Physicochemical Fluid Dynamics in Porous Media by Mikhail Panfilov

Author:Mikhail Panfilov
Language: eng
Format: epub
ISBN: 9783527806584
Publisher: John Wiley & Sons, Inc.
Published: 2018-11-27T14:07:02+00:00

These conditions determine the unique admissible shock for the Riemann problem.

In Figure 9.7, two shocks are shown by two straight lines: I contradicts the Lax condition, while II is admissible.

Figure 9.7 Nonunique discontinuous solution of the Riemann problem.

These conditions are called the entropy conditions, because they can be obtained from the second principle of thermodynamics, which says that the entropy of any process can only increase through a shock. Another method to deduce them consists of analyzing the fine structure of a shock by adding a small diffusion term to the Buckley–Leverett equation, similar to ((9.10)). As known, the diffusion is a natural mechanism that ensures spontaneous entropy growth.

Proof:

Let us analyze the limit properties of the Buckley–Leverett equation considered as the limit of the convection–diffusion Equation (9.10). The solution to this equation is a sharp but smooth function. Near the shock we can introduce the local extended space coordinate , where is the shock velocity. Let the shock be stable, i.e. it does not deform in time. Then the saturation near the shock is steady state (the traveling wave) and does not depend on : . Then Equation (9.10) becomes

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