Answered: An approximation for the boundary-layer…

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

See similar textbooks

Related questions

Concept explainers

Question

Fluid mechanics problem

$**Boundary-Layer Shape Approximation** An approximation for the boundary-layer shape in Figs. 1.5b and P1.51 is represented by the following formula: \[ u(y) \approx U \sin\left(\frac{\pi y}{2 \delta}\right), \quad 0 \le y \le \delta \] where: - $ U $ is the stream velocity far from the wall. - $ \delta $ is the boundary layer thickness. - $ y $ is the distance from the wall. If the fluid is helium at 20°C and 1 atm, and given that $ U = 10.8 \, \text{m/s} $ and $ \delta = 3 \, \text{cm} $, use the formula to: 1. Estimate the wall shear stress $ \tau_w $ in Pa. 2. Find the position in the boundary layer where $ \tau $ is one-half of $ \tau_w $. **Diagram Explanation:** The diagram below illustrates the boundary layer where the velocity $ u(y) $ varies from zero at the wall to the free stream velocity $ U $ at $ y = \delta $. - The x-axis represents the distance from the wall along the boundary layer. - The y-axis represents the distance perpendicular to the wall. \[ \begin{array}{c|c} &y \\ \hline 0 & \\ & \\ & \\ & y = \delta \\ \end{array} \] - The horizontal arrows denote the velocity profile $ u(y) $ which increases from zero at the wall (lower part of the graph) to $ U $ at $ y = \delta $ (upper part of the graph). **Steps for Calculations:** 1. **Wall Shear Stress $ \tau_w $:** To estimate the wall shear stress: \[ \tau_w = \mu \left. \frac{\partial u}{\partial y} \right|_{y=0} \] where $ \mu $ is the dynamic viscosity of helium at 20°C. 2. **Position in Boundary Layer:** To find the position $ y $ where the shear stress is half of $ \tau_w $: \[ \tau = \$

Transcribed Image Text:**Boundary-Layer Shape Approximation** An approximation for the boundary-layer shape in Figs. 1.5b and P1.51 is represented by the following formula: \[ u(y) \approx U \sin\left(\frac{\pi y}{2 \delta}\right), \quad 0 \le y \le \delta \] where: - $ U $ is the stream velocity far from the wall. - $ \delta $ is the boundary layer thickness. - $ y $ is the distance from the wall. If the fluid is helium at 20°C and 1 atm, and given that $ U = 10.8 \, \text{m/s} $ and $ \delta = 3 \, \text{cm} $, use the formula to: 1. Estimate the wall shear stress $ \tau_w $ in Pa. 2. Find the position in the boundary layer where $ \tau $ is one-half of $ \tau_w $. **Diagram Explanation:** The diagram below illustrates the boundary layer where the velocity $ u(y) $ varies from zero at the wall to the free stream velocity $ U $ at $ y = \delta $. - The x-axis represents the distance from the wall along the boundary layer. - The y-axis represents the distance perpendicular to the wall. \[ \begin{array}{c|c} &y \\ \hline 0 & \\ & \\ & \\ & y = \delta \\ \end{array} \] - The horizontal arrows denote the velocity profile $ u(y) $ which increases from zero at the wall (lower part of the graph) to $ U $ at $ y = \delta $ (upper part of the graph). **Steps for Calculations:** 1. **Wall Shear Stress $ \tau_w $:** To estimate the wall shear stress: \[ \tau_w = \mu \left. \frac{\partial u}{\partial y} \right|_{y=0} \] where $ \mu $ is the dynamic viscosity of helium at 20°C. 2. **Position in Boundary Layer:** To find the position $ y $ where the shear stress is half of $ \tau_w $: \[ \tau = \

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 6 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions