u v w x y z particles of constant volume but shape of volume can change STREAM

U v w x y z particles of constant volume but shape of

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u v w x y z + + = particles of constant volume, but shape of volume can change.
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STREAM FUNCTION ψ 2-D , steady flow : continuity ( ) ( ) 0 u v x y ρ ρ + = steady compressible or unsteady incompressible Define stream ψ function such that: , u v y x ψ ψ ρ ρ = = − 2 2 0 x y y x ψ ψ = ∂ ∂ ∂ ∂ continuity identically satisfied. ψ : first & second order der. exist & continuous Advantage • Continuity eq. discarded • # of unknowns (dependent variables) reduces by one. Disadvantage • Remaining velocity derivatives are increased by one order???
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( , ) d = . x y x y dx dy vdx udy V ds dm ψ ψ ψ ψ ψ ρ ρ ρ = + = − + = = JG G ¸ ( ) 0 dm = ¸ Physical significance of ψ Lines of constant ψ (d ψ =0) are lines across which mass flow They are stream lines of flow is zero ds dxi dy j = + G G G Along AB x=const. d s dyi = G G 0 . B B B B A A A A y y y B A y y y V ui v j m V ds udy dy d y ψ ψ ψ ρ ρ ψ ψ ψ = + = = = = = = JG G G JG G ¸ ρ =const.
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