On linearized theory for compressible viscous planar flow with slip
Author:
F. E. Fendell
Journal:
Quart. Appl. Math. 28 (1970), 37-55
DOI:
https://doi.org/10.1090/qam/99805
MathSciNet review:
QAM99805
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Abstract: Steady compressible supersonic planar flow of a viscous heat-conducting fluid is examined under linearization about freestream conditions. Although the results are derived here for a large-Prandtl-number fluid, Wu has shown that the conclusions hold for an order-unity-Prandtl-number fluid in the limits of large and small Reynolds numbers. First the compressible correction to the fundamental solution for flow past a singular flat plate (a point source of momentum directed antiparallel to the uniform freestreaming) is examined. Results valid near the point source are obtained by contour integration. Then these results are superposed to form an integral equation describing linearized slip flow past a finite flat plate at zero angle of attack. The large-slip limit for a short plate is characterized as a regular perturbation limit while the small-slip limit has singular-perturbation characteristics. For a long flat plate the integral equation can be approximated as Poisson type and is easily solved; for a short flat plate the integral equation can be approximated as a Carleman equation of the second kind and is less readily solved. However, an approximate solution for the shear near the leading edge of a short flat plate with small slip is given through the Wiener-Hopf technique. The integral equation describing thermal slip at a finite flat plate is seen to be like that describing velocity slip.
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C. W. Oseen, Neure Methoden und Ergebnisse in der Hyrdodynamik, Akad. Verlagsgesellschaft, Leipzig, 1927
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W. E. Olmstead and D. L. Hector, The lift and drag on a flat plate at low Reynolds number via variational methods, Quart. Appl. Math. 25, 415–422 (1968)
P. A. Largerstrom, J. D. Cole, and L. Trilling, Problems in the theory of viscous compressible fluids, Caltech Guggenheim Aeronautical Laboratory Report, Pasadena, California, 1949
M. Van Dyke, Impulsive motion of an infinite plate in a viscous compressible fluid, Z. angew. Math. Phys. 3, 343–353 (1952)
T. Y. Wu, On problems of heat conduction in a compressible fluid, Caltech Ph.D. thesis, Pasadena, California, 1952
---, Anemometry of a heated flat plate, Proc. 1952 Heat Transfer and Fluid Mechanics Institute, Stanford Univ. Press, Stanford, 1952, pp. 139–158
J. D. Cole and T. Y. Wu, Heat conduction in a compressible fluid, J. Appl. Mech. 19, 209–213 (1952)
T. Y. Wu, Small perturbations in the unsteady flow of a compressible, viscous, and heat-conducting fluid, J. Math. and Phys. 35, 13–27 (1956)
P. A. Lagerstrom, Laminar flow theory, in Theory of laminar flows, high speed aerodynamics and jet propulsion, Vol. 4, ed. by F. K. Moore, Princeton Univ. Press, Princeton, 1964, pp. 20–285
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a complex variable—theory and technique, McGraw-Hill, New York, 1966, Chaps. 6 and 8
I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series, and products, 4th ed., prepared by Yu. V. Geronimus and M. Yu. Tseytlin, transl. by A. Jeffrey, Academic Press, New York, 1965, p. 316
G. A. Campbell and R. M. Foster, Fourier integrals for practical application, D. Van Nostrand, Princeton, N. J., 1948, p. 122, pair 912.2
J. A. Laurmann, Linearized slip flow past a semi-infinite flat plate, J. Fluid Mech. 11, 82–96 (1961)
---, Structure of the boundary layer at the leading edge of a flat plate in hypersonic flip flow, AIAA J. 2, 1655–1657 (1964)
S. Chapman and T. G. Cowling, The mathematical theory of nonuniform gases, 2nd ed., Cambridge Univ. Press, Cambridge, pp. 107–117
S. G. Mikhlin, Integral equations, translated by A. H. Armstrong, Pergamon Press, New York, 1957, pp. 1–15
H. Mirels, Estimate of slip effects on compressible laminar-boundary-layer skin friction, NACA TN 2609, Washington, D. C., 1952
S. Bell, Studies of boundary-layer slip solutions and Alden’s method for boundary-layer correction, University of California Institute of Engineering Research Report He–150–133, 1955
J. D. Murray, Incompressible slip flow past a semi-infinite flat plate, J. Fluid Mech. 22, 463–469 (1965)
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Article copyright:
© Copyright 1970
American Mathematical Society
