Small regions in a flow where the density is different from that of the surrounding gas do not exactly follow accelerated motions of the latter, but move faster or slower depending on whether their density is smaller or larger than that of the main flow. This behaviour cannot be quantitatively explained by treating a gas ‘bubble’ as a hypothetical solid particle of the same density, because a gas bubble cannot move relative to the surrounding gas without being transformed into a vortex which absorbs part of the energy of the relative motion.To illustrate the acceleration effect, the flow velocity behind known pressure waves in a shock tube is compared with the observed velocity of a bubble produced by a spark discharge. The displacement of such a bubble by a wave exceeds that of a flow element by more than 20%, but the bubble density is not known. If the spark discharge is replaced by a small jet of another gas, a pressure wave cuts off a section of this jet which then represents a bubble of known density.A theory is developed which permits computing the response of such bubbles to accelerations. The ratio of the bubble velocity to the velocity of the surrounding gas depends on the density ratio for the two gases and on the shape of the bubble, but not on the acceleration. Experimental results with H2, He, and SF6bubbles in air, accelerated by shock waves of various strength, are presented and agree well with the theoretical predictions. The results apply regardless of whether accelerations are produced by pressure waves in a non-steady flow or by curvature of streamlines in a steady flow. Various aspects of the experimental observations are discussed.

Shock waves passing through uniform suspensions of droplets or solid particles in a gas upset the velocity and temperature equilibrium between the two phases, and a relaxation zone is created in which the equilibrium is gradually re-established. The effects of varying the shock strength or the properties of the mixture are investigated theoretically, and it is noted that some flow variables do not always change monotonically throughout the relaxation zone but may go through a maximum or minimum. The behavior of individual variables is discussed in some detail. A particularly interesting finding is that, for weak shock waves, the maximum particle drag and heat transfer may appear at some distance from, rather than immediately behind, the shock front. The behavior of the variables changes at critical conditions which depend on the thermodynamic properties of the materials involved, on the shock strength, and sometimes also on the assumptions made for the drag coefficient or the Nusselt number. For the gas velocity, these critical conditions can be expressed in simple analytical forms. In all calculations some assumptions must be made for particle drag and heat transfer. Calculations based on plausible variations of the customarily used formulas show that the results are significantly affected by the assumption made for particle drag, but only to a minor extent by that for heat transfer. It is concluded that experimental determination of the drag coefficient by shock-tube techniques appears feasible.

If the nonlinear equations for nonsteady blood flow are solved by the method of characteristics, shock discontinuities may develop as a result of omitting from the mathematical model some aspect of the system that becomes significant at rapid flow changes. As an illustration, the flow from the heart into the aorta at the beginning of systole is analyzed. An equation is derived which yields shock formation distances between a few centimeters and several meters depending on the elastic properties of the aorta. Since knowledge of the actual wave form would be useful for computer programming, a few exploratory experiments were performed with an unrestrained latex tube. They indicated wave transitions extending over several tube diameters, but maximum steepening of the wave has not yet been achieved.

No abstract

The techniques for solving problems of nonsteady, quasi-one-dimensional flows by the method of characteristics have been extended to flows of suspensions of small solid or liquid particles in a gas. The particles cannot follow the rapid velocity and temperature changes that are produced by pressure waves, and complicated relaxation processes result. Such flows are described by six simultaneous differential equations, and their characteristics and associated compatibility equations have been obtained. Terms are included to allow for variable duct area, for external forces that may act on the particles or on the gas, and for heat addition to the particles or the gas. To illustrate the application of the results, two flows in a pipe, filled with a suspension of small particles, are analyzed with the help of a small digital computer. In the first example, one end of the pipe is suddenly opened to a low-pressure reservoir, and the resulting expansion wave is computed. In the second example, a piston in the pipe is impulsively accelerated, and the flow between the resulting shock wave and the piston is obtained. A striking feature of this flow is the slow adjustment of the shock strength from its initial to its equilibrium value.