## Stream Function

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# Stream Function

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 (Δ

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

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 velocityMass 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

## Continuity Equation for Two-dimensional Flow in Term of Stream Function:

∂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

ω

ω

For irritation flow, ωz = 0

∂

Which is a Laplace equation for ψ.

∂ / ∂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 [ ∂^{2}ψ / ∂x^{2}+ ∂^{2}ψ / ∂y^{2}]For irritation flow, ωz = 0

∂

^{2}ψ / ∂x^{2}+ ∂^{2}2ψ / ∂y2^{2}= 0Which is a Laplace equation for ψ.

## Properties of Stream Function (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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2. If stream (ψ ) satisfies the Laplace equation, it is possible case of an irrotational flow.

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