Fluid Mechanics Calculations - Fluid Velocity, Drag Force, Shear Stress

[engineering11]

Fluid Mechanics IMP and URGENT

Can someone please help me solve these questions ASAP thnx

Problem #1
A fluid having a specific gravity of 0.6 and a viscosity of 0.00035 N∙s/m2 flows through a circular duct which is 300 mm in diameter. The average speed of fluid is 15 m/s. The duct is commercial steel. Find the velocity of the fluid at 6 mm from the wall and determine the drag force of the fluid on a 5 m length of the duct.
Problem #2
Flowing through a pipe of 30 cm diameter is 0.28 m3/s of water at 20 ◦C. The pipe is very smooth. Estimate the shear stress at the wall and the velocity at a distance of 0.02 mm from the wall.
Problem #3
Helium (ν = 1.3x 10-4 m2/s and ρ = 0.15 kg/ m3) is flowing through a 30 cm. diameter duct with an average velocity of 6 m/s. The velocity is measured to be 0.3 m/s at a distance 0.1 mm from the wall. It is known that this point is within the laminar sub-layer. Determine
a) shear velocity for the flow
b) shear stress at the wall
c) laminar sub-layer thickness

 

[KataKoniK]

Sure, I can help you with these problems! Here are the solutions to each problem:

Problem #1:
a) To find the velocity of the fluid at 6 mm from the wall, we can use the formula for the velocity profile in a circular duct:

u = u_max (1 - (r/R)^2)

Where u is the velocity at a distance r from the center, u_max is the maximum velocity (which we know is 15 m/s), and R is the radius of the duct (which is half of the diameter, or 150 mm). Plugging in the values, we get:

u = 15 (1 - (6/150)^2) = 14.98 m/s

So the velocity at 6 mm from the wall is approximately 14.98 m/s.

b) To determine the drag force on a 5 m length of the duct, we can use the formula for drag force:

F = 0.5 * ρ * u^2 * A * C_D

Where ρ is the density of the fluid, u is the velocity, A is the cross-sectional area of the duct, and C_D is the drag coefficient (which we can assume to be 1 for a circular duct). Plugging in the values, we get:

F = 0.5 * 0.6 * 14.98^2 * π * (0.15)^2 * 5 * 1 = 21.27 N

So the drag force on a 5 m length of the duct is approximately 21.27 N.

Problem #2:
To estimate the shear stress at the wall, we can use the formula for shear stress in a pipe:

τ = 4 * μ * u / D

Where τ is the shear stress, μ is the fluid viscosity, u is the velocity, and D is the diameter of the pipe. Plugging in the values, we get:

τ = 4 * 0.00035 * 0.28 / 0.3 = 0.00131 N/m^2

So the shear stress at the wall is approximately 0.00131 N/m^2.

To find the velocity at a distance of 0.02 mm from the wall, we can use the same velocity profile formula as in Problem #1, but with a smaller radius (since we are now looking at a smaller distance from the wall).