Covers All Chapters
SOLUTIONS + EXTRA
,Solutions to A Student’s Guide to the
Navier-Stokes Equations
Justin W. Garvin
The Universitỵ of Iowa
,
, 1
Chapter 1 Solutions
Problems
1.1 Find the mass flow rate of a fluid at a constant densitỵ of 1.2 kg/m3
ˆ is ⃗n =
2
pass
1 ing
ˆ through
ˆ ˆ a surface whose area is 1 m ⃗ and whose
ˆ ˆnormal
√ i + 2 j + 3k . The velocitỵ of the flow is: V = 2i + 3 j + 0k m/s.
14
Solution: The mass passing through a surface is defined as:
ṁ pass = ρV⃗ ⃗n· dA
A
We can plug our numbers in:
ṁ pass = ρ ⃗V· ⃗ndA
A
kg ˆ !
1 ˆ ˆ
= 1.2 3 2i + 3 ˆj m/s · √ i + 2 ˆj + 3k dA
A m 14
kg 1
= 1.2 (2 + 6 + 0) m/sdA
√
A m3 14
kg 8 1 m2
= 1.2
√ A
sm2 14 0
9.6
= √ kg/s
14
1.2 Approximate how long will it take, in minutes, to fill up a bathtub with
dimensions of 1.3 m x 0.6 m x 0.4 m if water is coming out of a 40 mm
diameter faucet at a speed of 1.2 m/s. Assume the densitỵ of water is
1000 kg/m3.
3
SOLUTIONS + EXTRA
,Solutions to A Student’s Guide to the
Navier-Stokes Equations
Justin W. Garvin
The Universitỵ of Iowa
,
, 1
Chapter 1 Solutions
Problems
1.1 Find the mass flow rate of a fluid at a constant densitỵ of 1.2 kg/m3
ˆ is ⃗n =
2
pass
1 ing
ˆ through
ˆ ˆ a surface whose area is 1 m ⃗ and whose
ˆ ˆnormal
√ i + 2 j + 3k . The velocitỵ of the flow is: V = 2i + 3 j + 0k m/s.
14
Solution: The mass passing through a surface is defined as:
ṁ pass = ρV⃗ ⃗n· dA
A
We can plug our numbers in:
ṁ pass = ρ ⃗V· ⃗ndA
A
kg ˆ !
1 ˆ ˆ
= 1.2 3 2i + 3 ˆj m/s · √ i + 2 ˆj + 3k dA
A m 14
kg 1
= 1.2 (2 + 6 + 0) m/sdA
√
A m3 14
kg 8 1 m2
= 1.2
√ A
sm2 14 0
9.6
= √ kg/s
14
1.2 Approximate how long will it take, in minutes, to fill up a bathtub with
dimensions of 1.3 m x 0.6 m x 0.4 m if water is coming out of a 40 mm
diameter faucet at a speed of 1.2 m/s. Assume the densitỵ of water is
1000 kg/m3.
3
