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190 PHYSICS (a) (b) (c) Fig 9.11 (a)Fluidstreamingpastastaticsphere.(b)Streamlinesforafluidaroundaspherespinning clockwise. (c) Air flowing past an aerofoil. ⊳ v = (v + v )/2 = 960 km/h = 267 m s , -1 Example 9.7 A fully loaded Boeing aircraft av 2 1 has a mass of 3.3 × 10 kg. Its total wing we have area is 500 m . It is in level flight with a P speed of 960 km/h. (a) Estimate the (v2 –v1)/ vav= ρv 2 ≈ 0.08 pressure difference between the lower and av The speed above the wing needs to be only 8 upper surfaces of the wings (b) Estimate % higher than that below. ⊳ the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface. [The density of air is ρ 9.5 VISCOSITY = 1.2 kg m ]3 Mostofthefluidsarenotidealonesandoffersome resistancetomotion.Thisresistancetofluidmotion islikeaninternalfrictionanalogoustofrictionwhen Answer (a) The weight of the Boeing aircraft is a solid moves on a surface. It is called viscosity. balanced by the upward force due to the This force exists when there is relative motion pressure difference 5 between layers of the liquid. Suppose we consider ∆P × A = 3.3 × 10 kg × 9.8 a fluid like oil enclosed between two glass plates 5 –2 2 asshowninFig.9.12 (a).Thebottomplateisfixed ∆P = (3.3 × 10 kg × 9.8 m s ) / 500 m 3 -2 while the top plate is moved with a constant = 6.5 ×10 Nm velocity v relative to the fixed plate. If oil is (b) We ignore the small height difference replaced by honey, a greater force is required to between the top and bottom sides in Eq. (9.12). move the plate with the same velocity. Hence The pressure difference between them is we say that honey is more viscous than oil. The then fluid in contact with a surface has the same ρ 2 2 velocity as that of the surfaces. Hence, the layer P = ( 2 v 1 ) 2 of the liquid in contact with top surface moves where v is the speed of air over the upper with a velocity v and the layer of the liquid in 2 contact with the fixed surface is stationary. The surface and v i1 the speed under the bottom surface. velocities of layers increase uniformly from bottom (zero velocity) to the top layer (velocity 2 P (v2– v1) ρ v + v v). For any layer of liquid, its upper layer pulls ( 2 1) it forward while lower layer pulls it backward. Taking the average speed This results in force between the layers. This Reprint 2026-27