A NOVEL CONCEPT FOR FINDING THE FUNDAMENTAL RELATIONS BETWEEN STREAM FUNCTION AND VELOCITY POTENTIAL IN REAL NUMBERS IN TWO-DIMENSIONAL FLUID MOTIONS

Authors:

Prabir Chandra Bhattacharyya ,

DOI NO:

https://doi.org/10.26782/jmcms.2023.02.00001

Keywords:

Riemann equations,Quadratic equations,Rectangular Bhattacharyya’s Coordinates,Stream function,Theory of Dynamics of Numbers,Velocity potential,

Abstract

In this paper, the author has presented the fundamental relations between stream function or current function,  and velocity potential or velocity function, φ which are ∂φ/∂x= ∂/∂y and ∂φ/∂y= - ∂/∂x where x,y,φ(x_(, ) y),  (x_(, ) y) are all real in two-dimensional fluid motions using real variables only whereas these relations had been established by using complex variables by Cauchy – Riemann which are known as Cauchy – Riemann equations in fluid dynamics.

Refference:

I. A. Roshko. 1993. Perspectives on Bluff Body Aerodynamics. Journal of Wind Engineering and Industrial Aerodynamics 49.1-3:79–100.
II. Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0-521-09817-3.
III. Cauchy, Augustin L. (1814). Mémoire sur les intégrales définies. Oeuvres complètes Ser. 1. Vol. 1. Paris (published 1882). pp. 319–506.
IV. Chanson, H. (2007). “Le Potentiel de Vitesse pour les Ecoulements de Fluides Réels: la Contribution de Joseph-Louis Lagrange” [Velocity Potential in Real Fluid Flows: Joseph-Louis Lagrange’s Contribution]. Journal la Houille Blanche. 93 (5): 127–131. doi:10.1051/lhb:2007072. ISSN 0018-6368. S2CID 110258050.
V. d’Alembert, Jean (1752). Essai d’une nouvelle théorie de la résistance des fluides. Paris: David l’aîné. Reprint 2018 by Hachette Livre-BNF ISBN 978-2012542839.
VI. D. Coles. 1965. Transition in Circular Couette Flow. Journal of Fluid Mechanics. 21(3):385–425.
VII. Dieudonné, Jean Alexandre (1969). Foundations of modern analysis. Academic Press. §9.10, Ex. 1.
VIII. Euler, Leonhard (1797). “Ulterior disquisitio de formulis integralibus imaginariis”. Nova Acta Academiae Scientiarum Imperialis Petropolitanae. 10: 3–19.
IX. G.I. Taylor. 1923. VIII. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Royal Soc. London. Series A, Containing Papers of a Mathematical or Physical Character 223:289–343.
X. G.I. Taylor. 1923. VIII. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Royal Soc. London. Series A, Containing Papers of a Mathematical or Physical Character 223:289–343.
XI. Gamelin, T. W. (2001), Complex Analysis, New York: Springer, ISBN 0-387-95093-1
XII. Gray, J. D.; Morris, S. A. (April 1978). “When is a Function that Satisfies the Cauchy–Riemann Equations Analytic?”. The American Mathematical Monthly. 85 (4): 246 – 256 doi:10.2307/2321164. JSTOR 2321164.
XIII. Gray & Morris 1978, Theorem 9.
XIV. Iwaniec, T.; Martin, G. (2001). Geometric function theory and non-linear analysis. Oxford. p. 32.
XV. Kobayashi, Shoshichi; Nomizu, Katsumi (1969). Foundations of differential geometry, volume 2. Wiley. Proposition IX.2.2.
XVI. Lagrange, J.-L. (1868), “Mémoire sur la théorie du mouvement des fluides (in: Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin, année 1781)”, Oevres de Lagrange, vol. Tome IV, pp. 695–748.
XVII. Lamb (1932, pp. 62–63) and Batchelor (1967, pp. 75–79)
XVIII. Lamb, H. (1932), Hydrodynamics (6th ed.), Cambridge University Press, republished by Dover Publications, ISBN 0-486-60256-7.
XIX. Looman 1923, p. 107.
XX. Looman, H. (1923). “Über die Cauchy–Riemannschen Differential gleichungen”. Göttinger Nachrichten (in German): 97–108.
XXI. Massey, B. S.; Ward-Smith, J. (1998), Mechanics of Fluids (7th ed.), UK: Nelson Thornes.
XXII. O. Reynolds. 1883. XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. Royal Soc. London 174: 935–982.
XXIII. P.R. Viswanath, R. Narasimha, A. Prabhu. 1978. Visualization of Relaminarizing Flows. Journal of the Indian Institute of Science. 60(3):159–166.
XXIV. Pólya, George; Szegő, Gábor (1978). Problems and theorems in analysis I. Springer. ISBN 3-540-63640-4.
XXV. Prabir Chandra Bhattacharyya, ‘AN INTRODUCTION TO RECTANGULAR BHATTACHARYYA’S CO-ORDINATES: A NEW CONCEPT’. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, November (2021). pp 76-86.
XXVI. Prabir Chandra Bhattacharyya, ‘AN INTRODUCTION TO THEORY OF DYNAMICS OF NUMBERS: A NEW CONCEPT’. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, January (2022). pp 37-53.
XXVII. Prabir Chandra Bhattacharyya, ‘A NOVEL CONCEPT IN THEORY OF QUADRATIC EQUATION’. J. Mech. Cont. & Math. Sci., Vol.-17, No.-3, March (2022) pp 41-63.
XXVIII. Prabir Chandra Bhattacharyya, ‘AN OPENING OF A NEW HORIZON IN THE THEORY OF QUADRATIC EQUATION: PURE AND PSEUDO QUADRATIC EQUATION – A NEW CONCEPT’, J. Mech. Cont. & Math. Sci., Vol.-17, No.-11, November (2022) pp 1-25
XXIX. R. Narasimha. 1957. On the Distribution of Intermittence in the Transition Region of a Boundary Layer. Journal of Aeronautical Science. 711–712.
XXX. R. Narasimha, V. Saxena S. and Kailas. 2002. Coherent Structures in Plumes with and without Off-Source Heating Using Wavelet Analysis of Flow Imagery. Experiments in Fluids. 33(1):196–201.
XXXI. Riemann, Bernhard (1851). “Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse”. In H. Weber (ed.). Riemann’s gesammelte math. Werke (in German). Dover (published 1953). pp. 3–48.
XXXII. Roddam Narasimha. ‘The search for new mathematics to solve the greatest unsolved problem in classical physics’. Bhāvanā, volume 3, issue 1, January 2019.
XXXIII. Rudin 1966.
XXXIV. Rudin 1966, Theorem 11.2.
XXXV. Rudin, Walter (1966). Real and complex analysis (3rd ed.). McGraw Hill (published 1987). ISBN 0-07-054234-1.
XXXVI. See Klein, Felix (1893). On Riemann’s theory of algebraic functions and their integrals. Translated by Frances Hardcastle. Cambridge: MacMillan and Bowes.
XXXVII. Stokes, G.G. (1842), “On the steady motion of incompressible fluids”, Transactions of the Cambridge Philosophical Society, 7: 439–453, Bibcode:1848TCaPS…7..439S.
Reprinted in: Stokes, G.G. (1880), Mathematical and Physical Papers, Volume I, Cambridge University Press, pp. 1–16.
XXXVIII. “Streamfunction”, AMS Glossary of Meteorology, American Meteorological Society, retrieved 2014-01-30
XXXIX. White, F. M. (2003), Fluid Mechanics (5th ed.), New York: McGraw-Hil

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