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Between (5) and (0) £ + ir; may be eliminated; u + iv then becomes a function of t and of .7; 4- iy, Hay
The equation with which wo started is of what is called in Hydrodynamics the Lagrangian type. We follow the motion of an individual particle. On the other hand, (7) i.s of the Kulerian type, expressing the velocities to be found at any time at a specified place. Keeping t fixed, •La. taking, an it were, an instantaneous view of the system, we see that u, v, as given by (7), satisfy
(d*l<h? + tP/dif) ( /t, yj) = 0,   ........................ (8)
equations which hold also for l,he irrotational motion of an incompressible liquid.
It is of interest to compare1 the present; motion with that of a highly viscous two-dimensional fluid, for which the equations an1.*
>V/'                 /      I-    »•
"i" U>
(f;t'.    ^ d,n d}> ,    , r/<?
^//V '
,     du     dv
v"'  ,   + ,   . du'     dy
If l-he prt'SHiiri* !H indi^pendcnt of density and if tlie iiusrtia tenns are neglected, tlitw et{iia(.i(»riH arc satisfied provided that
pX + p! tie id* « o, p y + f
In the case of real vistuniH fluids, there is reason to think that p! = \IJL. ImpreKsed forces are then recpiired so long an the fluid is moving. The supposition that j> IH coiiHtiuifc being already a large departure from the case of nature, we may perhapn as well suppose /A' — 0, and then no impressed bodily forces are called for either at rest or in motion.
If we suppose that this motion in (7) is steady in the hydrodynamical sense, n + iv must be independent of t, HO that the elimination of % + ir) between (5) and ((5) must carry with it the elimination of t, This requires that dfjdt in (C5) bo a function of/ ancl not otherwise of t and £ + t'»? ; ancl it follows that (5) must be of the form
l [t
* HfcokoB, Caml>. Trans. 1850 ; Mathematical and Phynleal Papert, Vol. iv. p. 11. It does not Hcem to be jjjflnerally known that the lawn of dynamical similarity for viscous fluids were formulated in this memoir. Beynolda's important application was 80 years later.b)J1(k1b) ' 4&1F0'(M)