

ration. For flow past a cylinder it turns out that the pres-

sure perturbation decays at a faster rate than the velocity

perturbation does over the top (and bottom) of the cyl-

inder. This appears to be a new result in theoretical fluid

dynamics [10].

REFERENCES

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Facts on File, New York.

[2] Kenyon, K.E. (1991) Cyclostrophic balance in surface

gravity waves. Journal of the Oceanographic Society of

Japan, 47, 45-48. doi:10.1007/BF02301754

[3] Einstein, A. (1916) Elementare theorie der wasserwellen

und des fluges. Naturwissenschaften, 4, 509-510.

doi:10.1007/BF01491471

[4] Kenyon, K.E. and Kenyon, D.C. (1989) An elementary

model of a tornado. Geophysical Research Letters, 16,

1281-1283. doi:10.1029/GL016i011p01281

[5] Kenyon, K.E. (1999) Newton’s hypothetical orbits inde-

pendently derived. Physics Essays, 12, 39-43.

doi:10.4006/1.3025371

[6] Kenyon, K.E. (1999) The n-body problem solved in two

dimensions. Physics Essays, 12, 197-203.

doi:10.4006/1.3025377

[7] Kenyon, K.E. (2001) Exciting a rotation mass on a spring

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Physics, 22, 471-475. doi:10.1088/0143-0807/22/5/301

[8] Kenyon, K.E. (2002) Equilibrium and perturbed forms of

a rotating self-gravitating mass. Physics Essays, 15, 290-296.

doi:10.4006/1.3025530

[9] Kenyon, K.E. (2003) Curvature boundary layer. Physics

Essays, 16, 74-85. doi:10.4006/1.3025569

[10] Kenyon, K.E. (2011) Flow past a cylinder (submitted for

publication).