Basics of Fluid Mechanics, 2014a

8.2. MASS CONSERVATION 233
Equation (8.I.e) is first order ODE with the boundary condition U y (y =0)=0which
can be arranged as
(
∂ U y α H (
0−y
))
H 0
1 − e
−βt
∂y
= −αβ
( )
H0 − y
e −βt
H 0
(8.I.f)
U y is a function of the time but not y. Equation (8.I.f) holds for any time and thus, it
can be treated for the solution of equation (8.I.f) as a constant 5 . Hence, the integration
with respect to y yields
(
U y α H 0 − y (
1 − e
−βt )) ( ) 2 H0 − y
= −αβ
e −βt y + c (8.I.g)
H 0
2 H 0
Utilizing the boundary condition U y (y =0)=0yields
(
U y α H 0 − y (
1 − e
−βt )) ( ) 2 H0 − y
= −αβ
e −βt (y − 1)
H 0
2 H 0
(8.I.h)
or the velocity is
U y = β
( )
2 H0 − y e −βt
2(H 0 − y) (1 − e −βt (1 − y) (8.I.i)
)
It can be noticed that indeed the velocity is a function of the time and space y.
End Solution
8.2.2 Simplified Continuity Equation
A simplified equation can be obtained for a steady state in which the transient term is
eliminated as (in a vector form)
∇ · (ρU) =0 (8.19)
If the fluid is incompressible then the governing equation is a volume conservation as
∇ · U =0 (8.20)
Note that this equation appropriate only for a single phase case.
Example 8.2:
In many coating processes a thin film is created by a continuous process in which liquid
injected into a moving belt which carries the material out as exhibited in Figure 8.4.
5 Since the time can be treated as a constant for y integration.