Streamlines

Streamlines are imaginary lines in the flow field such that the velocity at all points on these lines are always tangential to them. Flows are usually depicted graphically with the aid of streamlines. Streamlines proceeding through the periphery of an infinitesimal area at some time t forms a tube called streamtube, which is useful for the study of fluid flow phenomena. From the definition of streamlines, it can be inferred that:

• Flow cannot cross a streamline, and the mass flow between two streamlines is conserved.

• Based on the streamline concept, a function ф called stream function can be defined. The velocity components of a flow field can be obtained by differentiating the stream function.

In terms of stream function ф, the velocity components of a two-dimensional incompressible flow are given as:

It is important to note that the stream function is defined only for two-dimensional flows, and the definition does not exist for three-dimensional flows. Even though some books define ф for axisymmetric flow, they again prove to be equivalent to two-dimensional flow. We must realize that the definition of ф does not exist for three-dimensional flows, because such a definition demands a single tangent at any point on a streamline, which is not possible in three-dimensional flows.

2.8.1 Relationship between Stream Function and Velocity Potential

For irrotational flows (the fluid elements in the field are free from rotation), there exists a function ф called velocity potential or potential function. For a steady two-dimensional flow, ф must be a function of two space coordinates (say, x and y). The velocity components are given by:

From Equations (2.31) and (2.33), we can write:

(2.34)

These relations between stream function and potential function, given by Equation (2.34), are the famous Cauchy-Riemann equations of complex-variable theory. It can be shown that the lines of constant ф or potential lines form a family of curves which intersect the streamlines in such a manner as to have the tangents of the respective curves always at right angles at the point of intersection. Hence, the two sets of curves given by ф = constant and ф = constant form an orthogonal grid system or flow-net. That is, the streamlines and potential lines in flow field are orthogonal.

Unlike stream function, potential function exists for three-dimensional flows also, because there is no condition like the local flow velocity must be tangential to the potential lines imposed in the definition of ф. The only requirement for the existence of ф is that the flow must be potential.