Shock discontinuity

To permit a more general discussion, we can replace the snowplow with a piston, and replace the snow with any fluid (Fig. 2,3), We consider the example shown in a reference frame in which the undisturbed fluid has zero velocity. When the piston moves, it applies a planar stress, a, to the fluid. For a non-viscous, hydrodynamic fluid, the stress is numerically equal to the pressure, P, The pressure induces a shock discontinuity, denoted by which propagates through the fluid with velocity U. The velocity u of the piston, and the shocked material carried with it (with respect to the stationary frame of reference), is called the particle velocity, since that would be the velocity of a particle caught up in the flow, or of a particle of the fluid. 

To reiterate, the development of these relations, (2.1)-(2.3), expresses conservation of mass, momentum, and energy across a planar shock discontinuity between an initial and a final uniform state. They are frequently called the jump conditions" because the initial values jump to the final values as the idealized shock wave passes by. It should be pointed out that the assumption of a discontinuity was not required to derive them. They are equally valid for any steady compression wave, connecting two uniform states, whose profile does not change with time. It is important to note that the initial and final states achieved through the shock transition must be states of mechanical equilibrium for these relations to be valid. The time required to reach such equilibrium is arbitrary, providing the transition wave is steady. For a more rigorous discussion of steady compression waves, see Courant and Friedrichs (1948). 

Conservation equations Expressions that equate the mass, momentum, and energy across a steady wave or shock discontinuity ((2.1)-(2.3)). Also known as the jump conditions or the Rankine-Hugoniot relations. 

Since shock discontinuities move at supersonic speed into the fluid ahead, shocks overtake contact discontinuities and rarefaction waves. Since shocks move sub-sonically with respect to the fluid behind them, a shock will be overtaken by a shock or rarefaction behind it. When two shocks moving toward each other collide, two shocks moving away from each.other are produced together with two regions of different entropy separated by a contact discontinuity thru the point of collision. 

K. Ogi, N. Takeda and K. M. Prewo, Fracture Process ofThermally Shocked Discontinuous Fibre-Reinforced Glass Matrix Composites Under Tensile Loading, J. Mat. Sci. 32, 6153-6162 (1997). 

A computational procedure that marches forward in space must necessarily start from an initial condition that represents a deviation from equilibrium. For a partly dispersed shock wave, the difference in the vapour and liquid phase flow variables just downstream of the frozen shock discontinuity constitute the required initial departure from equilibrium. For a fully dispersed shock wave an initial, arbitrary perturbation of the flow must be specified. Step-by-... 

Collisionless shock Discontinuity formed on the sim-ward side of the earth in the solar wind because of the interposed obstacle of the magnetosphere. 

For the above assumptions and the case of continuous polymer water injection, Eqs. la and lb can be solved analytically using the method of characteristics and shock discontinuity theory. This analytical solution is illustrated here in the context of examples. Fig. 2 shows laboratory measured water-oil relative permeability curves for a 6,000-md unconsolidated, Nevada sand representative of a California viscous-oil reservoir. Fig. 3 shows die water fractional flow curves for the cases of normal water, = 1 cp, and polymer... 

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