Abstract
We write and analyse the equations of 3-D WNLRT and 3-D SRT. Then we discuss their many applications.
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Notes
- 1.
Printed in bold \(\varvec{\chi }= (\chi _1,\chi _2,...,\chi _d)\) is a ray velocity and not an angle.
- 2.
This continuous flow has a sonic line starting from a point \(P_G\) on the aerofoil, see Fig. 9.8-2.
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For the scalar conservation law \(u_t +(\frac{1}{2}u^2)_x=0\), we first note the transport equation \(\frac{du}{dt} \equiv (\frac{\partial }{\partial t} + u \frac{\partial }{\partial x})u = 0\) of a nonlinear wavefront (see Sect. 3.5). According to this the amplitude \(m=u\) remains constant \(=m_0\). Then, we refer to the transport equation (5.24) for a shock with \(u_r=\) constant, \(M-1=\frac{u_0-u_r}{u_r}\equiv \frac{u_\ell -u_r}{u_r}\) and \(v_1=\mathcal {V}\). This gives \(\frac{d(M-1)}{dt} = \{\frac{\partial }{\partial t} + \frac{1}{2}(M+1)\frac{\partial }{\partial x}\}(M-1) =-\frac{1}{2}(M-1) \mathcal {V}\). When \(\mathcal {V}>0, M-1\rightarrow 0+ ~as~ t\rightarrow \infty \).
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Prasad, P. (2017). 3-D WNLRT and SRT: Theory and Applications. In: Propagation of Multidimensional Nonlinear Waves and Kinematical Conservation Laws. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-10-7581-0_10
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DOI: https://doi.org/10.1007/978-981-10-7581-0_10
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