4. Air flows into a pipe from the region between a circular disk and a cone as shown in Fig. P4.55. The fluid velocity in the gap between the disk and the cone is closely approximated by V = V₁R²/r², where R is the radius of the disk, r is the radial coordinate, and is the fluid velocity at the edge of the disk. Determine the acceleration for r = 0.5 and 2 m if V0 = 5 m/s and R = 2 m. -Pipe sh Disk Cone

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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**Problem Description:**

Air flows into a pipe from the region between a circular disk and a cone as shown in Figure P4.55. The fluid velocity in the gap between the disk and the cone is closely approximated by the formula:

\[ V = V_0 R^2 / r^2 \]

Where:
- \( R \) is the radius of the disk.
- \( r \) is the radial coordinate.
- \( V \) is the fluid velocity at the edge of the disk.

**Objective:**

Determine the acceleration for \( r = 0.5 \, \text{m} \) and \( 2 \, \text{m} \) if \( V_0 = 5 \, \text{m/s} \) and \( R = 2 \, \text{m} \).

**Diagram Explanation:**

The illustration depicts a pipe where air enters from the space between a disk and a cone. The disk is oriented horizontally, while the cone is positioned such that its tip faces the disk, creating a gap through which the air flows. The air is then directed vertically into the pipe. The velocity \( V \) of the fluid varies with the radial coordinate \( r \), illustrating how the velocity distribution is influenced by the given parameters.

**Task:**

Given the relationship for velocity, calculate the acceleration of the fluid at the specified radial positions using the provided velocities and dimensions.
Transcribed Image Text:**Problem Description:** Air flows into a pipe from the region between a circular disk and a cone as shown in Figure P4.55. The fluid velocity in the gap between the disk and the cone is closely approximated by the formula: \[ V = V_0 R^2 / r^2 \] Where: - \( R \) is the radius of the disk. - \( r \) is the radial coordinate. - \( V \) is the fluid velocity at the edge of the disk. **Objective:** Determine the acceleration for \( r = 0.5 \, \text{m} \) and \( 2 \, \text{m} \) if \( V_0 = 5 \, \text{m/s} \) and \( R = 2 \, \text{m} \). **Diagram Explanation:** The illustration depicts a pipe where air enters from the space between a disk and a cone. The disk is oriented horizontally, while the cone is positioned such that its tip faces the disk, creating a gap through which the air flows. The air is then directed vertically into the pipe. The velocity \( V \) of the fluid varies with the radial coordinate \( r \), illustrating how the velocity distribution is influenced by the given parameters. **Task:** Given the relationship for velocity, calculate the acceleration of the fluid at the specified radial positions using the provided velocities and dimensions.
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