To calculate the pressure drop, average shear stress at the wall of the pipe, and the power required to maintain the flow, we can use the equations of fluid mechanics. Let's start with part (a), where the pipe is horizontal.
Pressure Drop:
The pressure drop in a pipe can be calculated using Bernoulli's equation:
P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
Since the pipe is horizontal, the height difference (h1 - h2) is zero and we can neglect the velocity head (1/2ρv1^2 - 1/2ρv2^2). Therefore the equation becomes:
P1 = P2
The pressure drop is zero.
Average Shear Stress:
The average shear stress at the wall of the pipe can be calculated using the formula:
τ = μdu/dy
where τ is the shear stress, μ is the dynamic viscosity, du/dy is the velocity gradient, and y is the distance from the wall. Since the flow is assumed to be laminar, the velocity gradient is constant and given by:
du/dy = (Δv/Δy)
Where Δv is the change in velocity and Δy is the distance from the wall. In this case, since the pipe is horizontal and the pressure drop is zero, there is no change in velocity, and therefore the shear stress at the wall is also zero.
Power Required:
The power required to maintain the flow can be calculated using the equation:
Power = Flow Rate * Pressure Drop
Since the pressure drop is zero, the power required to maintain the flow is also zero.
For part (b), we can repeat the same calculations for each case:
If the pipe is inclined at 15 degrees with the horizontal and the flow is in the upward direction: The pressure drop will be non-zero, the average shear stress at the wall will be non-zero, and the power required to maintain the flow will also be non-zero.
If the pipe is inclined at 15 degrees with the horizontal and the flow is in the downward direction: The pressure drop will be non-zero, the average shear stress at the wall will be non-zero, and the power required to maintain the flow will also be non-zero.