**3. Pressure Flow, Flat Plates.** Adverse pressure gradients in flow over bodies lead to flow reversal, separation, wakes, and increased drag. To examine such behavior, consider fully developed, laminar flow between two flat plates a distance δ apart. The bottom plate is fixed, and the top plate moves with speed U (representing the outer flow over a boundary layer of thickness δ). (a) Derive the critical adverse pressure gradient, \((dp/dx)^*\), above which there will be flow reversal, by solving the N-S equation in the coordinates shown, to get a relation for u(y), du/dy, and the critical \((dp/dx)^*\) in terms of δ, U, and μ above which there will be flow reversal at the lower plate. (b) If δ = 5 mm and U = 10 m/s, evaluate \((dp/dx)^*\) if the fluid is air. **Figures:** - **Figure 6.30** consists of two parts: - **(a)** Shows viscous flow between parallel plates with the bottom plate fixed and the upper plate moving (Couette flow). It includes the coordinate system and notation used in the analysis. - **(b)** Presents the velocity distribution as a function of parameter, P, where \(P = −(b^2/μU)dp/dx\). The graph illustrates different curves for values of P ranging from -3 to +3, displaying u/U on the x-axis and y/b on the y-axis. A backflow region is indicated for negative P values. **Assumptions and Constants:** - Patm = \(10^5\) Pa - Pa = 14.7 psi - ρwater ≈ 1000 kg/m\(^3\) - ρair ≈ 1.2 kg/m\(^3\) - μwater ≈ \(10^{-3}\) N·s/m\(^2\) - μair ≈ \(2 \times 10^{-5}\) N·s/m\(^2\) - νwater ≈ \(1.67 \times 10^{-6}\) m\(^2\)/s - g ≈ 9.8 m/s\(^2\) - 1 m/s = 2.24 mph - 1 lbf = 4.45 N
**3. Pressure Flow, Flat Plates.** Adverse pressure gradients in flow over bodies lead to flow reversal, separation, wakes, and increased drag. To examine such behavior, consider fully developed, laminar flow between two flat plates a distance δ apart. The bottom plate is fixed, and the top plate moves with speed U (representing the outer flow over a boundary layer of thickness δ). (a) Derive the critical adverse pressure gradient, \((dp/dx)^*\), above which there will be flow reversal, by solving the N-S equation in the coordinates shown, to get a relation for u(y), du/dy, and the critical \((dp/dx)^*\) in terms of δ, U, and μ above which there will be flow reversal at the lower plate. (b) If δ = 5 mm and U = 10 m/s, evaluate \((dp/dx)^*\) if the fluid is air. **Figures:** - **Figure 6.30** consists of two parts: - **(a)** Shows viscous flow between parallel plates with the bottom plate fixed and the upper plate moving (Couette flow). It includes the coordinate system and notation used in the analysis. - **(b)** Presents the velocity distribution as a function of parameter, P, where \(P = −(b^2/μU)dp/dx\). The graph illustrates different curves for values of P ranging from -3 to +3, displaying u/U on the x-axis and y/b on the y-axis. A backflow region is indicated for negative P values. **Assumptions and Constants:** - Patm = \(10^5\) Pa - Pa = 14.7 psi - ρwater ≈ 1000 kg/m\(^3\) - ρair ≈ 1.2 kg/m\(^3\) - μwater ≈ \(10^{-3}\) N·s/m\(^2\) - μair ≈ \(2 \times 10^{-5}\) N·s/m\(^2\) - νwater ≈ \(1.67 \times 10^{-6}\) m\(^2\)/s - g ≈ 9.8 m/s\(^2\) - 1 m/s = 2.24 mph - 1 lbf = 4.45 N
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:**3. Pressure Flow, Flat Plates.**
Adverse pressure gradients in flow over bodies lead to flow reversal, separation, wakes, and increased drag. To examine such behavior, consider fully developed, laminar flow between two flat plates a distance δ apart. The bottom plate is fixed, and the top plate moves with speed U (representing the outer flow over a boundary layer of thickness δ).
(a) Derive the critical adverse pressure gradient, \((dp/dx)^*\), above which there will be flow reversal, by solving the N-S equation in the coordinates shown, to get a relation for u(y), du/dy, and the critical \((dp/dx)^*\) in terms of δ, U, and μ above which there will be flow reversal at the lower plate.
(b) If δ = 5 mm and U = 10 m/s, evaluate \((dp/dx)^*\) if the fluid is air.
**Figures:**
- **Figure 6.30** consists of two parts:
- **(a)** Shows viscous flow between parallel plates with the bottom plate fixed and the upper plate moving (Couette flow). It includes the coordinate system and notation used in the analysis.
- **(b)** Presents the velocity distribution as a function of parameter, P, where \(P = −(b^2/μU)dp/dx\). The graph illustrates different curves for values of P ranging from -3 to +3, displaying u/U on the x-axis and y/b on the y-axis. A backflow region is indicated for negative P values.
**Assumptions and Constants:**
- Patm = \(10^5\) Pa
- Pa = 14.7 psi
- ρwater ≈ 1000 kg/m\(^3\)
- ρair ≈ 1.2 kg/m\(^3\)
- μwater ≈ \(10^{-3}\) N·s/m\(^2\)
- μair ≈ \(2 \times 10^{-5}\) N·s/m\(^2\)
- νwater ≈ \(1.67 \times 10^{-6}\) m\(^2\)/s
- g ≈ 9.8 m/s\(^2\)
- 1 m/s = 2.24 mph
- 1 lbf = 4.45 N
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