Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Fluid
![### Fluid Dynamics Problem: Volume Flow Computation Across a Control Volume
**Problem Statement:**
An incompressible fluid flows past an impermeable flat plate, as illustrated in Fig. P3.16, with a uniform inlet profile \( u = U_0 \) and a cubic polynomial exit profile:
\[ u \approx U_0 \left( \frac{3\eta - \eta^3}{2} \right) \text{ where } \eta = \frac{y}{\delta} \]
**Objective:**
Compute the volume flow \( Q \) across the top surface of the control volume.
---
**Diagram Explanation:**
- **Inlet Profile:**
- The fluid enters with a uniform velocity \( U_0 \).
- The inlet is shown with parallel horizontal arrows indicating the constant fluid velocity at this section.
- **Control Volume (CV):**
- The control volume is bounded by the impermeable flat plate at the bottom and extends upwards to \( y = \delta \), which defines the height of the control surface.
- The area of interest is the top surface of the control volume, where the fluid velocity varies according to the given cubic polynomial profile.
- **Exit Profile:**
- The exit at the right side shows a parabolic boundary, reflecting the non-uniform cubic velocity profile described by \( u = U_0 \left( \frac{3\eta - \eta^3}{2} \right) \).
- The velocity profile changes from \( u = U_0 \) at \( y = 0 \) to a different value determined by the polynomial at \( y = \delta \).
To better understand the volume flow, Fig. P3.16 is essential, which precisely outlines the flow profiles and control volume boundaries.
---
To solve this problem, the integration of the velocity profile over the height \( \delta \) on the control surface can be performed to find the total volume flow \( Q \).
---
**Figure P3.16 Explanation:**
- **Left Section (Inlet):**
- Uniform velocity profile \( U_0 \) shown by evenly spaced parallel, horizontal arrows.
- **Middle Section (Control Volume):**
- Top surface indicated by a dashed horizontal line.
- Height of control volume noted as \( y = \delta \).
- **Question Indicator:**
- The query \( Q? \) specifies where to](https://content.bartleby.com/qna-images/question/508ae4f5-8e9f-43cb-9ab2-37d719369fa8/7ff3c9b5-5622-434e-8273-a53f0eb1b776/v4kahh5.png)
Transcribed Image Text:### Fluid Dynamics Problem: Volume Flow Computation Across a Control Volume
**Problem Statement:**
An incompressible fluid flows past an impermeable flat plate, as illustrated in Fig. P3.16, with a uniform inlet profile \( u = U_0 \) and a cubic polynomial exit profile:
\[ u \approx U_0 \left( \frac{3\eta - \eta^3}{2} \right) \text{ where } \eta = \frac{y}{\delta} \]
**Objective:**
Compute the volume flow \( Q \) across the top surface of the control volume.
---
**Diagram Explanation:**
- **Inlet Profile:**
- The fluid enters with a uniform velocity \( U_0 \).
- The inlet is shown with parallel horizontal arrows indicating the constant fluid velocity at this section.
- **Control Volume (CV):**
- The control volume is bounded by the impermeable flat plate at the bottom and extends upwards to \( y = \delta \), which defines the height of the control surface.
- The area of interest is the top surface of the control volume, where the fluid velocity varies according to the given cubic polynomial profile.
- **Exit Profile:**
- The exit at the right side shows a parabolic boundary, reflecting the non-uniform cubic velocity profile described by \( u = U_0 \left( \frac{3\eta - \eta^3}{2} \right) \).
- The velocity profile changes from \( u = U_0 \) at \( y = 0 \) to a different value determined by the polynomial at \( y = \delta \).
To better understand the volume flow, Fig. P3.16 is essential, which precisely outlines the flow profiles and control volume boundaries.
---
To solve this problem, the integration of the velocity profile over the height \( \delta \) on the control surface can be performed to find the total volume flow \( Q \).
---
**Figure P3.16 Explanation:**
- **Left Section (Inlet):**
- Uniform velocity profile \( U_0 \) shown by evenly spaced parallel, horizontal arrows.
- **Middle Section (Control Volume):**
- Top surface indicated by a dashed horizontal line.
- Height of control volume noted as \( y = \delta \).
- **Question Indicator:**
- The query \( Q? \) specifies where to
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