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A thin film of oil of thickness Y and dynamic viscosity u flows uniformly down an inclined plane as shown in Figure Q2 (in which p is the pressure anč A is the cross-sectional area of the element of fluid shown) under laminar conditions. Given that the velocity gradient is zero at the free surface and that the pressure gradient must be zero (i.e. the flow is uniform and atmospherio conditions prevail) at the surface, show that the velocity profile is pg u = (Yy-y/2) sine where u is the local velocity at an elevation y above the bed, 0 is the inclination of the plane to the horizontal and pis the fluid density. Also, du T = u- dy where t is the shear stress and du/dy is the velocity gradient. A reservoir discharges through a sluice gate that is 1.2 m wide by 1.4 m deep. The top of the opening is 0.8 m below the water level in the reservoir. The water level just downstream of the sluice gate is below the bottom of the opening. Calculate the theoretical discharge. The theoretical discharge equation used should be developed from first principles. The formula for the velocity need not be derived. .... öy pA dt (T+ dy yk 8x

A thin film of oil of thickness Y and dynamic viscosity u flows uniformly down an inclined plane as shown in Figure Q2 (in which p is the pressure anč A is the cross-sectional area of the element of fluid shown) under laminar conditions. Given that the velocity gradient is zero at the free surface and that the pressure gradient must be zero (i.e. the flow is uniform and atmospherio conditions prevail) at the surface, show that the velocity profile is pg u = (Yy-y/2) sine where u is the local velocity at an elevation y above the bed, 0 is the inclination of the plane to the horizontal and pis the fluid density. Also, du T = u- dy where t is the shear stress and du/dy is the velocity gradient. A reservoir discharges through a sluice gate that is 1.2 m wide by 1.4 m deep. The top of the opening is 0.8 m below the water level in the reservoir. The water level just downstream of the sluice gate is below the bottom of the opening. Calculate the theoretical discharge. The theoretical discharge equation used should be developed from first principles. The formula for the velocity need not be derived. .... öy pA dt (T+ dy yk 8x

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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A thin film of oil of thickness Y and dynamic viscosity u flows uniformly down an inclined plane as shown in Figure Q2 (in which p is the pressure and A is the cross-sectional area of the element of fluid shown) under laminar conditions. Given that the velocity gradient is zero at the free surface and that the pressure gradient must be zero (i.e. the flow is uniform and atmospheric conditions prevail) at the surface, show that the velocity profile is pg u = (Yy- y/2) sine where u is the local velocity at an elevation y above the bed, 0 is the inclination of the plane to the horizontal and pis the fluid density. Also, du dy where t is the shear stress and du/dy is the velocity gradient. A reservoir discharges through a sluice gate that is 1.2 m wide by 1.4 m deep. The top of the opening is 0.8 m below the water level in the reservoir. The water level just downstream of the sluice gate is below the bottom of the opening. Calculate the theoretical discharge. The theoretical discharge equation used should be developed from first principles. The formula for the velocity need not be derived. pA dt (T+ dy y 8x
Transcribed Image Text:A thin film of oil of thickness Y and dynamic viscosity u flows uniformly down an inclined plane as shown in Figure Q2 (in which p is the pressure and A is the cross-sectional area of the element of fluid shown) under laminar conditions. Given that the velocity gradient is zero at the free surface and that the pressure gradient must be zero (i.e. the flow is uniform and atmospheric conditions prevail) at the surface, show that the velocity profile is pg u = (Yy- y/2) sine where u is the local velocity at an elevation y above the bed, 0 is the inclination of the plane to the horizontal and pis the fluid density. Also, du dy where t is the shear stress and du/dy is the velocity gradient. A reservoir discharges through a sluice gate that is 1.2 m wide by 1.4 m deep. The top of the opening is 0.8 m below the water level in the reservoir. The water level just downstream of the sluice gate is below the bottom of the opening. Calculate the theoretical discharge. The theoretical discharge equation used should be developed from first principles. The formula for the velocity need not be derived. pA dt (T+ dy y 8x
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