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Full text of "Scientific Papers - Vi"

354                        ON  THE  THEOBY  OF  THE  CAPILLABY TUBE                      [399
du    (c(l + 0)    J       c*x                 oe           .    c*
Thus         ,
CLOC       I      9/72             \   /^2__^2\ir      Sr/2r fr*__
V       j\Jj                '      \\j        tAj    )         tjLv w \\j    ^^
,                      ^                       I      ^^       \^     ~      W      1         /      o            rt\f,         O^/2        M^\
,______________________________________::_    __ ___   . /     \,  / rt& _ /g* y> ^^  y^Q f Q" __ /- J
,4(c'--A'2f L        2
1 ^2 log {c + V(c2 - 2)} + ic4 - c3 VO2 - ^2)   ,    ......(19)
u = '   '        '              ''        C
C
-                                   "1
log" (c + V(c2  2)}   + constant.  L         x         yjj
We have now to choose I, or rather (I + 0), and it may appear at first sight as though we might take it almost at pleasure. But this is not the case, at any rate if we wish our results to be applicable when c = r. For this purpose it is necessary that (du/dx\ x. (r* - x1} be a small quantity, and only a particular choice of (I + C) will make it so. For when as = c = r,
fdu\ r3 - of _        r        \r_(l -M7) _    _ _r*_      r*_ /            1\)       r*_
\dx)r     r-    ~ V(r2-^2) 1 ~~2oT~ ~        3^ + 60* (   g T + 2J f ~ 6o"4 + terms vanishing when ac=r. We must therefore take
making
1 6a4
It should be noticed that u so determined does not become infinite when c = r and % = r.    For we have
r1
--    ft4'4//.    TNloS   1 + "------^ + (7/C/.
6a4    6a4 V(r2  2)        (            ^       J
Also with the general value of c
21   rti                              /oo\
^     +......................I22)
As before                             h = l c + u0}
3
and            ra2 cos i =-----5-^- + ^^----- + I  (u-C')sc dx
A               o          Jo
r!l/l + c__^/'l_^l^     -f   - Cc2-r2)f-c3 as. At the wall,, where x = r, ty assumes a given value ($TT~ i), and (1) becomes