Applications of Conformal Mapping to Complex Velocity Potential of the Flow of an Ideal Fluid

Mohammed Mukhtar Mohammed Zabih; R.M. Lahurikar

doi:https://doi.org/10.14445/22315373/IJMTT-V52P528

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 52 | Number 3 | Year 2017 | Article Id. IJMTT-V52P528 | DOI : https://doi.org/10.14445/22315373/IJMTT-V52P528

Applications of Conformal Mapping to Complex Velocity Potential of the Flow of an Ideal Fluid

Mohammed Mukhtar Mohammed Zabih, R.M. Lahurikar

Citation :

Mohammed Mukhtar Mohammed Zabih, R.M. Lahurikar, "Applications of Conformal Mapping to Complex Velocity Potential of the Flow of an Ideal Fluid," International Journal of Mathematics Trends and Technology (IJMTT), vol. 52, no. 3, pp. 196-202, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V52P528

Abstract

In this paper we have discussed, an application of conformal mapping to the problems of finding complex velocity potential function Ω(z) for an irrotational flow of an incompressible fluid, that is, the flow of an ideal fluid in a domain D of the z-plane. In this application, our idea is to device an analytic mapping (in fact conformal mapping) from the z-plane to the w-plane, which maps the domain D conformally on to the domain D’ (precisely, either horizontal strip or vertical strip) in the w-plane, where the solution of problem is easy to find. The advantage of this technique is that, the theory of conformal mapping can be employed to reduce a problem to a simpler one whose solution is known. Determining the velocity potential Φ(u,v) in the w-plane and sending back to Φ(x,y) in the z-plane, gives the complex velocity potential Ω(z)= Φ + iψ, where ψ is a stream function. This technique is tested through examples.

Keywords

complex velocity potential, conformal mapping, Laplace equation, ideal fluid, analytic function.

References

1. J.L. Bansal, Viscous Fluid Dynamics, India Book House Pvt Ltd, (2003).
2. F. Chorlton, Textbook of Fluid Dynamics,Cbs Publishers & Distributors, (2004).
3. R. V. Churchrill, J. Brown and R. Verhey, Complex variable and application, McGraw Hill Co, New York, (1974).
4. J.B. Conway, Functions of One Complex Variable I, Springer-Verlag New York, second edition,(1978).
5. D. T.Piele, M. W. Firebaugh and R. Manulik, Application to conformal mapping to potential theory through computer graphics, American Mathematical month, (1977), 677-692.
6. Schliscting H. Boundary layer theory, McGraw Hill Co, New York, (1968).
7. I. N. Sneddon, Elements of Partial Differential Equations, McGraw- Hill Book Company, INC. Kogakusha Company, LTD. Tokyo.(1957).
8. D. G. Zill and Patrick D. Shanahan, A first Course in Complex Analysis with Applications, Second Edition. Jones and Bartlett Publishers.(2010).