It is defined as the scalar function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to that direction. it si denoted by ( ) and defined only two-dimensional flow.


For steady flow,  ψ = f(x,y) such that

∂ψ / ∂x  =  v

∂ψ / ∂y  =  -u

Consider two streamline in a flow at a distance of Δ with a unit depth.

Mass flow rate between two rate between two
streamlines ψ1 and ψ2 is ψ1 - ψ2

=  ρ. A. V

=  ρ  (Δ_n x 1) (v)                  where, v = total velocity

Mass flow rate in x-direction   =  ρ. Area . Velocity

= ρ (-dy x 1) u

Mass flow rate in direction = ρ dx v

Total mass flow rate,

                      = Mass flow rate in x direction + Mass flow rate in y-direction

=  - ρ dy u + ρ dx v

We know, ψ is a function of x and y i. e. ψ = f(x,y),

dψ  = ∂ψ / ∂x dx  + ∂ψ / ∂y dy

From Equation

∂ψ / ∂x  = +  ρ v            ∂ψ / ∂y = - ρu

For incompressible flow.

∂ψ / ∂x = v               ∂ψ / ∂y = -u

∂u / ∂x  +  ∂v / ∂y  =  0

∂ / ∂x  (- ∂ψ / ∂y) + ∂ / ∂y  (- ∂ψ / ∂x)  = 0

- ∂2ψ / ∂x / ∂y + ∂2ψ / ∂y / ∂x  = 0
Hence existence of ( ) means a possible case of fluid flow. The flow may be rotational or irrotational.

Rotational component,      ω_z  = 1/2 [ ∂v / ∂x - ∂u / ∂y ]

ω_z  =  1/2 [ ∂ / ∂x ( ∂ψ / ∂x) - ∂ / ∂y ( - ∂ψ / ∂y) ]

ω_z  = 1/2 [ ∂²ψ  / ∂x² + ∂²ψ / ∂y² ]

For irritation flow,     ωz  = 0

∂²ψ / ∂x² + ∂²2ψ / ∂y2²  =  0

Which is a Laplace equation for ψ.

1.    It stream function ( ψ) exists, it is a possible case of fluid flow which may be rotational or irrotational.
2.    If stream (ψ ) satisfies the Laplace equation, it is possible case of an irrotational flow.

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