Non-classical kinematic shocks in suspensions of particles in fluids

Kluwick, A.; Cox, E. A.; Scheichl, Stefan

doi:10.1007/bf01170175

Cited by 18 publications

(23 citation statements)

References 19 publications

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“…By accounting for inertial and other effects, an internal structure of kinematic shocks is obtained. In addition, splitting of kinematic shocks and other "strange properties" have been found [4,11,95]. …”

Section: Kinematic Shock Relationsmentioning

confidence: 94%

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On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces

Schneider

2017

Acta Mech

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Neglecting inertial and viscosity effects in the bulk flow is a common assumption in the analysis of separation processes in suspensions under the action of gravity or centrifugal and Coriolis forces. While there is a number of examples of particular solutions, the general form of the basic equations for three space dimensions, together with the appropriate boundary and initial conditions, is still uncertain and, with regard to certain aspects, even controversial. An essential point is a proper choice of the variables. Here it is proposed to introduce the mass density of the mixture, the mean mass velocity of the mixture and the total volume flux as a set of dependent variables. After some manipulations, a complete set of basic equations is obtained. It consists of two continuity equations, a generalized drift-flux relation, and two linearly independent components of a vector equation describing the total body force as irrotational. Then, by eliminating the mean mass velocity of the mixture from the set of unknowns, a generalized kinematic-wave equation is derived. It describes kinematic waves that are embedded in a bulk flow that may be one-, two-or three-dimensional. Concerning boundary conditions at solid walls, one has to ascertain whether the total body force at the wall points into the suspension or out of it. In the former case, a thin boundary layer of clear liquid is formed at the wall, whereas in the latter case a thin sediment layer may either stick at the wall or slide along it. Each of those three possibilities leads to a particular boundary condition for the bulk flow in terms of the dependent variables. In addition, initial conditions and kinematic shock relations are briefly discussed. Finally, the application of the kinematic-wave theory to the settling process in rotating tubes is outlined.

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Section: Kinematic Shock Relationsmentioning

confidence: 94%

“…Remarkably, the radial dependence of both the centrifugal force and the cross section can lead to kinematic shock waves with unusual properties [6][7][8]; cf. also [11].…”

Section: Introductionmentioning

confidence: 99%

On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces

Schneider

2017

Acta Mech

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“…18(a). Jump discontinuities characterized by this unconventional property are termed non-classical under-compressive shocks and have been predicted also for creeping flows, Bertozzi, Münch and Shearer [2], suspensions of particles in liquids, Kluwick, Cox and Scheichl [24], constrained two-layer flows, Segin, Tilley and Kondic [61], dense gas acoustics, Scheichl and Kluwick [57,58] and the continuum theory of pedestrian flow, Colombo and Rosini [5]. Readers interested in a more rigerous mathematical theory of under-compressive non-classical shocks are referred to the review papers by LeFloch [40] and El, Hoefer, Shearer [11] which also include applications from other fields such as nonlinear elasticity and nonlinear optics.…”

Section: Discussionmentioning

confidence: 89%

Shock discontinuities: from classical to non-classical shocks

Kluwick

2017

Acta Mech

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The possibility of shocks was first suggested by Stokes in 1848, but no consistent theory emerged for another almost 30 years. Indeed, severe criticism by Lord Kelvin and Lord Rayleigh led him to finally discard this idea. In the meantime, algebraic shock conditions had been derived by Rankine and Hugoniot, and their names have come to be associated with the shock conditions generally. As Thompson (Compressiblefluid dynamics, McGraw-Hill, New York, 1972) concludes, "Stokes's claim to recognition for his discovery has been diverted by circumstance." Starting with a brief discussion of these early developments, the present review paper will then concentrate on the important question of how to select from all formally possible solutions of the Rankine-Hugoniot jump relations those which are physically realizable solutions. This is at the core of the so-called admissibility problem. Of course, a necessary condition is provided by the second law of thermodynamics which states that the entropy must not decrease during adiabatic changes of state. For perfect gases, this requirement is also a sufficient condition to rule out "impossible" shocks. For fluids with an arbitrary equation of state and/or situations where in addition to thermoviscous effects, dispersive effects also come into play this is not the case in general. The selection of physically admissible solutions is then found to be a more delicate matter and may result in new types of shocks which differ distinctively from their classical counterparts and, therefore, are termed non-classical shocks.

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“…Kinematic shocks in suspensions of particles in fluids have been studied by Kluwick, Cox and Scheichl [8]. Thin liquid films on inclined planes with a fourth order diffusion term were analysed by Bertozzi, Münch and Shearer [1].…”

Section: Introductionmentioning

confidence: 99%

Waves and nonclassical shocks in a scalar conservation law with nonconvex flux

Cox

Kluwick

2001

Z. angew. Math. Phys.

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The propagation of waves in the nonlinear equationgenerates undercompressive shocks in the hyperbolic limit with dispersion and dissipation balanced. These shocks are undercompressive in type and the diversity of phenomena possible is illustrated for three different initial conditions: a propagating shock through a wave fan, a square pulse and a periodic pulse constructed from constant states. The rich variety of wave phenomena exhibited:-shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations produce waves that have no counterpart in classical shock theories. A mechanism for the formation of a nonclassical shock from a classical shock by wavefan interaction is presented.

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Non-classical kinematic shocks in suspensions of particles in fluids

Cited by 18 publications

References 19 publications

On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces

On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces

Shock discontinuities: from classical to non-classical shocks

Waves and nonclassical shocks in a scalar conservation law with nonconvex flux

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