Basics of Fluid Mechanics, 2014a

10.1. INTRODUCTION 331
The partial derivative of any vector, Υ, with respect to time is the same direction as
the unit vector. Hence, the product of multiplication of the partial derivative with an
unit vector is
∂Υ ̂(Υ )
∂l · = ∂Υ
(10.27)
Υ ∂l
where Υ is any vector and Υ its magnitude. The right hand side of equation (10.26)
U × Ω is perpendicular to both vectors U and Ω. Hence, the dot product of vector U
with a vector perpendicular to itself must be zero. Thus equation (10.26) becomes
or
U ·∇
{}}{
U
(
∂U
∂t + d U 2 ∫ ( dP
dl 2 + g l + ρ
) ) =
(
∂U
∂t + d U 2 ∫ ( dP
dl 2 + g l + ρ
=0
{ }} {
U
U · U × Ω (10.28)
) ) =0 (10.29)
The first time derivative of equation (10.28) can be manipulated as it was done before
to get into derivative as
∂U
∂t = d ∫ ∂U
dl (10.30)
dl ∂t
Substituting into equation (10.28) writes
(
d ∂U
dl ∂t + U 2 ∫ ( ) ) dP
2 + g l + =0 (10.31)
ρ
The integration with respect or along stream line, “l” is a function of time (similar
integration with respect x is a function of y.) and hence equation (10.28) becomes
Bernoulli On A Streamline
∂U
∂t + U 2 ∫ ( ) dP
2 + g l + = f(t)
ρ
(10.32)
In these derivations two cases where analyzed the first case, for irrotational
Bernoulli’s equation is applied any where in the flow field. This requirement means
that the flow field must obey U × Υ. The second requirement regardless whether the
flow is irrotational or not, must be along a streamline where the value is only function of
the time and not location. The confusion transpires because these two cases are referred
as the Bernoulli equation while they refer to two different conditions or situations 1 .For
both Bernoulli equations the viscosity must be zero.
1 It is interesting to point out that these equations where developed by Euler but credited to the last
D. Bernoulli. A discussion on this point can be found in Hunter’s book at Rouse, Hunter, and Simon
Ince. History of hydraulics. Vol. 214. Ann Arbor, MI: Iowa Institute of Hydraulic Research, State
University of Iowa, 1957.