Similarity solutions for the generalized equation of steady transonic gas flow with a singular source

Authors:
Hamid Bellout, Kuppalapalle Vajravelu and Robert A. Van Gorder

Journal:
Quart. Appl. Math. **73** (2015), 379-389

MSC (2010):
Primary 35Q53, 37K10, 35D30, 34E05

Published electronically:
March 31, 2015

MathSciNet review:
3357500

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Abstract | References | Similar Articles | Additional Information

Abstract: In this brief paper, we consider the generalized equation of steady transonic gas flow with the addition of a singular source term. While the addition of a source term often destroys self-similarity of such flows, we demonstrate that a self-similar solution can still exist in the case of a singular source. We first reduce the governing nonlinear partial differential equation into an ordinary differential equation for a class of similarity solutions. Then, we study the existence of solutions for this similarity equation. After that, several explicit solution forms are given. In constructing exact solutions analytically, we demonstrate that dual solution branches may exist for some parameter regimes. For those parameter regimes where exact or analytical solutions are not possible, we obtain numerical solutions. The results demonstrate interesting properties of the solutions which warrant further study.

- [1]
S. S. Titov, A method of finite dimensional rings for solving nonlinear equations of mathematical physics [in Russian]. In:
*Aerodynamics*(Editor T. P. Ivanova), Saratov Univ., Saratov, pp. 104-110, 1988. **[2]**S. R. Svirshchevskiĭ,*Lie-Bäcklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations*, Phys. Lett. A**199**(1995), no. 5-6, 344–348. MR**1327680**, 10.1016/0375-9601(95)00136-Q**[3]**Andrei D. Polyanin and Valentin F. Zaitsev,*Handbook of nonlinear partial differential equations*, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR**2042347****[4]**C. C. Lin, E. Reissner, and H. S. Tsien,*On two-dimensional non-steady motion of a slender body in a compressible fluid*, J. Math. Physics.**27**(1948), 220–231. MR**0026499****[5]**W. F. Ames and M. C. Nucci,*Analysis of fluid equations by group methods*, J. Engrg. Math.**20**(1986), no. 2, 181–187. MR**846957**, 10.1007/BF00042776- [6]
A. G. Kuz'min and A. V. Ivanova,
*The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions*, Fluid Dynamics**18**(2004), 335-344. **[7]**J. Haussermann, K. Vajravelu, and R. A. Van Gorder,*Self-similar solutions to Lin-Reissner-Tsien equation*, Appl. Math. Mech. (English Ed.)**32**(2011), no. 11, 1447–1456. MR**2896096**, 10.1007/s10483-011-1514-6**[8]**A. N. Bogdanov and V. N. Diesperov,*On the distinguishing features of asymptotic expansions for the parameters of unsteady transonic flows with axial symmetry*, Prikl. Mat. Mekh.**72**(2008), no. 1, 54–57 (Russian, with Russian summary); English transl., J. Appl. Math. Mech.**72**(2008), no. 1, 33–35. MR**2423436**, 10.1016/j.jappmathmech.2008.03.009**[9]**Yu. V. Bibik, V. N. Duesperov, and S. P. Popov,*Structure of time-dependent transonic flows in plane channels*, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza**2**(2005), 168–179 (Russian, with Russian summary); English transl., Fluid Dynam.**40**(2005), no. 2, 315–325. MR**2151851**, 10.1007/s10697-005-0071-y**[10]**S. N. Glazatov,*On the solvability of a spatially periodic problem for the Lin-Reissner-Tsien equation of transonic gas dynamics*, Mat. Zametki**87**(2010), no. 1, 137–140 (Russian); English transl., Math. Notes**87**(2010), no. 1-2, 130–134. MR**2730392**, 10.1134/S0001434610010177

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Additional Information

**Hamid Bellout**

Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115

Email:
bellout@math.niu.edu

**Kuppalapalle Vajravelu**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Email:
Kuppalapalle.Vajravelu@ucf.edu

**Robert A. Van Gorder**

Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

Email:
Robert.VanGorder@maths.ox.ac.uk

DOI:
https://doi.org/10.1090/qam/1381

Keywords:
Transonic gas flow,
singular source,
analytical solution,
exact solution,
existence result

Received by editor(s):
July 22, 2013

Published electronically:
March 31, 2015

Additional Notes:
The second author was supported in part by NSF grant #1144246.

Article copyright:
© Copyright 2015
Brown University