Discontinuous flow density

The solution expressed by Eq. (1.36) indicates that there is no discontinuous flow between the upstream 1 and the downstream 2. However, the solution given by Eq. (1.37) indicates the existence of a discontinuity of pressure, density, and temperature between 1 and 2. This discontinuity is called a normal shock wave , which is set-up in a flow field perpendicular to the flow direction. Discussions on the structures of normal shock waves and supersonic flow fields can be found in the relevant monographs. 

It is obvious that the entropy change will be positive in the region Mi > 1 and negative in the region Mi < 1 for gases with 1 < y < 1-67. Thus, Eq. (1.46) is valid only when Ml is greater than unity. In other words, a discontinuous flow is formed only when Ml > 1. This discontinuous surface perpendicular to the flow direction is the normal shock wave. The downstream Mach number, Mj, is always < 1, i. e. subsonic flow, and the stagnation pressure ratio is obtained as a function of Mi by Eqs. (1.37) and (1.41). The ratios of temperature, pressure, and density across the shock wave are obtained as a function of Mi by the use of Eqs. (1.38)-(1.40) and Eqs. (1.25)-(1.27). The characteristics of a normal shock wave are summarized as follows ... 

The discontinuities diagrammed in Fig. 5.4.1 are termed kinematic shocks in that they represent discontinuities in density. Let us calculate the speed at which the top discontinuity moves down and the bottom one up. For specificity consider a downward-moving shock. With respect to a coordinate system moving down with the speed of the discontinuity u (Fig. 5.4.2A), the flow is steady and conservation of mass for the one-dimensional picture considered gives... 

A detonation shock wave is an abrupt gas dynamic discontinuity across which properties such as gas pressure, density, temperature, and local flow velocities change discontinnonsly. Shockwaves are always characterized by the observation that the wave travels with a velocity that is faster than the local speed of sound in the undisturbed mixtnre ahead of the wave front. The ratio of the wave velocity to the speed of sound is called the Mach number. 

Among the wide choice of reactor designs, the biofilm reactor is one of the best suited for azo-dye conversion as it meets two important process requisites. The first is related to the hindered growth feature of bacterial metabolism under anaerobic conditions. The second is related to the necessity to increase cell densities (see previous section) with respect to those commonly harvested in liquid broths [55, 56]. Except for bacteria that forms aggregates spontaneously, immobilization of cells on granular carriers and membrane reactor technology are the two common pathways to achieve high-density confined cell cultures in either discontinuous or flow reactors. 

The point of intersection of I, R M is known as the triple point, TP. The resulting existence of the above three waves, causes a density discontinuity. The surface of this discontinuity, known as slipstream, S, represents a stream line for the flow relative to the intersection. Between this and the reflecting surface is the region of high pressure, known as Mach region here the pressure is approx twice that behind the incident wave. The top of this pressure region, the triple point, travels away from the reflected surface. As pressure and impulse appear to have their maximum values just above and below the triple point, respectively, the region of maximum blast effect is approximately that of the triple point... 

Evans St Ablow (Ref 2) defined the steady-flow as "a flow in which all partial derivatives with.respect to time are equal to zero . The five equations listed in their, paper (p 131), together with. appropriate initial and boundary conditions, are sufficient to solve for the dependent variables q (material or particle velocity factor), P (pressure), p (density), e (specific internal energy) and s (specific entropy) in regions which.are free of discontinuities. When dissipative irreversible effects are present, appropriate additional terms are required in the equations... 

There are two types of discontinuity surfaces contact surfaces and shock fronts. There is no flow between regions separated by a contact surface, while shock fronts are crossed by the flow. A contact surface moves with the fluid and separates two zones of different density and temperature, but the same pressure. The normal component of the flow velocity is the same on both sides of a contact discontinuity... 

In the last 25 years, calculations of the detonation properties of condensed explosives from their chemical compositions and densities have been approached in various ways.2 All have used the necessary conservation conditions for steady flow with the detonation discontinuity satisfying the Chapman-Jouguet hypothesis (minimum detonation velocity compatible with the conservation conditions or sonic flow behind the discontinuity in a reference frame where the discontinuity is at rest). In order to describe the product state and the thermodynamic variables which fix its composition, an equation of state applicable to a very dense state is required. To apply this equation to a mixture of gaseous and solid products, a mixing rule is also needed and the temperature must be explicitly defined. Of the equations of state for high-density molecular states which have been proposed, only three or four have been adapted to the calculation of equilibrium-product compositions as well as detonation parameters. These are briefly reviewed in order to introduce the equation used for the ruby computer code and show its relation to the others. 

The flow of combustion products behind the flame front has a non-zero vorticity. When the gas crosses the flame front, which represents the gas-dynamic discontinuity surface where the velocity, pressure, density and gas temperature are step-like changing, the vorticity of combustion products is generated (Zeldovich, 1944, 1966, 1979, 1980 Borisov, 1978 Zeldovich et al. 1979 Tsien, 1951 and Chernyi, 1954). 

Many polymers exhibit neither a measurable stick-slip transition nor flow oscillation. For example, commercial polystyrene (PS), polypropylene (PP), and low density polyethylene (LDPE) usually do not undergo a flow discontinuity transition nor oscillating flow. This does not mean that their extrudate would remain smooth. The often observed spiral-like extrudate distortion of PS, LDPE and PP, among other polymer melts, normally arises from a secondary (vortex) flow in the barrel due to a sharp die entry and is unrelated to interfacial slip. Section 11 discusses this type of extrudate distortion in some detail. Here we focus on the question of why polymers such as PS often do not exhibit interfacial flow instabilities and flow discontinuity. The answer is contained in the celebrated formula Eqs. (3) or (5). For a polymer to show an observable wall slip on a length scale of 1 mm requires a viscosity ratio q/q equal to 105 or larger. In other words, there should be a sufficient level of bulk chain entanglement at the critical stress for an interfacial breakdown (i.e., disentanglement transition between adsorbed and unbound chains). The above-mentioned commercial polymers do not meet this criterion. 

The above features of a sheared colloidal crystal appear to be similar in both BCC and FCC structures. However, there are differences in details, and perhaps even within a given symmetry the flow behavior might vary with particle concentration or charge density. For example, Chen et al. (1994) have shown that between the strained crystal and sliding-layer microstructures there can be a polycrystalline structure, the formation of which produces a discontinuous drop in shear stress (see Fig. 6-33). Ackerson and coworkers gave a detailed description of the fascinating shear-induced microstructures of these systems (Ackerson and Clark 1984 Ackerson et al. 1986 Chen et al. 1992, 1994). 

The GC route is particularly attractive for it requires no a priori information on the polymer. With the exception of X-ray measurements, most methods of measurement involve a comparison of some property of the polymer, such as density, with that of the totally amorphous or crystalline material. Furthermore neither the mass of polymer in the column nor the flow rate of carrier gas need to be measured since a ratio of retention volumes is computed in Eq. (21). It should be added, however, ttiat for the successful application of the method it is essential that the measured retention volumes correspond effectively to equilibrium bulk sorption, both above and below. Low molecular weight compounds are known to exhibit apparently similar discontinuities in retention diagrams at their melting points but this is to be ascribed to a change in retention mechanism, from surface adsorption for the solid to bulk sorption for the liquid stationary phase. For a detailed discussion of retention characteristics of low molecular weight substances near their transition temperatures the reader is referred to a recent review by McCrea (8J). 

The collection of various structures in nature or in engineering subjected to a flow of water or air will be extended and discussed in more details in the consequent chapters. The flows associated with them, despite their diversity, can be nevertheless united by the fact that one needs to account both for the internal flow within the permeable structure and for the external free flow over it. Deceleration of the flow within the obstructed but penetrable layer was found to depend significantly upon the closeness of the obstructions characterized by the density n, l/m3 or s, m2/m3. This fact prompts a uniform mathematical treatment of all the above-discussed different flows. It can be suggested to represent obstructions in mathematical models by individual forces Pj7 whereas their collective action on the flow can be described by a smeared (distributed) force (1.6)—(1.7) that acts within the layer but equals zero outside it. The force is discontinuous on the interface between the structure and the flow z = h, so that the interaction between the internal retarded flow and the free external one takes place. 

The solution of this simplest model just obtained brings the credible flow profiles for a duct with symmetric EPRs shown in Fig. 3.2. It can be seen that the velocity distribution varies from a parabolic shape (taking place in the absence of the EPR, A = 0 or <5 = 0) to a very distorted one which depends on the dimensionless density of the penetrable obstruction layer A. The shear stress keeps linear outside the obstruction layer but significantly bends within it. Both kinds of profiles quantitatively correspond to the experimental distributions measured in, for example, laboratory water flumes [231], Phenomenon of drag discontinuity (Chapter 6) can be observed at the top of the EPR that means that the profile r(z) is continuous but not differentiable. 

This equation is valid in the entire flow field even if the material properties vary discontinuously across phase boundaries. In Eq. (1), p and p are density and viscosity, v is the velocity field, p is pressure, and /is the body force. The effects of the interfacial tension are accounted for by the last term in Eq. (1). In this term, 5 is two or three dimensional delta function, cr is surface tension coefficient, k is the curvature of two-... 

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