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appendix_D - D Some simple ows and their potentials D.1 Two...

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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 con- venient. Reminder: (i) in Cartesians, u = ( u,v, 0) and u = ∂φ ∂x = ∂ψ ∂y , v = ∂φ ∂y = - ∂ψ ∂x where φ is the velocity potential and ψ is the streamfunction. (ii) In polars, u = ( u r ,u θ , 0) and u r = ∂φ ∂r = 1 r ∂ψ ∂θ , u θ = 1 r ∂φ ∂θ = - ∂ψ ∂r . Type of flow flow field u potential φ streamfunction ψ complex pot. w Uniform stream par- allel to x axis U ˆ x Ux Uy Uz Uniform stream at angle α to x axis U cos α ˆ x + U sin α ˆ y Ur cos( θ - α ) Ur sin( θ - α ) Uze - Stagnation point flow at origin u = kx v = - ky k 2 ( x 2 - y 2 ) kxy k 2 z 2 Line source, strength m , at r = 0 m ˆ r 2 πr m 2 π log r 2 π m 2 π ln z Line vortex, circula- tion Γ, at r = 0 Γ ˆ θ 2 πr Γ θ 2 π - Γ 2 π log r - i Γ 2 π ln z Horizontal dipole, strength μ , at r = 0 - μ 2 πr 2 parenleftBig ˆ z - 2 z r ˆ r parenrightBig - μ cos θ 2 πr μ sin θ 2 πr - μ 2 πz D.2 Axisymmetric flows We use cylindrical or spherical polars, whichever is more convenient. In cylindrical coor- dinates, ( r,θ,z ), u = ( u r ,u θ ,u z ); in terms of the potential φ ( r,z ) or the Stokes stream-
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