Answered: Water, assumed incompressible, flows…

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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Fluid mechanics question

$**Figure P3.15 Explanation** This figure represents the cross-sectional view of a cylindrical pipe with fluid flow. The diagram can be described in the following manner: 1. **Geometry and Axis:** - The pipe length is denoted from $ x = 0 $ to $ x = L $. - The radius of the pipe is represented by $ r $, with the maximum radius given as $ r = R $. - A dashed line extends along the pipe's central axis. 2. **Inlet and Outlet Flow:** - At $ x = 0 $, the fluid enters the pipe with a uniform flow velocity $ U_0 $. - At $ x = L $, the fluid exits the pipe with a velocity profile represented by $ u(r) $. - The arrows indicate the direction of the fluid flow, with fully developed flow exiting the pipe. 3. **Velocity Profile:** - The uniform velocity at the inlet, $ U_0 $, is depicted as a series of equally spaced arrows. - At the outlet, the velocity profile, $ u(r) $, varies across the pipe's radius, represented by arrows of varying lengths, indicating changes in fluid velocity. This figure is often used to illustrate the concepts of fluid dynamics within a cylindrical conduit, such as the development of the velocity profile and effects of viscosity. The transition from a uniform velocity profile at the inlet to a parabolic velocity profile at the outlet is a common phenomenon observed in laminar flow through pipes.$

Transcribed Image Text:**Figure P3.15 Explanation** This figure represents the cross-sectional view of a cylindrical pipe with fluid flow. The diagram can be described in the following manner: 1. **Geometry and Axis:** - The pipe length is denoted from $ x = 0 $ to $ x = L $. - The radius of the pipe is represented by $ r $, with the maximum radius given as $ r = R $. - A dashed line extends along the pipe's central axis. 2. **Inlet and Outlet Flow:** - At $ x = 0 $, the fluid enters the pipe with a uniform flow velocity $ U_0 $. - At $ x = L $, the fluid exits the pipe with a velocity profile represented by $ u(r) $. - The arrows indicate the direction of the fluid flow, with fully developed flow exiting the pipe. 3. **Velocity Profile:** - The uniform velocity at the inlet, $ U_0 $, is depicted as a series of equally spaced arrows. - At the outlet, the velocity profile, $ u(r) $, varies across the pipe's radius, represented by arrows of varying lengths, indicating changes in fluid velocity. This figure is often used to illustrate the concepts of fluid dynamics within a cylindrical conduit, such as the development of the velocity profile and effects of viscosity. The transition from a uniform velocity profile at the inlet to a parabolic velocity profile at the outlet is a common phenomenon observed in laminar flow through pipes.

$**Flow Through a Pipe: Velocity Ratio** In this exercise, we analyze the flow of water—considered incompressible—moving steadily through a round pipe, as depicted in Fig. P3.15. - **Entrance Velocity**: The velocity at the entrance of the pipe, denoted as $ u $, is constant and is given by: \[ u = U_0 \] where $ U_0 $ is a constant velocity. - **Exit Velocity**: At the exit of the pipe, the velocity approximates a turbulent flow, expressed as: \[ u = u_{\text{max}} \left(1 - \frac{r}{R}\right)^{1/7} \] Here: - $ u_{\text{max}} $ is the maximum velocity. - $ r $ is the radial position from the center of the pipe. - $ R $ is the radius of the pipe. **Task**: Determine the ratio of $ U_0 $ to $ u_{\text{max}} $ for this turbulent flow. **Explanation of Diagram (Fig. P3.15)**: Although Fig. P3.15 is not provided here, typically, it would illustrate a cylindrical pipe with an indication of flow direction. Velocities at different sections of the pipe (entrance and exit) would be marked, showing how the velocity profile changes from a constant $ U_0 $ at entrance to the turbulent profile at exit. To complete the analysis and find the ratio $ \frac{U_0}{u_{\text{max}}} $, further mathematical derivation or integration of the velocity profiles might be needed, which requires the given flow equations.$

Transcribed Image Text:**Flow Through a Pipe: Velocity Ratio** In this exercise, we analyze the flow of water—considered incompressible—moving steadily through a round pipe, as depicted in Fig. P3.15. - **Entrance Velocity**: The velocity at the entrance of the pipe, denoted as $ u $, is constant and is given by: \[ u = U_0 \] where $ U_0 $ is a constant velocity. - **Exit Velocity**: At the exit of the pipe, the velocity approximates a turbulent flow, expressed as: \[ u = u_{\text{max}} \left(1 - \frac{r}{R}\right)^{1/7} \] Here: - $ u_{\text{max}} $ is the maximum velocity. - $ r $ is the radial position from the center of the pipe. - $ R $ is the radius of the pipe. **Task**: Determine the ratio of $ U_0 $ to $ u_{\text{max}} $ for this turbulent flow. **Explanation of Diagram (Fig. P3.15)**: Although Fig. P3.15 is not provided here, typically, it would illustrate a cylindrical pipe with an indication of flow direction. Velocities at different sections of the pipe (entrance and exit) would be marked, showing how the velocity profile changes from a constant $ U_0 $ at entrance to the turbulent profile at exit. To complete the analysis and find the ratio $ \frac{U_0}{u_{\text{max}}} $, further mathematical derivation or integration of the velocity profiles might be needed, which requires the given flow equations.

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Water, assumed incompressible, flows steadily through the round pipe in Fig. P3.15. The entrance velocity is con- stant, u = Uo, and the exit velocity approximates turbulent flow, u = Umax(1 – r/R)". Determine the ratio UJumax for this flow.