Basics of Fluid Mechanics, 2014a

238 CHAPTER 8. DIFFERENTIAL ANALYSIS
The term e 5 αy is always positive, real value, and independent of y thus equation
(8.V.g) becomes
ρx+ ∂ρ
∂x = c 1
e 5 αy = c 3
(8.V.h)
Equation (8.V.h) is a constant coefficients first order ODE which its solution discussed
extensively in the appendix. The solution of (8.V.h) is given by
ρ = e − x2
2
⎛
⎜
⎝ c −
impossible solution ⎞
{ }} {
√ (
ix
πic3 erf
√
2
√
2
)
⎟
⎠
(8.V.i)
which indicates that the solution is a complex number thus the constant, c 3 , must
be zero and thus the constant, c 1 vanishes as well and the solution contain only the
homogeneous part and the private solution is dropped
ρ = c 2 e − x2
2 (8.V.j)
The solution is the multiplication of equation (8.V.j) by(8.V.f) transferred to
( )
ρ = c 2 e − x2 c1 sin (αt)
2
+ c 2
α
Where the constant, c 2 , is an arbitrary function of the y coordinate.
End Solution
(8.V.k)
8.3 Conservation of General Quantity
8.3.1 Generalization of Mathematical Approach for Derivations
In this section a general approach for the derivations for conservation of any quantity
e.g. scalar, vector or tensor, are presented. Suppose that the property φ is under a
study which is a function of the time and location as φ(x, y, z, t). The total amount of
quantity that exist in arbitrary system is
∫
Φ= φρdV (8.21)
sys
Where Φ is the total quantity of the system which has a volume V and a surface area
of A which is a function of time. A change with time is
DΦ
Dt = D ∫
φρdV (8.22)
Dt
Using RTT to change the system to a control volume (see equation (5.33)) yields
∫
D
φρdV = d ∫ ∫
φρdV + ρφU · dA (8.23)
Dt sys dt cv
A
sys