The most general case of plane wave propagation, when normal and shear stresses occur simultaneously, is considered in a material obeying the von Mises yield condition. The resulting nonlinear differential equations have not been solved previously for any boundary-value problem, except for special situations where the differential equations degenerate into linear ones. In the present paper, the stresses in a half-space, due to a uniformly distributed step load of pressure and shear on the surface, are obtained in closed form.
If a transparent test tank partially filled with water is strongly vibrated, small gas bubbles originated by surface disturbances appear in the lower parts of the vessel. These bubbles do not rise to the surface as would be expected from their buoyancy, but perform vibratory motions at the bottom or at the sides of the tank. As the rocket engine of a missile causes heavy vibrations, the possibility of the occurrence of similar bubble phenomena in fuel tanks is worth consideration. The present paper is a step toward an explanation of the complicated phenomena observed in the test tank. The basic equations for the motion of small gas bubbles in an inviscid liquid in the presence of harmonic vibrations are derived, and the mechanism which may make bubbles move contrary to gravity forces is explained. Applying the theory to cylindrical tanks, it is found that for sufficiently strong vibrations there are regions in the tank in which bubbles move downward and, further, that there are points where bubbles will collect. The paper does not treat all the phenomena observed in the test tank; e.g., the important question of the creation of the bubbles at the surface is left for future work. Nomenclature a = reference (nominal) radius of bubble C, C = coefficients in Equations [39, 41] defining the acceleration in an elastic vessel / = f(h,r) = distribution of the applied dynamic pressure (Eqs. [29,39]) g = gravitational constant h = depth of bubble below surface; specifically, depth at which bubble oscillates without rising IQ = modified Bessel function M = total mass of fluid in vessel N = number defining the applied acceleration x = Ng cos cot p = gas pressure in bubble p 0 -ullage pressure in vessel Pi = p 0 -f hg static pressure at depth h p -dynamic pressure due to vibration P = potential energy r = radial coordinate (Fig. 4) R = radius of cylindrical tank t = time T -kinetic energy u = vertical component of fluid velocity U ~v = velocity (vector) of a fluid particle v = volume; specifically, volume of bubble x(t)= displacement of vessel with respect to a reference line z(t) -relative depth of bubble below the surface which moves with the velocity x(t)
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