制作张昆实 Yangtze University

BilingualMechanics Chapter 10 Fluids 制作张昆实 Yangtze University

10-1 Fluids and the World Around Us 10-2 What Is a Fluid? 10-3 Density and Pressure 10-4 Fluids at Rest 10-5 Measuring Pressure 10-6 Pascal's Principle 10-7 Archimedes' Principle 10-8 Ideal Fluids in Motion 10-9 The Equation of Continuity 10-10 Bernoulli's Equation Chapter 10 Fluids

Fluids, which include both liquids and gases, play a central role in our daily lives. We breath and drink them, and a rather vital fluid circulates in the human cardiovascularsystem. There are the fluid ocean and the fluid atmosphere. A fluid is a substance that can flow. Fluids conform to the boundaries of any containerin which we put them. 10-1 Fluids and the World Around Us10-2 What Is a Fluid?

With fluids, we are more interested in the extended substance, and in properties that can varyfrom point to point in that substance. It is more useful to speak of density and pressure than of mass and force. Density: To find thedensity of a fluid at any point, we isolate a small volume element around that point and measure the mass of the fluid contained within that element. The density is then (10-1) In theory, the density at any point in a fluid is the limit of this ratio as the volume element at that point is made smaller and smaller. In practice, for a “smooth” (with uniform density) fluid, its density can be written as (10-2) ( uniform density ) 10-3 Density and Pressure

Pressure: To find thePressureat any point in a fluid, we isolate a small area element around that point and measure the magnitude of the force that acts normal to that element. The Pressure is then (10-3) In theory, the Pressure at any point in a fluid is the limit of this ratio as the area element at that point is made smaller and smaller. However, if the force is uniform over a flat area , the Pressure can be written as ( Pressure of uniform force on flat area ) (10-4) The SI unit of pressure: Atmosphere (at sea lever) 10-3 Density and Pressure Pascal 1Pa=1N/m2 Millimeter of mercury (mmHg)

Set up a vertical axis in a tank of water with its origin at the surface. Consider an imaginary column of water. and are the depths below the surface of the upper and lower column fases, respectively. Three forces act on the column: acts at the top of the column; acts at the bottem of the column; The gravitational force of the column 10-4 Fluids at Rest The pressureincreaseswithdepthin water. The pressuredecreases with altitudeinatmosphere.

The column is in static equilibrium, these three forces balanced. (10-5) (10-6) or (10-7) ★Pressure in a liquid and Eq. 10-7 : (10-8) 10-4Fluids at Rest level 1: surface; level 2: h below it

Eq. 10-7 : (Atmospheric density is uniform) This case is different from the example in 漆安慎力学 P387。 and There the atmospheric density is proportional to thepressure! 10-4 Fluids at Rest (10-7) Level 2 ★Pressure in atmosphere level 1: surface; level 2: d above it d

The load put apressure on the piston and thus on the liquid. The pressure at any point P in the liquid is then (10-11) Add more shot to increase by , the and unchanged so the pressure change at P (10-12) 10-6 Pascal's Principle Pascal's Principle A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminishedto every portion of the fluid and to the wall of its container. and

Hydraulic Lever Pistoni : Pistono : and and The pressures on both sides are equal (10-13) The same volume of incompressible liquid is displaced at both pistons (10-11) 10-6 Pascal's Principle output forces

Hydraulic Lever Pistoni : Pistono : and and The output work (10-13) With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance. 10-6 Pascal's Principle output forces

Archimedes' Principle Apparent Weight in a Fluid (10-19) (apparent weight) 10-7 Archimedes' Principle When a body is partially or whollyimmersed in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight of the fluid that has been displaced by the body.

10-8 Ideal Fluids in Motion Ideal fluid. There are four assumptions about ideal fluid: 1. Steady Flow. In steady flow the velocity of the moving fluid at any given point does not change as time goes on. 2. Incompressible Flow. The ideal fluid is incompressible means its density has a constant value. 3. Nonviscous Flow. An object moving through a nonviscous fluid would experience no viscous drag force. 4. Irrotational Flow. In irrotational flow a test body will not rotate about an axis through its own center of mass.

streamline fluid element 10-8 Ideal Fluids in Motion Streamlines Figure 10-12 shows streamlines traced out by injecting dye into the moving fluid. A streamlineis the path traced out by a tiny fluid element which we may call a fluid “particle”. Fig.10-12 streamlines As the fluid particle moves, its velocity may change, both in magnitude and in direction. The velocity vector at any point will alwaysbe tangent tothe streamline at that point.

Streamlines Streamlines never cross because, if they did, a fluid particle arriving at the intersection would have to assume two different velocities simultaneously, an impossibility. tube of flow a tube of flow We can build up a tube of flow whose boundary is made up of streamlines. Such a tube acts like a pipe because any fluid particle that enters it cannot escape through its walls; if it did, we would have a case of streamlines crossing each other. streamline fluid element 10-8 Ideal Fluids in Motion

Consider a tube segment (L) through which an idea fluid flows toward the right. Left endRight end Cross-sec-tional area Fluid speed In a time intervala volumea of fluid enters the tube at its left end. Then because the fluid is incompressible, an identical volume must emerge from the right end. ( equation of continuity ) (10-23) For an idea fluid, when 10-9 The Equation of Continuity

( Equation of continuity ) (10-23) For an idea fluid, when Closer streamlines lower speed a constant (10-24) (volume flow rate, equation of continuity) is thevolume flow rate ( volume per unit time ) greatest speed a constant is themass flow rate ( mass per unit time ) (10-25) ( Mass flow rate ) 10-9 The Equation of Continuity If the density of the fluid is uniform, multiply Eq.10-24 by that density to get

An idea fluid is flowing through a tube segment with a steady rate. In a time interval , a volume of fluidenters the tube at the left end and an identical volumeemerges at the right end because the fluid is incompressible. Left endRight end elevation speed pressure By applying the principle of conserva-tion of energy to the fluid, these quantities are related by (10-28) 10-10 Bernoulli's Equation

Eq.10-28 can be written as a constant (10-29) Bernoulli's Equation (only for ideal fluid ) If the fluid doesn’t change its eleva-tion as it flows in a horizontal tube, take , Bernoulli's Equation is nowin the following form (10-30) If the speed of fluid element increases as it travels along a horizontal stream-line, the pressure of the fluid must decrease, and conversely. 10-10 Bernoulli's Equation Ideal fluid. There are four assumptions about ideal fluid: (10-28)

Proof ofBernoulli's Equation Take the entire volume of the fluid as our system; Apply the principle of con-servation of energy to this system as it moves from initial state (Fig.(a)) to the final state (Fig.(b)). We need be concerned only with chan-ges that take place at the input and out-put ends. Applyenergy conservation in the form of the work-kinetic energy theorem (10-31) (10-32) 10-10 Bernoulli's Equation

Proof ofBernoulli's Equation (10-31) (10-32) The work done by the gravitational force on the fluid from the input level to the output level is (10-33) Work must also be doneat the input endto push the entering fluid into thetubeand by the system at the output endto push forward the fluid ahead of the emerging fluid. 10-10 Bernoulli's Equation

Proof ofBernoulli's Equation (10-31) (10-32) (10-33) Generally, the work done by a force F on an area A through , is The work done at the input end is The work done at the output end from the system is (10-34) 10-10 Bernoulli's Equation

Proof ofBernoulli's Equation (10-31) (10-32) (10-33) (10-34) The work-kinetic energy theoremnow becomes (10-28) 10-10 Bernoulli's Equation Substituting from (10-32), (10-33) and (10-34) yields Bernoulli's Equation

制作张昆实 Yangtze University