By A. B. Basset
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The equation of continuity is du dx -j- dv + -7- = 0, dy which shows that the left-hand side of (22) ential cfr/r, is a perfect differ- whence u ^d = _d v The function ......... dx dy . (23) . Earnshaw's current function. ty is called the motion takes place in planes passing through the axis of z the equation of the lines of flow may be written When t w(wdvr-udz) = The equation of continuity ..................... (24). cr dz _Q is a perfect differential ety -r- ............... (25), which shows that the left-hand side of (24) cfyr, whence 1 cty r w = --5 w a-cr where i|r is , u 1 = --is dz Stokes' current function.
Bernoulli's Theorem. , ^ = const, be two surfaces stream line in other words if is ^ : whose intersections determine the stream of A|T and 2. lines, A is a function Let us now consider the steady motion of a liquid 1 which symmetrical with respect to the axis of z. The vortex lines 38. is will evidently be perpendicular to every plane of z, hence by 19 (41) the molecular rotation G> through the axis be determined will by the equation ~ 2o> Stokes, "On the steady Soc. vol. vn. p. 439. 1 = du dw dz dur , 7 .
Have 6 KINETIC ENERGY. On the Application 31 of the Principles of Energy and Least Action. 8 be any imaginary closed surface, which is fixed in a small interval St upon the the within contained 8, by impressed forces which act liquid with the work done by the pressure upon upon its mass, together be to the increase during the of must the boundary 8, equal of the liquid contained within 8, interval Bt of the kinetic energy Let 35. the fluid. The work done during together with the kinetic energy which, during the same interval, flows into 8 across the boundary.
A Treatise on Hydrodynamics, Vol. 1 by A. B. Basset