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4.8 In addition to the customary horizontal velocity compo- nents of the air in the atmosphere (the "wind"), there often are vertical air currents (thermals) caused by buoyant effects due to uneven heating of the air as indicated in Fig. P4.8. Assume that the velocity field in a certain region is approximated by u = uo, v = vo(1y/h) for 0 < y h. De- termine the equation for the streamlines and plot the streamline that passes through the origin for values of u/vo= 0.5, 1, and 2.

4.8 In addition to the customary horizontal velocity compo- nents of the air in the atmosphere (the "wind"), there often are vertical air currents (thermals) caused by buoyant effects due to uneven heating of the air as indicated in Fig. P4.8. Assume that the velocity field in a certain region is approximated by u = uo, v = vo(1y/h) for 0 < y h. De- termine the equation for the streamlines and plot the streamline that passes through the origin for values of u/vo= 0.5, 1, and 2.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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**4.8** In addition to the customary horizontal velocity components of the air in the atmosphere (the “wind”), there often are vertical air currents (thermals) caused by buoyant effects due to uneven heating of the air as indicated in Fig. P4.8. Assume that the velocity field in a certain region is approximated by \( u = u_0, \; v = v_0(1 - y/h) \) for \( 0 < y < h \), and \( u = u_0, \; v = 0 \) for \( y > h \). Determine the equation for the streamlines and plot the streamline that passes through the origin for values of \( u_0/v_0 = 0.5, 1, \) and \( 2 \). **Figure P4.8 Explanation:** - The figure is a graph illustrating air velocity components with axes labeled \( x \) (horizontal) and \( y \) (vertical). - The horizontal component of the air velocity (\( u \)) is represented by a rightward arrow labeled \( u_0 \). - The vertical component of the air velocity (\( v \)) varies with height \( y \) from 0 to \( h \) and is zero above \( y = h \). - Streamlines are shown as dashed curved lines that start at the origin and curve upwards. - The figure highlights different streamline paths for \( u_0/v_0 \) ratios of 0.5, 1, and 2, where the steepness of each path varies according to these ratios.
Transcribed Image Text:**4.8** In addition to the customary horizontal velocity components of the air in the atmosphere (the “wind”), there often are vertical air currents (thermals) caused by buoyant effects due to uneven heating of the air as indicated in Fig. P4.8. Assume that the velocity field in a certain region is approximated by \( u = u_0, \; v = v_0(1 - y/h) \) for \( 0 < y < h \), and \( u = u_0, \; v = 0 \) for \( y > h \). Determine the equation for the streamlines and plot the streamline that passes through the origin for values of \( u_0/v_0 = 0.5, 1, \) and \( 2 \). **Figure P4.8 Explanation:** - The figure is a graph illustrating air velocity components with axes labeled \( x \) (horizontal) and \( y \) (vertical). - The horizontal component of the air velocity (\( u \)) is represented by a rightward arrow labeled \( u_0 \). - The vertical component of the air velocity (\( v \)) varies with height \( y \) from 0 to \( h \) and is zero above \( y = h \). - Streamlines are shown as dashed curved lines that start at the origin and curve upwards. - The figure highlights different streamline paths for \( u_0/v_0 \) ratios of 0.5, 1, and 2, where the steepness of each path varies according to these ratios.
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