In regions far from the entrance, fluid flow through a circular pipe is one-dimensional and the velocity profile for laminar flow is given by

Name :_________________ Quiz: No. 1 Time: 15 minutes Student ID# : ____________ Course: ME 5160, Fall 2021

--- The exam is closed book and closed notes.

In regions far from the entrance, fluid flow through a circular pipe is one-dimensional and the velocity profile for laminar flow is given by

𝑢(𝑟) = 𝑢_𝑚𝑎𝑥(1 −𝑟² 𝑅²)

where 𝑅 is the radius of the pipe, 𝑟 is the radial distance from the center of the pipe, and 𝑢_𝑚𝑎𝑥 is the maximum flow velocity, which occurs at the center. Obtain (a) a relation for the drag force applied by the fluid on a section of the pipe of length 𝐿 and (b) the value of the drag force for water flow at 20°C with 𝑅 = 0.095 m, 𝐿 = 17 m, 𝑢_𝑚𝑎𝑥 = 2.5 m/s, and 𝜇 = 0.0010 kg/m · s.

Equation: 𝜏 = 𝜇^𝑑𝑢

𝑑𝑦

Name :_________________ Quiz: No. 1 Time: 15 minutes Student ID# : ____________ Course: ME 5160, Fall 2021

--- Solution

KNOWN: u(r) FIND: FD

ASSUMPTIONS: The flow is one-dimensional; the fluid is Newtonian ANALYSIS:

(a)

The shear stress at the pipe surface can be expressed as:

𝜏_𝑤 = 𝜇𝑑𝑢 𝑑𝑦|

𝑤𝑎𝑙𝑙

= −𝜇𝑑𝑢 𝑑𝑟|

𝑟=𝑅

𝜏_𝑤 = −𝜇𝑢_𝑚𝑎𝑥 ^𝑑

𝑑𝑟(1 −𝑟² 𝑅²)

𝑟=𝑅

= −𝜇𝑢_𝑚𝑎𝑥−2𝑟 𝑅² |

𝑟=𝑅

=2𝜇𝑢_𝑚𝑎𝑥 𝑅

The friction drag force exerted by the fluid on the inner surface of the pipe becomes:

𝐹_𝐷 =𝜏_𝑤𝐴 =(2𝜇𝑢_𝑚𝑎𝑥

𝑅 ) (2𝜋𝑅𝐿)= 4𝜋𝜇𝐿𝑢_𝑚𝑎𝑥 (b)

Substituting,

𝐹_𝐷 =4𝜋𝜇𝐿𝑢_𝑚𝑎𝑥 =4𝜋(0.0010 kg/m · s)(17 m)(2.5 m/s)= 0.534 N +2

+2 +2

+2

+2