Consider incompressible flow in the entrance of a circular...

Consider incompressible flow in the entrance of a circular tube, as in Fig. P3.53. The inlet flow is uniform, $u_{1}=U_{0}$. The flow at section 2 is developed pipe flow. Find the wall drag force $F$ as a function of $\left(p_{1}, p_{2}, \rho, U_{0}, R\right)$ if the flow at section 2 is (a) Laminar: $u_{2}=u_{\max }\left(1-\frac{r^{2}}{R^{2}}\right)$ (b) Turbulent: $u_{2} \approx u_{\max }\left(1-\frac{r}{R}\right)^{1 / 7}$

Consider incompressible flow in the entrance of a circular tube, as in Fig. P3.53. The inlet flow is uniform, $u_{1}=U_{0}$. The flow at section 2 is developed pipe flow. Find the wall drag force $F$ as a function of $\left(p_{1}, p_{2}, \rho, U_{0}, R\right)$ if the flow at section 2 is
(a) Laminar: $u_{2}=u_{\max }\left(1-\frac{r^{2}}{R^{2}}\right)$
(b) Turbulent: $u_{2} \approx u_{\max }\left(1-\frac{r}{R}\right)^{1 / 7}$

Key Concepts

Pressure Forces and Wall Drag

The pressure difference between the inlet and the developed flow section drives the fluid motion and is balanced by the wall drag force, which arises due to shear stress at the tube wall. This concept ties together the macroscopic pressure drop with the microscopic frictional effects, allowing one to relate the overall drag force to the given parameters such as pressures, density, velocity, and geometry of the tube.

Laminar and Turbulent Flow Characteristics

These two flow regimes exhibit distinct velocity distributions, pressure drops, and wall shear stresses. In laminar flow, the parabolic velocity profile results in a well?defined analytical expression for momentum flux, whereas the turbulent flow with a flatter profile near the center leads to different integration results. Understanding these differences is vital for correctly formulating the wall drag force in each case.

Velocity Profiles in Pipe Flow

The velocity profile describes how the fluid speed varies across the cross-section of the pipe. In laminar flow, the profile is parabolic (Poiseuille flow), while in turbulent flow, it follows a 1/7th power law approximation. These profiles are crucial because they determine the distribution of momentum across the section and directly affect the integrated momentum flux and wall shear stress calculations.

Conservation of Linear Momentum

This concept involves applying Newton’s second law to a control volume in fluid flow, balancing the pressure forces and the momentum flux change across the control volume boundaries. In this problem, it is used to equate the net pressure force driving the flow with the integrated momentum changes and the drag force due to wall friction. This analysis forms the foundation for determining the wall drag force in terms of pressure differences and fluid properties.

Control Volume Analysis

A control volume is a defined region in space through which fluid flows, and the analysis involves summing forces and fluxes across its boundaries. Here, a control volume is chosen that spans from the inlet to the developed flow section of the tube to account for the momentum flux differences between uniform and developed velocity profiles. This method is essential for connecting macroscopic quantities like wall drag with local flow parameters.

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