appendix_D - D Some simple flows and their potentials D.1...

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Unformatted text preview: D Some simple flows and their potentials D.1 Two dimensional flows We will give the answer either in Cartesians or polars, depending on which is more convenient. Reminder: ∂ψ ∂φ ∂ψ ∂φ = , v= =− where φ is the ∂x ∂y ∂y ∂x velocity potential and ψ is the streamfunction. (i) in Cartesians, u = (u, v, 0) and u = (ii) In polars, u = (ur , uθ , 0) and ur = Type of flow 1 ∂ψ 1 ∂φ ∂ψ ∂φ = , uθ = =− . ∂r r ∂θ r ∂θ ∂r flow field u Horizontal dipole, strength µ, at r = 0 − Uz Ur cos(θ − α) Ur sin(θ − α) Uze−iα k2 (x − y 2 ) 2 kxy k2 z 2 m log r 2π mθ 2π m ln z 2π ˆ Γθ 2πr Line vortex, circulation Γ, at r = 0 Uy mˆ r 2πr Line source, strength m, at r = 0 Ux u = kx v = −ky Stagnation point flow at origin complex pot. w ˆ U cos αx+ ˆ U sin αy Uniform stream at angle α to x axis streamfunction ψ ˆ Ux Uniform stream parallel to x axis potential φ Γθ 2π µ z ˆ r z−2 ˆ 2 2πr r −µ cos θ 2πr − Γ log r 2π µ sin θ 2πr − − D.2 Axisymmetric flows We use cylindrical or spherical polars, whichever is more convenient. In cylindrical coordinates, (r, θ, z ), u = (ur , uθ , uz ); in terms of the potential φ(r, z ) or the Stokes streamfunction Ψ(r, z ), ∂φ 1 ∂Ψ ∂φ 1 ∂φ ur = =− , uz = = . ∂r r ∂z ∂z r ∂r In spherical polar coordinates, (r, θ, ϕ), u = (ur , uθ , uϕ ); in terms of the potential φ(r, θ) or the Stokes streamfunction Ψ(r, θ), ur = 1 ∂Ψ ∂φ =2 , ∂r r sin θ ∂θ uθ = 1 ∂φ 1 ∂Ψ =− . r ∂θ r sin θ ∂r iΓ ln z 2π µ 2πz Type of flow Uniform stream aligned with axis of symmetry Stagnation point flow at origin Point strength r=0 source, m, at flow field u potential φ ˆ Uz Uz k r 2 uz = −kz k2 (r − 2 z 2 ) 4 ur = m ˆ r 4πr 2 µ z ˆ−3 ˆ Dipole, strength µ, − r z 3 4πr r in ˆ-direction z − m 4πr −µ cos θ 4πr 2 streamfunction Ψ coordinate system 1 Ur 2 2 − kr 2 z 2 cylindrical cylindrical m (1 − cos θ) 4π spherical sin2 θ 4πr spherical µ ...
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This note was uploaded on 01/20/2012 for the course MATH 33200 taught by Professor Eggers during the Fall '11 term at University of Bristol.

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appendix_D - D Some simple flows and their potentials D.1...

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