Mechanical Engineering Student with Fluid Mechanics Problem

[Spimon]

Hey everyone! I'm having difficulty and wondering if someone would be kind enough to help me figure out how to approach this problem. I'm a mechanical engineering student and reasonably new to fluid mechanics, and quite confused.
Thanks for any help anyone can give me

Homework Statement

http://img522.imageshack.us/img522/6383/pipedg4.jpg
http://g.imageshack.us/img522/pipedg4.jpg/1/

An incompressible fluid enters a horizontal circular pipe (at inlet 1) with uniform velocity, Uo, as shown. Pipe radius R. At some distance downstream (section 2), the fluid velocity becomes parabolic as per the equation below.

Q.Obtain an expression for the drag force by the flowing fluid on the pipe wall over the length 1-2 in terms of pressure at sections 1 and 2, fluid density p, pipe radius R and inlet velcity, Uo.

Homework Equations

Fluid Velocity @ 2.
U = U(centre) [1-(r/R)^2]
where U is the velocity at radial distance r, and U(centre) is the velocity along the centre line.

From the mention of pressure and velocity, I would guess Bernoulli's equations is needed.

The Attempt at a Solution

I'm not sure how to approach this problem. If someone could give me some pointers I'll attempt it, but at the moment I'm very lost.

 

[nilesh_pat]

Hint:The drag force on the pipe wall can be defined as a sum of the pressure forces and shear stress forces. You'll need to calculate the pressure forces first, followed by the shear stress forces. To do this, you'll need to use Bernoulli's equation to relate the pressure at each section to the velocity at that section. Once you have the pressure forces, calculate the shear stress forces using the equation for dynamic viscosity. Then you can add them together to get the total drag force.

 

[piercas]

Dear mechanical engineering student,

I understand your struggle with fluid mechanics problems and I am more than happy to help you out. First, let's start by understanding the problem and the given information. The problem describes an incompressible fluid entering a horizontal circular pipe with uniform velocity at the inlet. This means that the fluid velocity at the inlet (section 1) is equal to the velocity of the fluid at any cross-section along the pipe. We are also given the equation for the fluid velocity at section 2, which is parabolic in nature.

To solve this problem, we will need to use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid in a system. We will also need to use the continuity equation, which states that the mass flow rate at any cross-section in a pipe is constant.

Now, let's focus on the specific question of obtaining an expression for the drag force on the pipe wall. The drag force is caused by the pressure difference between the two sections (1 and 2) and the frictional force between the fluid and the pipe wall. To find the pressure difference, we can use Bernoulli's equation at sections 1 and 2:

P1 + 1/2ρU1^2 = P2 + 1/2ρU2^2

where P is the pressure, ρ is the fluid density, and U is the velocity at each section. Since the density is constant, we can rearrange this equation to solve for the pressure difference:

P1 - P2 = 1/2ρ(U2^2 - U1^2)

Next, we can use the continuity equation to relate the velocities at sections 1 and 2:

A1U1 = A2U2

where A is the cross-sectional area of the pipe. Since the pipe is circular, we can substitute A = πR^2. Now, we can solve for U2 in terms of U1 and plug it into our equation for the pressure difference:

P1 - P2 = 1/2ρ(U1^2[(r/R)^2 - 1])

Finally, we can use the definition of drag force (Fd = P1 - P2) and substitute our expression for the pressure difference to obtain the final equation for the drag force on the pipe wall:

Fd = 1/2ρ(U1^2[(r/R)^2 - 1])