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MECHANICAL PROPERTIES OF FLUIDS 189 from its parabolic trajectory as it moves through air. This deviation can be partly explained on the basis of Bernoulli’s principle. (i) Ball moving without spin: Fig. 9.11(a) shows the streamlines around a non-spinning ball moving relative to a fluid. From the symmetry of streamlines it is clear that the velocity of fluid (air) above and below the ball at corresponding points is the same resulting in zero pressure difference. The air therefore, exerts no upward or downward force on the ball. Fig. 9.10 Torricelli’s law. The speed of efflu1, v , from the side of the container is given byii) Ball moving with spin: A ball which is the application of Bernoulli’s equation. spinning drags air along with it. If the If the container is open at the top to the surface is rough more air will be dragged. atmosphere then 1 = 2 g h Fig 9.11(b) shows the streamlines of air for a ball which is moving and spinning at the same time. The ball is moving If the cross-sectional area of the tank A 2s forward and relative to it the air is moving much larger than that of the hole (A >>A ), then 2 1 backwards. Therefore, the velocity of air we may take the fluid to be approximately at rest above the ball relative to the ball is larger at the top, i.e.2v = 0. Now, applying the Bernoulli equation at points 1 and 2 and noting that at and below it is smaller (see Section 9.3). The stream lines, thus, get crowded above the hole P1= P a the atmospheric pressure, we have from Eq. (9.12) and rarified below. This difference in the velocities of air results P + 1 ρv + ρg y = P + ρ g y a 2 1 1 2 in the pressure difference between the lower and upper faces and there is a net upward force on Taking y 2 y 1 h we have the ball. This dynamic lift due to spining is called 2(P − Pa) Magnus effect. v1= 2g h + (9.14) ρ Aerofoil or lift on aircraft wing: Figure 9.11 When P >>P and 2 g h may be ignored, the (c) shows an aerofoil, which is a solid piece a speed of efflux is determined by the container shaped to provide an upward dynamic lift pressure. Such a situation occurs in rocket when it moves horizontally through air. The propulsion. On the other hand, if the tank is cross-section of the wings of an aeroplane open to the atmosphere, then P = P ana looks somewhat like the aerofoil shown in Fig. v1= 2gh (9.15) 9.11 (c) with streamlines around it. When the This is also the speed of a freely falling body. aerofoil moves against the wind, the Equation (9.15) represents Torricelli’s law. orientation of the wing relative to flow direction causes the streamlines to crowd together 9.4.2 Dynamic Lift above the wing more than those below it. The Dynamic lift is the force that acts on a body, flow speed on top is higher than that below it. such as airplane wing, a hydrofoil or a spinning There is an upward force resulting in a ball, by virtue of its motion through a fluid. In dynamic lift of the wings and this balances many games such as cricket, tennis, baseball, the weight of the plane. The following example or golf, we notice that a spinning ball deviates illustrates this. Reprint 2026-27