Chapter 9 Viscous flow

10 CHAPTER 9. VISCOUS FLOW
x
3
x 1
x 2
g
U
Figure 9.3: Sphere settling in a fluid.
condition (∂u i /∂x i ) = 0 can be simplified as
∂u i
∂x i
(
∂ 1 ∂
= A 1 U i
∂x i r + A δ ij
2U j
∂x i r − 3 ∂ )
x i x j
+ A 3U j
3 ∂x i r 5 2µ
= − A 1U i x i
r 3 + A 2 U j
(
+ A 3U j
2µ
= − A 1U i x i
r 3
(
δii x j
r 3
+ δ ijx i
r 3
− 3δ ijx i
r 5
− 3δ ijx i
r 5
− 3x2 i x )
j
r 5
− 3δ iix j
r 5
∂ x i x j
∂x i r 3
)
+ 15x2 ix j
r 7
+ A 3U j x j
2µr 3 (9.30)
In simplifying equation 9.30, we have used x 2 i = (x 2 1+x 2 2+x 2 3) = r 2 . Therefore,
from the incompressibility condition, we get A 1 = (A 3 /2µ). Inserting this
into equation 9.30, the velocity field is
(
δij
u i = A 1 U j
r + x ) (
ix j δij
+ A
r 3 2 U j
r − 3x )
ix j
(9.31)
3 r 5
The constants A 1 and A 2 are determined from the boundary condition at the
surface of the sphere (r = 1),
u i | r=1
= A 1 U i + A 1 U j x i x j + A 2 U i − 3A 2 U j x i x j
= U i (9.32)
From this, we get two equations for A 1 and A 2 ,
A 1 + A 2 = 1
A 1 − 3A 2 = 0 (9.33)