Continuum and Kinematics
Change of Volume/Area, ATP
Solution:
Solution 1
Consider the originally rectangular differential particle with sides
δ
x and
δ
y, formed by the points a,b,c and
d, located at (x,y), (x+
δ
x,y), (x, y+
δ
y) and (x+
δ
x, y +
δ
y) respectively, shown in figure 1.
Figure 1: Differential Area Element
As the differential mass element moves accordingly to the velocity field, it’s area and geometrical shape
changes. The corners of the original differential element, a,b,c,d, become a’, b’, c’ and d’.
The distances travelled by the corners of the rectangle, a’-a, b’-b, c’-c, and d’-d, can be calculated from
the velocity field (after all, the velocity field is the instantaneous velocity of the particles at the given point)
and adding this amount to the original position, the final position of the points can be calculated.
First, for the distances:
A
= (
u
(
x, y
)
, v
(
x, y
))
δt
+
O
(
δt
2
)
(1)
B
= (
u
(
x
+
δx, y
)
, v
(
x
+
δx, y
))
δt
+
O
(
δt
2
)
(2)
C
= (
u
(
x, y
+
δy
)
, v
(
x, y
+
δy
))
δt
+
O
(
δt
2
)
(3)
2.25 Advanced Fluid Mechanics
2
Copyright
c
2010, MIT