Proving Routh s Theorem Using a Different Approach

Transcription

1 Proving Routh s Theorem Using a Different Approach Muhammad Mushtaq Nawazish Shah, Ph.D. Ghulam Muhammad Abstract The Routh s theorem which has important applications in Fluid mechanics already exists in literature e.g. A.S. Ramsey [], L.M. Milne Thomson [], C.C. Lin [3], and P.G. Saffman [4]. In this paper, Routh s theorem has been proved using a different approach. ROUTH S THEOREM Routh s Stream Function Let ψ ( ξ, η ) be the stream function of a vortex of strength Γ at the point Q ( ζ )in the ζ plane and ψ 0 ( ξ, η ) be the stream function of the flow field (i.e. a uniform stream or any other vortex), then the stream function of the combined flow called the Routh s stream function Ψ is defined as: Ψ = ψ 0 ( ξ, η ) + Γ ψ (ξ, η ) STATEMENT Under a conformal transformation z = f (ζ), which gives motion in the z plane from that in the ζ plane, the Routh function for the new motion is given by Ψ = Ψ + Γ 4 π log d ζ d z DERIVATION Let there be a vortex of strengths Γ placed at the point P ( z ) in the z plane. Then the complex velocity potential w ( z ) in the z plane at the point ( x, y ) is given by w ( z ) = w ( z ) + i Γ π log ( z z ) () where w ( z ) is the complex velocity potential of an other flow field (i.e. a uniform stream or any other vortex). Let there be corresponding vortex of strength Γ placed at the point Q ( ζ ) in the ζ plane under the conformal transformation ζ = f ( z ). Journal of Mathematical Sciences & Mathematics Education, Vol. 3, No. 8

2 Then the complex velocity potential w ( ζ ) in the ζ plane at point ( ξ, η ) is given by: w ( ζ ) = w ( ζ ) + i Γ π log ( ζ ζ ) () Now taking the conjugate on both sides of equation (), we get log ( z z ) (3) w ( z ) = w ( z) i Γ π Subtracting equation (3) from equation (), we have w (z) w ( z) = w (z) w ( z) + i Γ π log (z z ) + i Γ log ( z z π ) i ψ (x, y) = i ψ (x, y) + i Γ π log (z z ) ( z z ) or i ψ (x, y) = i ψ (x, y) + i Γ π log z z or i ψ (x, y) = i ψ (x, y) + i Γ π log z z or ψ (x, y) = ψ (x, y) + Γ π log z z (4) Similarly, ψ (ξ, η) = ψ (ξ, η) + Γ π log ζ ζ (5) Since the complex velocity potentials w are equal at the corresponding points in the two planes, it follows from equation () and (), that w ( z ) = w ( ζ ) therefore ψ (x, y) = ψ (ξ, η) (6) From equations (4), (5), and (6), we get ψ (x, y) + Γ π log z z = ψ (ξ, η) + Γ π log ζ ζ ψ (x, y)= ψ (ξ, η) + Γ π log ζ ζ Γ π log z z = ψ (ξ, η) + Γ π log ζ ζ z z or ψ (x, y) = ψ (ξ, η) + Γ π log ζ ζ z z (7) Taking limit as z z, ζ ζ, we have ψ (x, y ) = ψ (ξ, η ) Journal of Mathematical Sciences & Mathematics Education, Vol. 3, No. 9

3 + lim z z ζ ζ Γ π log ζ ζ z z = ψ (ξ, η ) + Γ π log lim z z ζ ζ ζ ζ (8) z z Since f ( z ) = f ( z + z z ) = f [ z + ( z z ) ] or f ( z ) = f ( z ) + ( z z )! f ( z ) + ( z z )! f ( z ) + (9) But f ( z ) = ζ and f ( z ) = ζ Also f ( z ) = z = z Thus from equation (9), we get, f d ζ ( z ) = d z, z = z ζ = ζ + ( z z ) z = z + ( z z ) d ζ d z + z = z or ζ ζ = ( z z ) z = z Dividing both sides by ( z z ), we get + ( z z ) d ζ d z + z = z ζ ζ z z = z = z + ( z z d ζ ) d z + z = z Now taking limit as z z, ζ ζ, we have lim z z ζ ζ z z ζ ζ = lim z z d ζ d z + (z z ) d ζ d z +. ζ ζ = d z = z = z ζ = ζ d ζ or d z lim z z ζ ζ ζ ζ = d ζ z z (0) d z Journal of Mathematical Sciences & Mathematics Education, Vol. 3, No. 0

4 From equations (8) and (0), we get ψ (x, y ) = ψ (ξ, η ) + Γ π log d ζ () Multiplying both sides of equation () by Γ, we have Γ ψ (x, y ) = Γ ψ (ξ, η ) + Γ 4 π log d ζ () Adding ψ 0 (x, y ) on both sides of equation (), we have ψ 0 (x, y ) + Γ ψ (x, y ) = ψ 0 (x, y ) + Γ ψ (ξ, η ) + Γ 4 π log d ζ (3) But ψ 0 (x, y ) = ψ 0 (ξ, η ) Then equation (3) becomes ψ 0 (x, y ) + Γ ψ (x, y ) = ψ 0 (ξ, η ) + Γ ψ (ξ, η ) + Γ 4 π log d ζ (4) Since Ψ = ψ 0 (x, y ) + Γ ψ (x, y ) (5) Similarly Ψ = ψ 0 (ξ, η ) + Γ ψ (ξ, η ) (6) From equations (4), (5), and (6), we get Ψ = Ψ + Γ 4 π log d ζ (7)which is the Routh s theorem. GENERALIZATION Let there be n vortices of strengths Γ, Γ,., Γ n placed at the points P ( z ), P ( z ), P n ( z n ) respectively in the z plane. Let there be corresponding vortices of strengths Γ, Γ,., Γ n placed at the points Q ( ζ ), Q ( ζ )., Q n ( ζ n ) in the ζ plane under the conformal transformation ζ = f ( z ). Journal of Mathematical Sciences & Mathematics Education, Vol. 3, No.

5 Then the generalized form of the Routh s theorem is Ψ = Ψ + n Σ i = Γ i 4 π log d ζ i i (8) Muhammad Mushtaq, University of Engineering & technology, Pakistan Nawazish Shah, Ph.D, University of Engineering & technology, Pakistan Ghulam Muhammad, University of Engineering & technology, Pakistan References [] A.S. Ramsey, M.A., A Treatise on Hydrodynamics, G. Bell and Sons, Ltd. 94. [] L.M. Milne Thomson, C.B.E. Theoretical Hydrodynamics, Robert Maclehose and Co. Ltd. The University Press, Glasgow, 974. [3] C.C. Lin, On the Motion of Vortices in two Dimensions II Some Further Investigations on the Kirchhoff Routh Function, PNAS, U.S.A., Vol. 7, 94. [4] P.G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Second Edition, 99. Journal of Mathematical Sciences & Mathematics Education, Vol. 3, No.