Mathematical Analysis of Shock Wave Reflection by Shuxing Chen

Author:Shuxing Chen
Language: eng
Format: epub, pdf
ISBN: 9789811577529
Publisher: Springer Singapore

with being a small positive number, while the norm of is preserved in the process of extension. Then people can compute

and define

and use to measure the size of perturbation (i.e.“consequence”)

Therefore, the problem on stability is to study whether Q can be controlled by .

Here we have to emphasize that in the discussion of stability of Mach configuration it is not enough to set the condition on upstream flow. Indeed, in the discussion on E-E type Mach configuration the flow behind the reflected shock and the Mach stem are subsonic, then we must add some restriction to downstream of the flow, because the downstream always influences the upstream in subsonic flow. A reasonable choice is to give pressure of the downstream flow at somewhere. To this end we take a straight line with small , such that the interval of the line cut by is located in the circle , then the pressure p of the flow on is required to remain unchanged. Later, we will see that such an additional requirement is suitable to ensure the stability for E-E type Mach configuration. As for the treatment for E-H type Mach configuration will leave to Sect. 4.5.

Denote , then to confirm the stability of Mach configuration amounts to prove the following theorem.

Theorem 4.1

Assume that and C form a flat Mach configuration as mentioned above. If is a sufficiently small number, and with , then there exist and functions defined on (i = 2,3), such that

(1) satisfies (4.1) in ().

(2) satisfies Rankine-Hugoniot conditions (4.2) and entropy condition on .

(3) on .

(4) on , on .

(5) The estimate

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