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A Treatise on Hydrodynamics, Vol. 1 by A. B. Basset PDF

By A. B. Basset

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For and Whence, omitting the double integrals which refer to the boundary, we obtain - (f) -/// ^Y db by (46) and (47) ; Sb * I (f) * + -y- Sc dc [ (f)} *** dxdydz j whence which proves the proposition. Impulsive Motion. 42. before, Let u, v, w and u', v', w' be the velocities of a fluid, just and just after the impulse p the impulsive pressure. Then ; be any closed surface, the change of momentum parallel to x, of the fluid contained within 8, must be equal to the component parallel to a; of the impulsive pressure upon the surface of S.

39 In the case of a liquid p is constant, whence differentiating (48) with respect to x, y, z respectively, and taking account of the equation of continuity, we obtain If the liquid were originally at rest it is clear that the produced by the impulse must be irrotational, whence velocity potential, if motion be its (f> we must have p= (50). p(f> EXAMPLES AND APPLICATIONS. A 43. radius a, mass of liquid whose external surface and which is subject to a constant pressure is II, a sphere of surrounds a The solid sphere is annihilated, solid sphere of radius b.

The equal to first term in brackets vanishes, and the second term JdK/dx y - whence is xS d T dt dadb dc (a, b, c) d\ -j- ax Hence the conditions of the ~ Bx + d\ ~ -j- 8y ^ dy I T7 7 77 7 z\ Jdtdadbdc. (24). p dy __ dz P, problem require that __, dy d\ + -jdz = p dz Now ti, y, z are the component accelerations of the element whose coordinates are x, y, z, and are therefore equal to du/dt, and when we interpret dv/dt, and dw/dt respectively \p which must represent the pressure, equations (24) are the equations of motion in the ordinary form.

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A Treatise on Hydrodynamics, Vol. 1 by A. B. Basset

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