Answered: (";") '3η -η y where n u = U. 2

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 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$

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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