Basics of Fluid Mechanics, 2014a

9.4. SUMMARY OF DIMENSIONLESS NUMBERS 313
In Table 10.2 Mach and Eckert numbers are defined as
Ec = U 2
M = √ U
C p ΔT
P
ρ
(9.XIII.a)
The material which obeys the ideal flow model 18 (P/ρ = RT and P = C 1 ρ k ) can be
written that
/√
P
M = U
ρ = U
√ (9.XIII.b)
kRT
For the comparison, the reference temperature used to be equal to zero. Thus Eckert
number can be written as
√ U Ec = √
Cp T = U
( )
Rk
T
√ k − 1
} {{ }
C p
=
√
k − 1 U
√
kRT
= √ k − 1 M
(9.XIII.c)
The Eckert number and Mach number are related under ideal gas model and isentropic
relationship.
End Solution
Brinkman number measures of the importance of the viscous heating relative the
conductive heat transfer. This number is important in cases when a large velocity
change occurs over short distances such as lubricant, supersonic flow in rocket mechanics
creating large heat effect in the head due to large velocity (in many place it is a
combination of Eckert number with Brinkman number. The Mach number is based on
different equations depending on the property of the medium in which pressure disturbance
moves through. Cauchy number and Mach number are related as well and see
Example 9.15 for explanation.
Example 9.14:
For historical reason some fields prefer to use certain numbers and not other ones.
For example in Mechanical engineers prefer to use the combination Re and We number
while Chemical engineers prefers to use the combination of Re and the Capillary number.
While in some instances this combination is justified, other cases it is arbitrary. Show
what the relationship between these dimensionless numbers.
Solution
The definitions of these number in Table 10.2
We = ρU2 l
σ
Re = ρUl
μ
Ca = μU
σ
= U σ
μ
(9.XIV.a)
18 See for more details http://www.potto.org/gasDynamics/node70.html