Osborne Reynolds.

Papers on mechanical and physical subjects (Volume 1) online

. (page 34 of 40)
Online LibraryOsborne ReynoldsPapers on mechanical and physical subjects (Volume 1) → online text (page 34 of 40)
Font size
QR-code for this ebook



f ir dz "


(Mu} = ^ 4- - dpaU dpa 2

4 VTT \/TT d# 4 d#

Ty\Mu) =

_s_ d/j?7 s dpa 2

4 VTT VTT dx 4> dx '
paU _s c?/3a?7 s dpa? s dpaV
2V-7T 2V7T dy 2?r da? 2V^ dx

_ paU s dpa. U s


f ir 2\/7r dy ZTT dx

dx '

/TIT \ P a2 2s dpaf7
.(Mu) = t - j^- -,

2 VTT aa;

r y (Mu) = -^=


VTT da; V?

,,, , s dpaW s

<r z (Mu}= - j^~ 7= -


with corresponding equation for Q = Mv, Q = Mw,

and similar equations.





The values of U, V, W.

Hitherto U, V, W have been treated merely as functions of x, y, z. They
are, however, completely expressed by equation (43).

For remembering that <r x (M), a- y (M), a z (M) are respectively equivalent
to pu, pv, pw, we have

_ s dpa \

s dpa

r -t-

\7r dy


W= w-\




O. R.



The neighbourhood of a solid surface.
83. In this case we have by the corollary to theorem (II.)

Ba (Q) = Ba' (Q) + s {1 -f(lmn)e-7} l + m + n
or as in equation (39)


a- a (Q) = A + s f f 2 {1 -f(lmn) e~^} ( I %- + m ^~ + n 8A sin
J o J o \ ax ay


In this case it is clear that the coefficients which correspond to LI, L 2 , &c.,
Art. 82, will be functions of the position of the point with respect to the
solid surface, and will depend on the value of f(lmn). f(lmn) will obviously
depend to some extent on the action between the molecules and the solid
surface. It appears, however, that when the solid surface extends in all
directions in its own plane to distances which are great as compared with s,
and the variation Q is perpendicular to this plane, the result of the integration
of equation (50) is the same as that of equation (39). For taking the solid
surface parallel to xy

and by symmetry, since Q varies only in the direction z, for two opposite
groups such as a, 6

= A+B (51).

Therefore the integral of the last term of equation (50) for A will have
the same value but the opposite sign as for B. Hence since the solid surface
can only be on one side of the element, say the side ACGE, fig. 10, Art. 66,
and r/s will be infinite for the group B, or for this group equation 50 is
identical with equation 39, therefore for either of the opposite groups the
results of the integration of (50) are the same as of (39).

Near the edge of a solid surface, or when Q varies in some direction
parallel to the surface, equation (51) no longer holds good, and then the
coefficients corresponding to L 1} L 2 , &c., will depend on the position of the
element with respect to the solid surface and on the action between the
molecules and the solid surface.

In dealing with such cases two courses were open the one was to try and
find some form for f(lmn} which would satisfy the equations, the other course,


and that which is here adopted, is to introduce arbitrary functions s lt s 2> in
place of s, and subsequently to determine the form of s ly s 2 , so as to satisfy
the experimental results.

84. The only case of importance in this investigation is that in which
the temperature varies along a solid surface and is constant at right angles.

Taking z = c as the equation to the solid surface, and supposing the
gas uniform in the direction y, and that -^- = -J- = 0, then if x, y, z are the


coordinates of a point P and z is greater than c the effect of the solid
surface will be to alter the values of s in the terms involving the differentials
of p and a. Using a suffix to indicate that the values of s for such terms is
arbitrary, we may proceed to determine the values of a- (Q), as in Art. 82.
The important cases are Q = Mu and Q = M.

Remembering that s l refers to such terms in A, C, E, G, as involve -^- or ~ ,
and that W = 0, we have by the method of Art. 82

,,, . s Sidpa? s dU
^(Jft,). !JE - - /-/>- ..................... (52)

ZTT dx VTT dz

dx 2 dz

, \ .................. (53).

dpa so dU
J - tL



Further, to adapt these equations to the form required, put Wj and u 2 for
the mean velocity of the opposite groups w + and w , so that

Then since u may be taken as constant in the direction of cc, we have by
corollary to theorem (II.) and equation (51)

Subtracting equations (53)

du s j dpa d U

sp = j= -* sp ,
dz 2 VTT dx dz

and substituting for - in equation (52)

.,_ , s Si da s du /?A\

o- z (Mu) = pa - - -= pa .................. (54).

2?r dx VTT dz



If the point P lies between two surfaces, then putting s z as an arbitrary
function we have

,, r . So Si da. s du /et \

<7 Z (Mu) = -=- - pa = pa (54 a).

2?r dx I^/TT dz

For further consideration of s 2 ^ see Art. 96.


85. If Q is a quantity of such a nature that 2(Q) cannot change on account
of any mutual action between the molecules within a unit of volume ; and
further, if we assume that the molecules within a unit of volume at any
instant are not subject to any influence other than those which they exert on
one another, then whatever change may take place in an elementary volume
must be on account of the excess of Q carried into the unit of volume over
and above that which is carried out ; and we have

= d<r x (Q) d<r y (Q) d<r z (Q)

dt dx dy dz

j- is the rate at which 2 (Q) is increasing at a point fixed in space.
Hence if the condition of the gas is steady

#2, (V) A /K>\

~w (56) -

Therefore if the condition of the gas is steady, we have

_^i5B + __w + w) = ^


86. If, therefore, we put Q = M, equation (57) gives us the condition of
steady density.

Whereas if we put successively Q = Mu, Q = Mv, Q = Mw, we have from
equation (57) the conditions of steady momentum in the directions of the

And if we put Q = M(v? + v 2 + w*) we have the condition of steady

The condition that the gas may be subject to no distortion or shear stress.

87. In order that <r x (Mv), <r x (Mw), <r y (Mw), <r y (Mu), <r z (Mu), and
<r z (Mv) may respectively be zero for all positions of the axes, we must have

<r x (Mu) = o- y (Mv)=<r z (Mw) (58).


Therefore from the first of equations (46) and like equations

s dp*U =s dpV = s dj,W
ax ay dz

These are the conditions that there shall be no tangential stress within
the gas at a distance from a solid.

Coupled with the conditions for steady density, steady momentum, and
steady pressure, these equations are, within the limits of our approximation,
equivalent to

da? df dz*

pa* _
2 =

where p the pressure is constant throughout the gas.

2 =P (61)

88. The important condition in this investigation is that the tangential
force on a solid surface shall be zero.

This condition can only be obtained by the aid of some assumption as to
the action between the molecules and the surface. An extremely obvious
assumption will suffice, viz. : that the tangential force on the surface has the
same direction as the momentum, parallel to the surface, of all the molecules
which reach the surface in a unit of time.

The condition that there shall be no force on the surface is, then, that the
momentum parallel to the surface which is carried up to the surface shall be

Thus, if the axial planes be solid surfaces, we have from the values of

w- w-

<r z (Mu), cr z (Mv), &c., equations (45), that

U=V=W=Q (62)

at the surface.

If, further, there is no tangential stress within the gas, it appears from
equations (59), (60), and (61), that equation (62) must hold throughout the

The condition that there shall be no tangential stress on a particular solid
surface, say, the plane of xy, is satisfied if at that surface pa? is constant and

U=0, F=0 (63)


dU = dV = dW = dW =() (64)

dz dz dx dy


w- v>-

This appears at once from the values of a- z (Mu), <r z (Mv) obtained as
equations (45).


89. It will be sufficient to consider the simplest cases ; hence it is sup-
posed that the gas is transpiring through a tube of uniform section, and
further that the tube is of unlimited breadth, the surfaces being planes
parallel to the plane xy ; the axis of x is taken for the axis of the tube, and
it is assumed that all perpendicular sections of the tube are surfaces of equal
pressure and temperature, the variation of temperature and pressure being in
the direction x.

The equations to the surfaces of the tube are taken

z=c (65).

90. From equation (57) we have for steady momentum

for steady density

j j

= (67),

dee 1

and for steady pressure

j J

= (68).

Steady pressure not important.

91. In a tube, since heat may be communicated from the surface to the
gas, the temperature may be maintained constant ; and if the density be
steady the pressure will also be steady, hence the condition of steady pressure
ceases to be important. The law of variation of temperature is determined
by the sides of the tube.

Transpiration when s is small as compared with c.

92. If s is so small that it is unnecessary to consider the layer of gas
throughout which the direct influence arising from the discontinuity at the
surface extends, substituting in equation (66) from equations (46), and


putting j- = -j- (cr x (Mu)), which we may do within the limits of our
approximation, we have for steady momentum, since W=0 in the tube,

dp = ld L ( s dp*U\

dx VTT dz\ dz /

And from equations (43) and (67), since p and a do not vary across the tube,
we have for steady density


Since pu = pU - - we have from equation (70)
VTT dx



And since the action of the tube is symmetrical about the plane xy, we have
at this plane

Therefore, integrating between the limits z and 0, we have from equation (69)

dp s dU

= ~~


Also, since s is constant across the tube, except within the layer over
which the influence of the surface of the tube extends, and which is not
taken into account, we have, integrating from z to c, and putting U c for U at
the surface,

From equation (43) we have, since s -~ does not vary with z,

p(u-u c } = P (U-U c ) ........................ (75).

Therefore, from equation (74),

= _^I (c2 _, 2) ^ + Sc ..................... (76),

2s pa.^ ' dx

or putting

re '





so that fl is the mean velocity of the gas along the tube, we have, integrating

pa 2
(76) and putting p = ~ ,

= _
a 6 s p dx "

The relation between s and p.

93. The only respect in which equation (78) differs from the usual
equation between the motion of gas and the variation of pressure in a tube
is that instead of //, we have

2 p


For, putting

dp _ d f du\
dx dz\ dz)

we have for the usual equation

and comparing (78) and (79)

u - -

U c ^r- -y-

3i dx

2 p

= 77- s

The difference between equations (78) and (79) is, however, very important.
For whereas p is usually supposed to be constant, i.e., independent of the
diameter of the tube, it appears from (78) that such can only be the case so
long as c is large as compared with s : s being a distance measured across the
tube which by no variation in the condition of the gas can be made larger
than the mean diameter of the tube.

This fact that s cannot increase beyond the diameter of the tube at once
explains the anomalies (as they appeared to Graham) between the times of
transpiration for fine and coarse plugs.

The mean diameter of the interstices of Graham's coarse plugs was so
large, that with gas in the condition in which he used it, s was less than this
diameter, and not being limited to the diameter of the tube was different for
different gases and for different conditions of the same gas ; whereas with the
fine plugs, s being limited to the diameter of the tube, could no longer vary
with the nature of the gas.

The limit to the value of s also indicates, what has been verified by
the experiments described in Part I. of this paper, that the results which


Graham obtained with fine plates only, are to be obtained with coarse plates
when the condition of the gas is such that s is limited by the diameter of
the interstices.

The relation between s and the other properties of the gas.

94. The experiments made by Graham and by Maxwell, in which the
distances between the surfaces were such that there was no chance of s being
limited by this distance, give consistent results, from which it has been found

Hence taking fi = I - and substituting in equation (80) we have

From which it appears that in the same gas

soc - (82)


when not limited by the solid objects.

The general case of transpiration.

95. The equation (78) is obtained on the assumption that s is so small
compared with the diameter of the tube, that the layer of gas through which
the influence of the surface of the tube extends may be neglected, and hence
this equation cannot be taken as the law of transpiration when s comes to be
limited by the diameter of the tube. And besides this, it is necessary to
consider the value of w c , which cannot be done without considering the layer
of gas throughout which the effect of discontinuity at the surface extends.

In order to take the discontinuity at the surfaces z = c into account,
the values of <r x (M) and <r z (Mu) must be taken from equations (53) and
(54 a). These values substituted in equations (66) and (67) give equations
which correspond to equations (69) and (70), but which involve the quantity
Si s 2 , which quantity it will be well to examine before proceeding to the

* Added Dec. 1879. Subsequent observers have found that /* oc f - J so that Maxwell's

conclusions are not borne out. See Phil. Trans., Part i., 1879, p. 240. This makes no
difference to the subsequent part of this investigation, as no further use is made of equation


The value of Si s 2 .

96. Remembering that s x and s 2 are taken respectively to represent the
mean range of the quantity Q for the groups of molecules which have w
respectively positive and negative, and taking s/, s 2 to represent the values of
Si, s 2 at the surface z = c, we may express s l s 2 as a function of s, c, and z.

The fact that s 1 s 2 = s when the point considered is without the range
of the influence of the surface, shows that whatever may be the value of
Si s 2 , s 1 s 2 gradually diminishes as the point considered recedes from the
solid surface, until at some distance depending on s at which the mean range
is unaffected by the surface s 1 -s 2 = 0. It also appears from the fact of the
gas being symmetrical about the axis of the tube that s l s 2 is zero at the
axis, so that even if the value of Sj is limited by the surface, s 2 approximates
to Sj as the point considered approaches the axis of the tube.

The definite manner in which s x s 2 varies across the tube could only be
deduced by taking into account the distribution of velocities amongst the
molecules; but as s 1 s 2 must change after a continuous manner from one
surface to another, we may take for an illustration, or even for an approxi-
mation, any law of variation which fits the extremes.

Such a law is given by

c-z c+z

e a,s e i
,-*, = (/-,') - -5 ..................... (83)

1 e i*
in which a^ is a numerical factor depending only on the nature of the gas.

For the sake of distinctness it will for the present be assumed that Sj - s 2
has the values given by equation (83).

The velocity of the gas at the solid surface.

97. Putting q for the tangential force on the solid surfaces z = c, we


q = <r z (Mu) ................................. (84),

and by equations (53 a) and (54 a)

a r -TV *i' *' da

Also since -j- is constant over the section, we have for the equilibrium of

the fluid between two perpendicular sections of the tube at distance dx

................................ ( 86 >.


where me is the hydraulic mean depth of the tube (in the case of a flat tube
m= 1); therefore

pa. . _ , - /x Si s 2 ' da dp

(Ui w 2 ) = = pa -j me -f- (87)

2 VTT ^ <7r dx dx

Then if u c is the velocity along the solid surface, we have

V'J/ ftl I nl /CQ\

^ "c * i ** a ' ^oo^.

And since w/ is the mean velocity after encounter at the surface of the
tube, we may put

<=/< (89).

where f is a factor depending on the nature of the impact at the surface.

or putting

^ = \^f (90 a)

2u e = \(ui w 2 ') (90 6).

And from equation (87)

pa _ jV-*a' da dp] , ,

^w c = X-^ - - pa-T- -mc-f}- ... (91).

VTT I 2?r a aicj

TAe coefficient of friction at the solid surface.

98. Since f, or \, is important as regards that which is to follow, it is
necessary to determine, as far as possible, on what these factors depend. I
am not aware that any very definite idea has hitherto been arrived at as to
the action between the molecules of a gas and a solid surface over which the
gas may be in motion. It appears to have been thought sufficient in most
cases to assume that the gas in immediate contact with the surface is at
rest, which supposition is equivalent to neglecting any small motion there
may be.

We see at once that the gas at the surface must have a velocity when the
gas further away is in motion. For by our fundamental assumption the
molecules which approach the surface will partake of the motion further
away ; so that, even supposing the surface to be perfectly rough, the entire
group, consisting of the approaching and receding molecules, would have a
velocity equal to half that of the approaching molecules.

If the surface be less than perfectly rough, we have, as in equation (89),

u' z =fu\
where f~ l may be considered to be the coefficient of roughness.


Since we have nothing in nature analogous to perfect roughness, we may
assume that / is not zero, and the question arises whether / may not largely
depend on the angle at which the molecules approach the plane.

Even if the solid surface were a perfectly even plane, -^ would not be the

simple coefficient of friction, but must also be a function of the force with which
the molecules strike the surface, and the more nearly perpendicular to the
surface was the direction of approach the smaller would be the value off.

Whereas if, as seems highly probable, the action between the molecules
and the surface is closely analogous to that between a ball and an uneven but
perfectly smooth elastic surface, then for molecules approaching the surface
at very small angles f would be unity, while for those approaching in a
manner nearly perpendicular f would be zero, or nearly so.

The variation of / with the angle of approach can be of no particular
moment so long as there is a sufficient thickness of gas between the surface
considered, and any surface which may be opposite, for in that case the mean
angle of approach must be the same, whatever may be the condition of the
gas. But when the gas is between two surfaces, as in a tube, and these
surfaces are so near that the molecules range across the interval, then the
fact, that if small, the angle of reflection (measured from the normal) will
always be less than the angle of incidence, must cause the molecules to
assume directions more and more nearly perpendicular to the surface as the
tube becomes narrower, until some limit is reached.

The case of a billiard ball started obliquely along the table will serve to
illustrate this. Each time the ball leaves the side cushions its path will be
more nearly perpendicular, and if it could maintain its velocity, and the table
was sufficiently long, it would eventually be moving directly across the table.
This, however, would not be the final condition if the cushions were zigzag,
for then a number of balls, in whatever direction they might be started, would
finally attain a certain mean obliquity, depending on the unevenness of the
cushions. And it would seem probable that the latter case must be that of
the molecules in a tube so narrow that they can range across.

The ability of the molecules to range across the tube wilFdepend on the


value of - ; hence it would appear that the most probable assumption with

regard to the nature of X is that

c /c\

where i f- and / a (-) are functions of some such form as e


having respectively the values unity and zero when - = 0, and zero and


unity when - = oo ; and X x is a coefficient independent of the nature of the


gas on which X 2 may depend.

That there is good reason for making this assumption appears from the
comparison of the results for hydrogen and air (see result VIII., Art. 106).

The equations of motion as affected by discontinuity.
99. Substituting in equation (66) from equation (54 a), and putting -^-
for j- (<r x (Mu)} as in Art. 92, we have for steady momentum along the tube

) (92).

dx dz { VTT dz 2ir dx
Whence integrating between the limits and z

_ = |

dx VTT dz ZTT dx
And substituting for 5j s 2 from equation (83)

c-g e+z

spa du dp e~ i* e~, , , , pa da

-! = z (Si S 2 ) r~ (94).

Vwd* dx i-_V5 2-n-dx

Integrating equation (94) between the limits c and z we have

{J5. c-g c+z\

l + e~w e'^s+e~^s\ .. pa da

:, 5 > (s/ s 2 ) ~ j-
_ -~ .. _ . 2?r ax

1 - e a lS 1 e a,* J

(94 a).


j w< ^

Integrating again between the limits and c and putting H = , we


have, substituting for w c from equation (91),



, f q ^ T" P a '* **! " o> I/,' Q /\A' Jll * M ' /QKX

+ 1 a l*~ ~2? 6/V J^ 1 ^/ 51 V^/-

spa /c 2 \ dp

**= n = - K + -

VTT \3

l + e~<*,s a^
*- jer^
1 e ,


100. Equation (95) is the equation of transpiration in a flat tube on the
assumption that

c-z c+z

_e~M e~^s , ,_ ,.


A slight modification however is all that is necessary to render the
equation perfectly general.

The only way in which the shape of the tube enters into the equation is
in the coefficient of the first line on the right-hand side, i.e., the coefficient

of -^ , and whatever may be the shape of the tube this coefficient will be of

the same form as far as the linear dimensions of the tube are involved, the
only possible difference being in the numerical coefficients of c 2 and sc\.
Therefore if c 2 be multiplied by a coefficient A, which depends on the shape
of the tube, since m also varies with the shape of the tube, we have for the

general coefficient of -

o fi m*


(C \

A - + m\ I .
s )

As regards the coefficient of ^- , this is affected by the assumption as to

the particular form of (s 1 s< 2 ); and if we assume a general form for s 1 s 2 ,
such as

f _( c .zf) w _(*))

-^ )6 ^ a ' s ' e \ a,s > I . , ,

*i S 2 * < / ., x > \ s \ #2 ;

the coefficient of the last term would still be of the form

(c\ c

- } varies continuously as - varies from to oo , having a finite
S / S

(* (* {*

value when - is infinite and being zero of the order - when - is zero.
s s s


101. The factor s/ s 2 ' is clearly a function of c, - and X 3 , where Xj


depends on the nature of the impacts between the gas and the tube. And,


moreover, when - is small and the molecules cross the tube without encounter,

81 s 2 ' is proportional to c it may be shown that in the case of a flat tube


/ 2\
s l s z = Trmc, and in the case of a round tube s l s 2 = 7rmc I 1 H I , for tubes

V 7T/

of other shapes 5j s 2 would have an intermediate value so in this case we


Si S 2 ' = nrm'c.


Again, where - is large, then s/ s 2 ' is equal to sX 3 .


Hence, as a perfectly general form for s/ s 2 ', we have


wherein ./$( ] is zero when - is large, and unity when - is small ; while

(c\ c c

- j is unity when is large, and zero when is small.
s / s s

The general equation of transpiration.

102. Substituting in equation (95) from equations (96) and (97) we

pan = - VTTC \ A - + raXJ -f
{ s j da;


H T- \Trm'f 3 [ - ) + -\ 3 f 4 ( -}[ \ f ( - } + X| pa -=-

2\/7rl \sJ c J \sJ)\ J \sj }^ dx

Online LibraryOsborne ReynoldsPapers on mechanical and physical subjects (Volume 1) → online text (page 34 of 40)