Basics of Fluid Mechanics, 2014a

338 CHAPTER 10. POTENTIAL FLOW
∇U =0and vector identity of ∇·∇U =0where in this case U is any vector. As
opposed to two dimensional case, the stream function is defined as a vector function as
B = ψ ∇ξ (10.61)
The idea behind this definition is to build stream function based on two scalar functions
one provide the “direction” and one provides the the magnitude. In that case, the
velocity (to satisfy the continuity equation)
U = ∇×(ψ ∇χ) (10.62)
where ψ and χ are scalar functions. Note while ψ is used here is not the same stream
functions that were used in previous cases. The velocity can be obtained by expanding
equation (10.62) to obtained
U = ∇ψ ×∇χ + ψ
=0
{ }} {
∇×(∇χ) (10.63)
The second term is zero for any operation of scalar function and hence equation (10.63)
becomes
U = ∇ψ ×∇χ (10.64)
These derivations demonstrates that the velocity is orthogonal to two gradient vectors.
In another words, the velocity is tangent to the surfaces defined by ψ = constant and
χ = constant. Hence, these functions, ψ and χ are possible stream functions in three
dimensions fields. It can be shown that the flow rate is
˙Q =(ψ 2 − ψ 1 )(χ − χ 1 ) (10.65)
The answer to the question whether this method is useful and effective is that in some
limited situations it could help. In fact, very few research papers deals this method and
currently there is not analytical alternative. Hence, this method will not be expanded
here.
End Advance material
10.2.3 The Connection Between the Stream Function and the
Potential Function
For this discussion, the situation of two dimensional incompressible is assumed. It was
shown that
and
U x = ∂φ
∂x = ∂ψ
∂y
U y = ∂φ
∂y = −∂ψ ∂x
(10.66)
(10.67)