Basics of Fluid Mechanics, 2014a

6.1. MOMENTUM GOVERNING EQUATION 177
The velocity profile is
U
( r
) [ ( r
) ] 2
= U max 1 −
R R
(6.I.a)
Substituting equation (6.I.a) into equation (6.18)
results in
Thus,
U 2 = 1 ∫ R
2 πR 2 [U(r)] 2 2 πrdr
U 2 =(U max ) 2 ∫ 1
0
0
(6.I.b)
(
1 − ¯r
2 ) 2
¯rd¯r =
1
6 (U max) 2 (6.I.c)
U = U max
√
6
End Solution
U i
y
x
U o
θ
U o
F
U i
F
Fig a. Schematics of area impinged by a jet
for example 6.2.
Fig b. Schematics of maximum angle for
impinged by a jet.
Fig. -6.2. Schematics of area impinged by a jet and angle effects.
Example 6.2:
A jet is impinging on a stationary surface by changing only the jet direction (see Figure
6.2). Neglect the friction, calculate the force and the angle which the support has to
apply to keep the system in equilibrium. What is the angle for which maximum force
will be created?
Solution
Equation (6.11) can be reduced, because it is a steady state, to
∫
F =
out
ρU
U { }}
rn
{ ∫ U { }}
rn
{
(U · ˆn) dA − ρU (U · ˆn) dA = ṁU U o − ṁU U i
in
(6.II.a)