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51. Solve Example IIIa.3.13 by direct derivation. For this purpose, consider the elemental control volume inside the boundary layer as shown in the figure. Write a steady state energy balance for this control volume by considering the net con- vection in the flow direction, net conduction perpendicular to the flow direction, and viscous work. Convection dy dx Conduction ↑ Viscous work

51. Solve Example IIIa.3.13 by direct derivation. For this purpose, consider the elemental control volume inside the boundary layer as shown in the figure. Write a steady state energy balance for this control volume by considering the net con- vection in the flow direction, net conduction perpendicular to the flow direction, and viscous work. Convection dy dx Conduction ↑ Viscous work

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
Problem 1.1MA
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REFERENCE: FROM BOOK - ENGINEERING THERMOFLUIDS, M. MASSOUD

51. Solve Example III..3.13 by direct derivation. For this purpose, consider the elemental control volume inside the boundary layer as shown in the figure. Write a steady state energy balance for this control volume by considering the net con- vection in the flow direction, net conduction perpendicular to the flow direction, and viscous work. Convection dy dx Conduction TViscous work
Transcribed Image Text:51. Solve Example III..3.13 by direct derivation. For this purpose, consider the elemental control volume inside the boundary layer as shown in the figure. Write a steady state energy balance for this control volume by considering the net con- vection in the flow direction, net conduction perpendicular to the flow direction, and viscous work. Convection dy dx Conduction TViscous work
Example IIIa.3.13. Find the governing equation for steady state incompressible flow over a flat plate. Solution: Starting with Equation IIIa.3.23, the first term in the left-hand side and the fourth term in the right-hand side are canceled due to the steady and incom- pressible fluid assumptions, respectively. The equation then reduces to: pv Vu° =V · (kỹT)-V ğ; +ġ* ++V ·B If the effect of all body forces is also negligible, there is no internal heat genera- tion, and we ignore contribution by thermal radiation then we get: pv.Vu = V · (kỹT)+O Substituting for u in terms of temperature, developing terms, substituting for vis- cous-dissipation function, and considering only two-dimensional flow, the equa- tion becomes: v( av, əv ду ƏT Vx +V, ax ду c( dr ду dx In the boundary layer, variation in V, in the x-direction is much less than variation in V, in the y-direction and the first term in the second parenthesis can be ne- glected. Also neglecting dV,/dy and temperature variations in the x-direction, the above equation simplifies to: ƏT ƏT Vx +V dx ду vav, -= - y dy? c dy Ignoring viscous dissipation, the above equation further simplifies to: a?T dy ƏT +V. Ша.3.23-1 ax ду Note the striking resemblance between Equations IIIa.3.20-1 (in the absence of the pressure gradient term) and IIIa.3.23-1.
Transcribed Image Text:Example IIIa.3.13. Find the governing equation for steady state incompressible flow over a flat plate. Solution: Starting with Equation IIIa.3.23, the first term in the left-hand side and the fourth term in the right-hand side are canceled due to the steady and incom- pressible fluid assumptions, respectively. The equation then reduces to: pv Vu° =V · (kỹT)-V ğ; +ġ* ++V ·B If the effect of all body forces is also negligible, there is no internal heat genera- tion, and we ignore contribution by thermal radiation then we get: pv.Vu = V · (kỹT)+O Substituting for u in terms of temperature, developing terms, substituting for vis- cous-dissipation function, and considering only two-dimensional flow, the equa- tion becomes: v( av, əv ду ƏT Vx +V, ax ду c( dr ду dx In the boundary layer, variation in V, in the x-direction is much less than variation in V, in the y-direction and the first term in the second parenthesis can be ne- glected. Also neglecting dV,/dy and temperature variations in the x-direction, the above equation simplifies to: ƏT ƏT Vx +V dx ду vav, -= - y dy? c dy Ignoring viscous dissipation, the above equation further simplifies to: a?T dy ƏT +V. Ша.3.23-1 ax ду Note the striking resemblance between Equations IIIa.3.20-1 (in the absence of the pressure gradient term) and IIIa.3.23-1.
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