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

Cox, E. A.; Kluwick, A.

doi:10.1007/pl00001588

Cited by 5 publications

(5 citation statements)

References 10 publications

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“…18(b). For suspensions, this has been demonstrated by Cox and Kluwick [6], but if this new mechanism applies for other types of flows as well remains an open question. For hydraulic jumps in single-layer and two-layer flows, however, a first step has been taken more recently by Viertl [71], Kluwick, Szeywerth, Braun and Cox [34,35] who showed that in the limit of weak interaction and strong dispersion the perturbation displacement thickness due to the presence of the viscous wall layer can be calculated analytically and takes the form A ∼ (2/3)…”

Section: Discussionmentioning

confidence: 89%

“…In his Habilitationsschrift mentioned before, he writes 5 : "In einem gegebenen Medium können mechanisch nur Versdichtungsstöße oder Verdünnungsstöße entstehen je nachdem ob (∂ 2 p/∂v 2 ) s positiv oder negativ ist". Also he adds 6 : "Genau dieses Kriterium wird uns später (Sect. 6) bei der Betrachtung der thermodynamischen Möglichkeiten wieder begegnen."…”

Section: Admissibility In Arbitrary Fluidsunclassified

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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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Section: Discussionmentioning

confidence: 89%

Section: Admissibility In Arbitrary Fluidsunclassified

Shock discontinuities: from classical to non-classical shocks

Kluwick

2017

Acta Mech

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“…3, namely that a classical shock of maximum strength can be viewed as a compound structure made up of a nonclassical and associated nonclassical shock, Cox and Kluwick [4]. Namely, the question arises how can such shocks be generated.…”

Section: Nonclassical Shock Formationmentioning

confidence: 99%

“…the solution of the limiting hyperbolic form of the transport equation which can be calculated analytically and wave profiles calculated numerically on the basis of the full modified Burgers-Korteweg-de Vries equation with small diffusive and dispersive terms has recently be carried out by Cox and Kluwick [4]. the solution of the limiting hyperbolic form of the transport equation which can be calculated analytically and wave profiles calculated numerically on the basis of the full modified Burgers-Korteweg-de Vries equation with small diffusive and dispersive terms has recently be carried out by Cox and Kluwick [4].…”

Section: Nonclassical Shock Formationmentioning

confidence: 99%

Introduction to nonclassical shock solutions of the modified Burgers-Korteweg-de Vries equation

Kluwick

2002

Proc. Appl. Math. Mech.

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A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg-de Vries-Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two-layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.A. Kluwick,

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“…There has been widespread interest in the nonlinear wave phenomena mainly through the so-called evolution equations, derived from a large system of PDEs, representing an important aspect of the original system. [4][5][6][7][8] When the quadratic nonlinearity parameter that appears in the evolution up to time of order O(1/δ) is small, the nonlinear distortion of the disturbance is noticeable over time scale larger than O(1/δ); see Cox and Kluwick, 9 Nariboli and Lin, 10 and Sharma. 11 Here, δ ≪ 1 is a measure of the wave amplitude.…”

Section: Introductionmentioning

confidence: 99%

Undercompressive shock waves in Hall‐magnetohydrodynamics

Shukla

Sharma

2018

Math Methods in App Sciences

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Funding informationUniversity Grants Commission (UGC), India JEL Classification: 35C20; 35G25; 35L67; 76N15; 76W05We study the propagation of nonlinear waves in a Hall-magnetohydrodynamic model. An asymptotic method is used to derive the Gardner-Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second-and third-order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splitting-a phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave-fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.

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

Cited by 5 publications

References 10 publications

Shock discontinuities: from classical to non-classical shocks

Shock discontinuities: from classical to non-classical shocks

Introduction to nonclassical shock solutions of the modified Burgers-Korteweg-de Vries equation

Undercompressive shock waves in Hall‐magnetohydrodynamics

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