On the effect of the internal friction of fluids on the motion of pendulums / by G.G. Stokes.

  • Stokes, George Gabriel, 1819-1903.
Date:
1851
Catalogue details

Licence: Public Domain Mark

Credit: On the effect of the internal friction of fluids on the motion of pendulums / by G.G. Stokes. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

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    of equations (39), (40), (41) under the form U + \/- 1 V, and reject the imaginary part. Putting for shortness / n V —/ = (42) 2fj. we have m = v(l + \X— l), and we obtain V- t- . ^ — sin n^, n (43) \|/ = i a^c sin* 01 + ') cos nt + —^ f l + —'j sin n^l - ^ ^ \L\ 2va) 2va \ va) Jr --—e“ j^cos (w^ - vr + ra) + H ^ sin (n# - i/r + i/a)J . (44) p =-ipacw |fl +sin w# — fl + —'l cos cos 0 . — (45) ^ 2val 2va \ vaj \ ^ The reader will remark that the \|r, p of the present article are not the same as the p o{ the preceding. The latter are the imaginary expressions, of which the real parts constitute the former. It did not appear necessary to change the notation. When fjL = 0, 1/ = 00 , and reduces itself to a^c . . dp — sin^ Q cos nt, or — sin* 9 — . 2r 2r dt In this case we get from (26) d^cosG ^ d^sinO and Rdr + QrdO is an exact differential d<p where d^cosQ m = — i a , ^ ^ dt which agrees with the result deduced directly from the ordinary equations of hydrodynamics*. 18. Let us now form the expression for the resultant of the pressures of the fluid on the several elements of the surface of the sphere. Let be the normal, and Tg the tangential, component of the pressure at any point in the direction of a plane drawn perpendicular to the radius vector. The formulae (4), (5) are general, and therefore we may replace x, y in these formulae by x\ y, where x\ y are measured in any tv/o rectangular directions we please. Let the plane of oo y pass through the axis of x and the radius vector, and let the axis of x' be inclined to that of x at an angle $•, which after differentiation is made equal to 9. Then Pi, 7*3 will become P,., T,, respectively. We have u R cos (0 - 9-) - 0 sin (0 - Q-)? v = R sin (0 - 9) + 0 cos (0 - 9), See Camb. Phil. Trans. Vol. Vlll. p. 119.