Answered: An incompressible viscous liquid (with…

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

Question

$**Text Transcription:** An incompressible viscous liquid (with density $\rho$ and dynamic viscosity $\mu$) slides down an inclined plane as seen in the figure. The film thickness is $h$, and the solid substrate moves with a speed $U$ up the incline. Assume steady, fully developed, laminar flow in the $x$ direction with no pressure gradient (in the $x$ direction). Furthermore, assume that the atmosphere at $y = h$ imposes no shear stress on the liquid. (a) Derive an expression for the velocity profile using the Navier-Stokes equations in terms of the parameters given. (b) If the magnitude of the substrate velocity is given as $U = \frac{\rho g h^2 \sin \theta}{4 \mu}$, determine the $y$-coordinate, normalized by the thickness, where the velocity is zero, i.e., find $y/h$ for $u(y) = 0$. (c) Draw a qualitative sketch of the velocity profile, including a zero-velocity reference line, assuming the substrate velocity given in part (b). **Diagram Explanation:** The diagram shows an inclined plane with a liquid film of thickness $h$. Key elements include: - An inclined plane with an angle $\theta$. - A coordinate system with axes $x$ and $y$, where $x$ is along the incline and $y$ is perpendicular. - The liquid has properties $\rho$ (density) and $\mu$ (dynamic viscosity). - A gravitational force $g$ acts downward. - The substrate moves upward with speed $U$. - Atmospheric pressure, $p_{\text{atm}}$, is noted at the upper boundary of the liquid film. The diagram aids in visualizing how the liquid moves on the inclined plane under the given conditions.$

Transcribed Image Text:**Text Transcription:** An incompressible viscous liquid (with density $\rho$ and dynamic viscosity $\mu$) slides down an inclined plane as seen in the figure. The film thickness is $h$, and the solid substrate moves with a speed $U$ up the incline. Assume steady, fully developed, laminar flow in the $x$ direction with no pressure gradient (in the $x$ direction). Furthermore, assume that the atmosphere at $y = h$ imposes no shear stress on the liquid. (a) Derive an expression for the velocity profile using the Navier-Stokes equations in terms of the parameters given. (b) If the magnitude of the substrate velocity is given as $U = \frac{\rho g h^2 \sin \theta}{4 \mu}$, determine the $y$-coordinate, normalized by the thickness, where the velocity is zero, i.e., find $y/h$ for $u(y) = 0$. (c) Draw a qualitative sketch of the velocity profile, including a zero-velocity reference line, assuming the substrate velocity given in part (b). **Diagram Explanation:** The diagram shows an inclined plane with a liquid film of thickness $h$. Key elements include: - An inclined plane with an angle $\theta$. - A coordinate system with axes $x$ and $y$, where $x$ is along the incline and $y$ is perpendicular. - The liquid has properties $\rho$ (density) and $\mu$ (dynamic viscosity). - A gravitational force $g$ acts downward. - The substrate moves upward with speed $U$. - Atmospheric pressure, $p_{\text{atm}}$, is noted at the upper boundary of the liquid film. The diagram aids in visualizing how the liquid moves on the inclined plane under the given conditions.

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 4 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