By A. B. Basset
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Extra resources for A Treatise on Hydrodynamics, Vol. 1
F is 32. dx Integrating, + where dw in the present case du 5r ot where q - ^ = ^(0 we obtain ................. (11), an arbitrary function. DEF. A vortex line is a line whose direction coincides with the direction of the instantaneous axis of molecular rotation. The differential equations of a vortex line are thus dx dy dz EQUATIONS OF MOTION. 28 1 Clebsch's Transformation . When a velocity potential does not exist, a first integral of the general equations of motion can be obtained by means of a method which depends upon the analytical theorem, that if u, v, w 33.
Rotational motion. B. 3 EQUATIONS OF MOTION. (25). (26). In this case the quantity When the motion line, is C is evidently an absolute constant. rotational, let ds be an element of a stream then dx dz dy Multiplying the general equations by , dQ = -^ ds~ . we obtain , 1 -| 2 dn u, v, w and adding, z ds> whence This is (27). 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.
Z, t) 2 - 22 is K, prove is the velocity at the point F (x, y, 2a> in a fine tube of variable section that the equation of continuity where v 2 + dv/dz, s. the equation of a velocity of the surface normal to itself moving surface the is + (dF/dz}\ Hence deduce equation and u, v y and z are given functions of a, 6, c and t, where a, constants for any particular element of fluid, and if are the values of x, y, z when a, 6, c are eliminated, If x, 9. 6 (19). c are and w prove analytically that d*x 10.