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Uniform Flow Sink Concept

Uniform Flow

Velocity does not change with y-coordinate

There exists only one component of velocity which is in the x direction.

Magnitude of the velocity is U₀.

Using stream function ψ for uniform flow

The constants of integration C₁ and K₁ are arbitrary.

The values of ψ and Φ for different streamlines and velocity potential lines may change but flow pattern is unaltered. The constants of integration may be omitted, without any loss of generality and it is possible to write

These are plotted in Fig. 1 (a) and consist of a rectangular mesh of straight streamlines and orthogonal straight potential-lines (remember streamlines and potential lines are always orthogonal). It is conventional to put arrows on the streamlines showing the direction of flow.

In terms of polar (r - θ) coordinate, Eq. (3) becomes

Flow at an angle

If we consider a uniform stream at an angle α to the x-axis as shown in Fig. 1b. we require that

Integrating: We obtain for a uniform velocity U₀ at an angle α, the stream function and velocity potential respectively as

Source or Sink

Source flow -

(i) A flow with straight streamlines emerging from a point.

(ii) Velocity along each streamline varies inversely with distance from the point (shown in Fig. 2).

(iii) Only the radial component of velocity is non-trivial. (v_θ=0, v_z=0 ).

In a steady source flow the amount of fluid crossing any given cylindrical surface of radius r and unit length is constant

That is

(Which shows that velocity is inversely proportional to the distance )

Where, K is the source strength and ⋀is the volume flow rate

The definition of stream function in cylindrical polar coordinate states that

For the source flow,

Combining Eqs (9) and (10), we get

Because the flow is irrotational, we can write

The integration constants C₁ and C₂ in Eqs (11) and (12) have no effect on the basic structure of velocity and pressure in the flow.

The equations for streamlines and velocity potential lines for source flow become

Ø = Klnr          ψ = – k θ   (13)

K = source strength and is proportional to ⋀

⋀ = the rate of volume flow from the source per unit depth perpendicular to the page

Sink flow

(i) When ⋀ is negative , we get sink flow,

(ii) Here the flow is in the opposite direction of the source flow.

In Fig. 20.3, the point 0 is the origin of the radial streamlines. We visualize that point O is a point source or sink that induces radial flow in the neighbourhood.

The point source or sink is a point of singularity in the flow field (because vr becomes infinite).

The stream function and velocity potential function are

Ø = Klnr          ψ = – k θ (14)