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Fluid MechanicsA Geometrical Point of View$
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S. G. Rajeev

Print publication date: 2018

Print ISBN-13: 9780198805021

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198805021.001.0001

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(p.72) 6 Shocks
Fluid Mechanics

S. G. Rajeev

Oxford University Press

When the speed of a fluid exceeds that of sound, discontinuities in density occur, called shocks.The opposite limit from incompressibility (constant density) is constant pressure. In this limit, we get Burgers equation. It can be solved exactly in one dimension using the Cole–Hopf transformation. The limit of small viscosity is found not to be the same as zero viscosity: there is a residual drag no matter how small it is. The Maxwell construction of thermodynamics was adapted by Lax and Oleneik to derive rules for shocks in this limit. The Riemann problem of time evolution with a discontinuous initial density is solved in one dimension. These simple solutions provide the basic intuition for more complicated shocks.

Keywords:   Shocks, Burgers equation, Cole–Hopf transformation, Lax–Oleneik rule, Maxwell construction, Riemann problem

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