An incompressible fluid flows steadily in the entrance...

An incompressible fluid flows steadily in the entrance region of a circular tube of radius $R=75 \mathrm{mm}$. The flow rate is $Q=0.1 \mathrm{m}^{3} / \mathrm{s}$. Find the uniform velocity $U_{1}$ at the entrance. The velocity distribution at a section downstream is \[ \frac{u}{u_{\max }}=1-\left(\frac{r}{R}\right)^{2} \] Evaluate the maximum velocity at the downstream section. Calculate the pressure drop that would exist in the channel if viscous friction at the walls could be neglected.

An incompressible fluid flows steadily in the entrance region of a circular tube of radius $R=75 \mathrm{mm}$. The flow rate is $Q=0.1 \mathrm{m}^{3} / \mathrm{s}$. Find the uniform velocity $U_{1}$ at the entrance. The velocity distribution at a section downstream is
\[
\frac{u}{u_{\max }}=1-\left(\frac{r}{R}\right)^{2}
\]
Evaluate the maximum velocity at the downstream section. Calculate the pressure drop that would exist in the channel if viscous friction at the walls could be neglected.

Key Concepts

Bernoulli’s Principle

Bernoulli’s principle is a statement of energy conservation in fluid flow, indicating that for an inviscid and steady flow, the sum of pressure energy, kinetic energy, and potential energy remains constant along a streamline. When viscous friction is neglected, pressure differences between sections of the pipe can be assessed by equating the changes in kinetic energy to changes in pressure, leading to calculations of pressure drops under acceleration or deceleration in the flow.

Relationship Between Average and Maximum Velocity

For a parabolic velocity profile in circular pipes, there exists a specific relation between the maximum velocity at the center and the average velocity across the cross-section. This relationship is essential in determining the highest speed within the pipe when the average flow rate is known, and it is derived from integrating the velocity profile over the tube's cross-sectional area.

Flow Rate and Average Velocity

The flow rate is the volume of fluid passing per unit time and is linked to the average velocity and the cross-sectional area of the tube. For uniform flow, the average velocity is simply the total flow divided by the area. This concept is crucial for calculating the entrance velocity of the fluid from the given flow rate and tube diameter.

Continuity Equation

The continuity equation represents the conservation of mass in fluid mechanics. For an incompressible fluid, the flow rate remains constant along the tube, meaning that the integral of the velocity over the cross-sectional area is invariant. This principle is used to relate the average velocity at the tube entrance, where the velocity is uniform, to the known volumetric flow rate.

Velocity Profiles in Circular Pipes

In fully developed pipe flow, the velocity distribution is typically non-uniform and can be described by a parabolic profile for laminar flow. The provided expression, which relates the local velocity to the maximum velocity at the center, reflects how the velocity decreases from the centerline to the walls due to viscous effects, even if in certain sections the flow may be approximated as uniform.

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