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THE CONTINUITY EQUATION AND THE STREAM FUNCTION

1. The mathematical expression for the conservation of mass in flows is known as the

continuity equation:

t+ (V) = 0. (1)

2. For an incompressible flow, = constant and (1) reduces to

V = 0. (2)

3. For a two-dimensional incompressible flow in Cartesian coordinates, if{u, v}are the

x and y-components of the velocity V, then (2) reduces to

u

x

+v

y

= 0. (3)

4. This leads to the definition of the stream function ,

u=

y, v=

x. (4)

5. For a two-dimensional incompressible flow in polar coordinates, if{ur, u} are the

radial and circumferential -components of the velocity V, i.e., V =urer+ ue, then (2)reduces to

ur

r +

ur

r +

u

r = 0. (5)

6. This leads to the definition of the stream function ,

ur =

r, u =

r. (6)

7. A flow is said to be irrotational if the vorticity= V= 0. For a two-dimensionalflow, the vorticity is in the direction z perpendicular to the flow plane and its magnitudeis, in Cartesian coordinates,

z = v

x

u

y. (7)

In polar coordinates,

z = ur

+ ur

urr

. (8)

8. A potential functioncan be defined if the flow is irrotational. In cartesian coordinates:

u=

x, v =

y

In polar coordinates:

Vr =

r, V =

1

r

. (9)

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TWO-DIMENSIONAL IRROTATIONAL INCOMPRESSIBLE FLOWS

1. Uniform Flow V

= Vx

= Vy

2. Source at Origin

= Q

2lnr

= Q

2

3. Vortex Flow at Origin

=

2

=

2lnr

4. Doublet at Origin

= k

2

cos

r

= k

2

sin

r

5. Source at Point x0= (x0, y0)

= Q

2ln|xx0|

= Q

2tan1(

yy0xx0

)

6. Vortex Flow at Point x0= (x0, y0)

=

2tan1(

yy0xx0

)

= 2

ln|xx0|

7. Doublet at point x0= (x0, y0)

= k

2

xx0(xx0)2 + (yy0)2

= k

2

yy0(xx0)2 + (yy0)2

2