Viscous Flows, Homework Sets 1
Due on Oct. 18, 2018
Q1:
Prove the following vector calculus identities using the index notation
(i)
∇ ×
(
∇
φ
) = 0, where
φ
is a scalar function.
(ii)
∇ ·
(
∇ ×
u
) = 0, where
u
is a vector function.
(iii)
u
×
(
∇ ×
u
) =
1
2
∇
(
u
·
u
)

(
u
· ∇
)
u
, where
u
is a vector function.
(iv)
∇ ×
(
∇ ×
u
) =
∇
(
∇ ·
u
)
 ∇
2
u
, where
u
is a vector function and
∇
2
is
the Laplacian operator.
Q2:
A steady, twodimensional velocity field is given by
u
(
x, y
) =(
ay
)
ˆ
i
+ (
bx
)
ˆ
j
,
where
a
and
b
are positive constants.
ˆ
i
and
ˆ
j
are unit vectors in the
x
and
y
direction, respectively.
(a) Find the angular velocity of the fluid elements in this flow field.
(b) Compute the twodimensional strain rate tensor ˙ of this flow field.
(c) Find the principal strain rates and the directions of principal axes of the
strain rate tensor.
(d) Compute the total derivate of the velocity field in a frame of reference
moving with a velocity,
V
=
c
ˆ
j
, where
c
is a positive constant.
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 Fall '13
 Fluid Dynamics, Derivative, Work, Vector field