George M. Minchin- Hydrostatics and Elementary Hydrokinetics

HYDROSTATICS AND

ELEMENTARY HYDROKINETICS MINCHIN

HENRY

F

ROW BE

OXFORD UNIVERSITY PRESS WAKEIIOUSE

AMEN CORNER,

ii2

E.C.

FOURTH AVENUE

HYDROSTATICS AND

ELEMENTARY HYDROIINETICS

GEORGE

M.

MINCHIN,

MjV.

HI

PROFESSOR OF APPLIED MATHEMATICS IN

THE ROYAL

INDIAH EN&INEEBING COLLEGE, COOPEBS HILL

Ojcforfc

PKINTEU AT THE

C

I.

A REN

PON PRESS

BY HORACE HART, PRINTER TO THli UNIVERSITY

PREFACE IN

thin

work no previous acquaintance

and properties of a StaticB,

suming

fluid is assumed.

As

with, the nature

in

my treatise

on

I have begun with the very elements, and, asthat the student's reading- in pure mathematics

in

advancing- simultaneously with his study of Hydrostatics,

1

have endeavoured to lead him into the advanced portions

of the subject.

It will be noticed, however, that the

in which the reader

way

introduced to the notion of a perfect fluid is very different from that which is usually adopted in -similar treatises. definition of a perfect fluid founded is

A

upon the elementary facts and principles of the theory of strain and stress is not calculated to produce the impression of simplicity, more especially when the symbols of the Dillerential Calculus are employed in the process. I maintain, however, that in such a presentation of the basis of the subject there

who

is

is

1

really nothing

which a beginner

familiar with the elements of Geometry, Algebra,

and Trigonometry cannot

readily understand.

The preva-

lent view that the fundamental notions of the Differential

Calculus are a mystery which the beginner should not dare which cannot be unveiled until great

to approach, and

experience in mathematics has been attained, has long

thorn

u ennlliet hetwcen the alijvhni and the phv4e

in

of tho Hiibjcet, or

iliJlifiill

u.

y or iniNcrinrrpfinn \\hifJi in vrv { have hern at jaiiin

likely 1o arise in the student's mind,

emphasise and enlarge on the difficult y, and to invite un attempt at explanation on the part of (he student. This am ctinvinei'il that more than one-hull" have, done because to

I

I

of the

eHiotential

line 13, omit the

words "

i.e., if

the external forces hi

"

jtfinchin's liydrustutics

body is said to be in a state of strain. For example, let A 23 (Fig. i) represent an is with the end A fixed while the end

by any force. Consider the state of affairs at any point, P, in the substance of the string. If at P we imagine a very small plane

pulled

p q (represented in the left-hand figure) perpendicular to the direction of the string, it is clear that the molecules of

having the position

the body at the under side of this elementplane experience an upward pull from the

elastic string

_

2

and Elementary

Hydrostatics

Hydrokinetics.

instance., consists of the increase of natural distances

between

molecules.

At the same time the molecules at the upper side ojp q experience a downward pull, exactly equal in magnitude to the previously named pull. The

stress

on the element-plane has, therefore, two

it is an upward force when considered with reaspects ference to the molecules on its under face, and a downward :

force in reference tp the molecules on its upper face. This double aspect is a characteristic of every stress,

and

of every force in the Universe, however exerted whether within the sxibstancc of what we call a single body or s

between two bodies influencing each other by attraction or repulsion. The double aspect is just as necessary a feature of every force as it is of every surface, which we are compelled to recognise as having two sides.

Let us now, in imagination, consider a little elementplane having the position r t, and separating molecules moment's reflection shows that the right and left. molecules at the right experience no foi'ce (or only an in-

A

finitesimal force) from those at the left side. Practically we may say that the stress on the element-plane r t is zero.

In the same way, if we consider an element-plane at P having a position m n, intermediate to p q and r t, the force exerted on the molecules at the under side by the substance on the upper side is an upward pull whose direction is oblique to the plane. Hence in this case of a stretched string

nn

tibfi filf>mfvn t;-nln:nn l

n

ft

f-.Tio

s^voca

i'a

we see that -nAvmnl f.nai/-vr

while a load

is

put on B,

it is

evident that the previous sign ; that, for ex-

state of stress is exactly reversed in

ample, the molecules on the under side of pq experience a normal pressure from those on the upper side, and that the

on in n is oblique pressure. This reversal of tension into pressure could not practically take place if the "body / J5 were a perfectly flexible stringin other words, the stress

;

body would at once collapse pressure at

if

we attempted

to produce

J3.

Again, if AB is an iron column whose base A is fixed on the ground while a great horizontal pressure is exerted

from right to left at the top J5, the column will be slightly bent and its different horizontal sections have evidently a tendency to slip on each other in other words, the molecules at the under side of an element-plane having- the position p g, experience a force from right to left in their :

own plane from the substance above The

pq.

on an element-plane at any point body such as iron may have any direction in it may be normal pressure, normal reference to the plane stress, therefore,

inside a solid

tension, or force wholly tangential to the plane, according to the manner in which the body is strained by externally

Inside a body such as a flexible string the applied force. nature of any possible stress is, as we have said, more limited,

We

inasmuch

cannot be normal pressure. a body in which the nature of still more limited namely, a body

as it

now imagine

shall

any possible which the

stress is

stress on every element-plane, however imagined at a point, can never be otherwise than normal. Such a body is a perfect fluid and then the stress is, in

in

;

all

ordinary circumstances, pressure

such in the sequel

we

4 is

Hydrostatics a lody suck

and Elementary

Hydrokinetics.

that, whatever forces aot

upon

thus pro-'

it,

clucing strain, the stress on every element-plane throughout, it is

normal,

If we take any element-plane, and take the whole amount of the stress exerted on either side of the plane, and then divide the 2.

Intensity of Stress.

m?i, at a point

stress by the area of the element-plane, we obtain the average stress on the little plane. Thus, if the area of mn is -ooi square inches, and the stress on either side is -02 pounds' weight, the rate of stress on the plane

amount of the

*OCl

is

,

or 20 pounds' weight per square inch.

on the plane

is

suming the

stress

The

stress

not iiniformly distributed but the smaller the area of the element-plane., the less the error in as;

to

be uniformly distributed over

it.

Hence, according to the usual method of the Differential Calculus, if we take an element-plane of indefinitely small area., bs and if S/'is the amount of the stress exerted on }

either side, the limiting value of the fraction

r, 56'

when

Ss (and therefore also 8/) is indefinitely diminished, is the rate, or intensity, of stress at on a plane in the n direction

P

m

.

It is obvious from what has been explained that such an in a expression as the intensity of stress at a point strained body' is indefinite, because different element-

P

'

planes at the same point l-o-r.c.^l'nc ,vP efvoc-a ^v^-rl

P may ,-,

4-\

have very ,,,-v,

W

different in-

~ ^1^11

,,,,

per square inch, or kilogrammes' weight per square or dynes per square centimetre, or generally, in units offorce per unit area, 3. Principle of Separate Equilibrium. The following' principle is very largely employed in the consideration of the equilibrium or motion of a fluid, or, indeed, of any

material system

:

We may

always consider the equilibrium or motion of any limited portion of a system, apartfrom the remainder provided

'.

,

we imagine as applied exerted on

Thus,

to it all the forces which are actually by the parts imagined to be removed. suppose Fig. 2 to it

represent a fluid, or other mass, at rest under the action of

any

forces,

and

\

/

let us trace

out in imagination

any closed ~

surface

M,

enclosing a portion, Then all the

\.

-Ff^E\~^