CHAPTER 14

1. CHAPTER 14 FLUIDS

2. 14: Fluids Fluids (liquids and gases), by contrast with solids, have the ability to FLOW. Fluids push to the boundary of the object which holds them. Density has SI units of kg/m3. In general, the density of liquids does not vary (they are incompressible); gases are readily compressible. Pressure: The pressure at any point in a fluid is defined by the limit of the expression, p = DF /DAas DA is made as small as possible. If the force is UNIFORM over a FLAT AREA, A, we can writep=F/A The pressure in a fluid has the same value no matter what direction the pressure WITHIN the fluid is measured. Pressure is a SCALAR quantity (i.e.,independent of direction). The SI unit of pressure is the PASCAL (Pa) where 1 Pa=1Nm-2. Other units of pressure include ‘atmospheres’ (atm), torr (mmHg) and lbs/in2 where 1 atm = 1.01x105Pa=760 torr = 760 mm Hg =14.7 lb/in2 REGAN PHY34210

3. Fluids at Rest For a tank of water open to air. The water pressure increases with depth below the air-water interface, while air pressure decreases with height above the water. If the water and air are at rest, their pressures are called HYDROSTATIC PRESSURES. AIR y=0 F1 A y1 y2 WATER F2 mg The pressure at a point in a fluid in static equilibrium depends on the depth of that point but NOT on any horizontal dimension of the fluid. REGAN PHY34210

4. Fi Fo do Ao Ai di Pascal’s Principle ‘ a change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container ’ i.e. squeezing a tube of toothpaste at one end pushes it out the other. We can write Pascal’s principle as Dp=Dpext, i.e. the change in pressure in the liquid equals the change in the applied external pressure. This is the basis behind the concept of the HYDRAULIC LEVER. A downward force on one platform (the ‘input piston’) causes a change in pressure of the INCOMPRESSIBLE LIQUID, resulting in the movement of a second platform (the ‘output piston’). For equilibrium, there must be a downward force due to a load on the output piston which balances the upward force, Fo. load Fo REGAN PHY34210

5. Fi Fo load do Ao Ai di Fo The Hydraulic Lever With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force over a smaller distance. REGAN PHY34210

6. Archimedes’ Principle ‘when a body is fully or partially submerged in a fluid, a BUOYANT FORCE, Fb from the surrounding fluid acts on the body. The force is directed upwards and has a magnitude equal to the weight, mfg of the fluid that has been displaced by the body. This net upward bouyant force exists because the water pressure around the submerged body increases with depth below the surface (Dp=rgh). Thus the pressure at the bottom of the object is larger than at the top. If a body submerged in a fluid has a greater density that then fluid, there is a net force downwards (Fg>Fb), while if the density is less than the fluid, there will be net force upwards (since Fb<Fg). For a body to float in a fluid, the magnitude of the bouyant force Fb, equals to the magnitude of the gravitational force Fg or, the magnitude of the gravitational force on the body is equal to the weight, mfg of the fluid which has been displaced by the body. The weight of a body in fluid is the APPARENT WEIGHT (Wapp) where Wapp= actual weight - magnitude of bouyant force. Since floating bodies have Fb=mg, their apparent weight is ZERO! REGAN PHY34210

7. Flow of Ideal Fluids In Motion The flow of real fluids is very complicated mathematically. Often matters are simplified by assuming an IDEAL FLUID. This requires 4 basic assumptions: A) STEADY FLOW: in steady flow, at a fixed point, the velocity of the moving fluid does not change in magnitude or direction. B) INCOMPRESSIBLE FLOW: This assumes the fluid has constant and fixed density (i.e. it is incompressible). C) NONVISCOUS FLOW: Viscosity is a measure of how resistive a fluid is to flow and is analogous to friction in solids. For example, honey has a higher viscosity than water). An object moving through an ideal, non-viscous fluid experiences NO VISCOUS DRAG force (i.e. no resistive force due to the viscosity of the fluid). D) IRROTATIONAL FLOW: In irrotational flow, a body can not rotate about its own centre of mass as it flows in the fluid. (Note that this does not mean that it can not move in a circular path). REGAN PHY34210

8. v1 v2 The Equation of Continuity Everyday experience tells us that the velocity of a fluid emerging from a tube depends on the cross-sectional areas of the tube. (For example, you can speed up the water exiting a hose by squeezing the end). If we have a tube of cross-sectional area, A1, which narrows to area A2. In a time interval, Dt, a volume DV of fluid enters the tube, with velocity v1. Since the fluid is ideal, and thus incompressible, the same volume of fluid must exit the smaller end of the tube with velocity, v2, some time interval later. The volume of fluid element at both ends (DV) is given by the product of the cross-sectional area (A) and the length it flows (Dx). Also, by definition,v=Dx/Dtthus, A2 A1 REGAN PHY34210

9. p2 ,v2 p1 ,v1 y2 y1 Bernoulli’s Equation If an ideal (incompressible) fluid flows through a tube at a steady rate. If in time Dt , a volume of fluid, DV enters the tube and an identical volume emerges from the other end. If y1, v1 and p1 are elevation, speed and pressure of the fluid entering the tube and y2, v2and p2 are the same quantities for the fluid emerging from the other end of the tube. These quantities are related by the BERNOULLI’S EQUATION, which states, If the speed of a fluid element increases as it travels along a horizontal streamline, the pressure of the fluid must decrease and vice versa. REGAN PHY34210

10. p2 ,v2 Proof of Bernoulli’s Equation p1 ,v1 y2 y1 REGAN PHY34210