E-Book Content
VOL,. 9, 1923
13
MA THEMA TICS: R. S. WOOD WARD
SOME EXTENSIONS IN THE MA THEMA TICS OF HYDROMECHANICS By ROBERT S. WOODWARD CARNIGIE INSTITUTION OF WASHINGTON, D. C.
Read before the Academy, April 25, 1922
Present Status and Needs of Hydrokinetics.-Precise description of fluid motion when viscosity is taken into account is a matter of much complexity. No less than twenty symbols and fifteen equations are required for a complete specification of the linear and angular movements of a fluid element, of its expansion and contraction, and of the changes in pressure and density to which it is subject. The formation of what Poinsot called an "image sensible" of these characteristics is a matter of corresponding difficulty which has taxed the resources of the ablest analysts from the time of Euler down to the present day. The nature of the difficulty seems to lie in an inadequacy of our mathematical machinery rather than in defects of our physical concepts. The following abstract, therefore, aims to suggest some extensions and improvements in that machinery, with the hope that greater generality and uniformity may be ultimately realized in the treatment of concrete problems in this branch of physics. The general equations of fluid motion are commonly written in the following simplest forms, namely:'
Pdtu
+ p-
dt
= bx
PX +13U- .ab
+ +
A
A
A2U
1 (66 dv 6p P dt + a- = pY + 3- y '- + ,U 2v2, by by
p
dw ap -wdt + ?az -p
=
1 aaO
pZ + S -0 + 3 ?z 1
A
(1)
A2W.
In these x, y, z are the rectangular coordinates of any element of the fluid at the time t; u, v, w are the linear velocity components of the element; p is the internal stress and p is the density of the fluid at x, y, z, at the time t; X, Y, Z are the force components per unit mass of the fluid at x, y, z; ,u is the coefficient of viscosity of the fluid; and 0 is the time rate of expansion of the element at x, y, z at the time t. This rate is given by the equation expressing the conservation of the mass of the element, that is, by +
?x
v w i+ bz by
_ d log p. dt
(2)
14
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PRoC. N. A. S.
MA THEMA TICS: R. S. WOOD WARD
The symbol A2 which appears in the second members of (1) is defined, for example, by
2262+2U
62U
by2
az2'
bx2
and the operation here indicated is now commonly called the Laplacian of u. Along with the equations (1) and (2) three others are needed to express the relations between the linear velocity components u, v, w and the angular velocity components (or spin components) of the fluid element. These latter components with reference to the axes of X, Y, Z, respectively, are given by av au 2, = auz- awVxf ay- CJv 2t = aw 2t = __ _
z,
and these are subject to the obvious condition at
ax
+
+ ar by az 71
=
O.
(4)
Moreover, three additional equations, introducing four new functions (potentials), are required to express u, v, w in forms to meet the requirements of the conditions (2) and (4) These equations are u=-
?ax
av +-aw z by
afP + av_ aw by
ao
?z 6v
az
ax
(5)
bx
av by
in which qp is the velocity potential, representing the irrotational part of the motion, and U, V, W may be called the spin potentials, since they represent the rotational part of the motion, these two parts being independent of one another. Finally the conditions (2) and (4) lead to the further relations2
A2V =-2t,
A2p = 0, A2V = - 2n,
A2W
2t,
(6)
- + w = 0. )z