How is a two-dimensional irrotational flow expressed in cylindrical coordinates?

Two-Dimensional Irrotational Flow in Cylindrical Coordinates In a two-dimensional flow pattern, we can automatically satisfy the incompressibility constraint, , by expressing the pattern in terms of a stream function. Suppose, however, that, in addition to being incompressible, the flow pattern is also irrotational.

Is the irrotational flow pattern also incompressible?

Suppose, however, that, in addition to being incompressible, the flow pattern is also irrotational. In this case, Equation ( 5.10) yields Let us search for a separable solution of Equation ( 5.61) of the form

How is the velocity potential of an irrotational flow defined?

It is to be noted, however, that the velocity potential can be defined for a general three-dimensional flow, whereas the stream function is restricted to two-dimensional flows. For an incompressible flow we know from the conservation of mass: ∇⋅=V0 and therefore for incompressible, irrotational flow, it follows that ∇2φ=0

What is the continuity equation for incompressible flow?

where q is the volumetric flow rate per unit width between streamlines ψ1 and ψ2 . If ( ψ2 – ψ1 ) is positive, the flow is from left to right; whereas if ( ψ2 – ψ1 ) is negative, the flow is in the opposite direction (from right to left). For incompressible flow in the two-dimensional cylindrical coordinate system, the continuity equation is

Why are cylindrical coordinates used for axisymmetric flow?

Axisymmetric flow Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangential velocity ((u_{theta}=0)), and the remaining quantities are independent of (theta).