制作 张昆实 制作 张昆实 Yangtze University 制作 张昆实 制作 张昆实 Yangtze University Bilingual Mechanics Chapter 10 Fluids Fluids Chapter 10 Fluids Fluids

Chapter 10 Fluids 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 Bernoulli's Equation 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 Bernoulli's Equation

10-1 Fluids and the World Around Us 10-2 What Is a Fluid? Fluidsliquidsgases vital fluid cardiovascular systemthe fluid oceanthe fluid atmosphere 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 cardiovascular system. There are the fluid ocean and the fluid atmosphere. A fluidsubstancecan flow conform A fluid is a substance that can flow. Fluids conform to the boundaries of any container in which we put them.

fluids in propertiescan varyfrom point to pointmore usefuldensity pressure massforce With fluids, we are more interested in the extended substance, and in properties that can vary from point to point in that substance. It is more useful to speak of density and pressure than of mass and force Density and Pressure densityat any point volume element the mass density Density: To find the density 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) the limit of this ratio In practicesmooth uniform densitydensity 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 Pressureat any point area element the magnitude Pressure: To find the Pressure at 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. Pressure The Pressure is then (10-3) Pressurethe limit of this ratio Pressure 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 (10-4) ( Pressure of uniform force on flat area ) force on flat area ) The SI unit of pressure: Atmosphere (at sea lever) Millimeter of mercury Millimeter of mercury (mmHg) Pascal Pascal 1Pa=1N/m 2

10-4 Fluids at Rest Three forces act on the column Three forces act on the column: pressureincreasesdepth in waterpressuredecreases altitudein atmosphere. The pressure increases with depth in water. The pressure decreases with altitude in atmosphere. originthe surface an imaginary column the depths the upper lower column fases 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. the top acts at the top of the column; gravitational force The gravitational force of the column at the bottem acts at the bottem of the column;

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

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

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

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

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

10-7 Archimedes' Principle (10-19) (apparent weight) Archimedes' Principle partially or whollyimmersed a buoyant force acts on the bodydirected upwardequal to the weight of the fluiddisplaced by the body When a body is partially or wholly immersed 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. Apparent Weight in a Fluid

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

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

10-8 Ideal Fluids in Motion StreamlinesStreamlines Streamlines never cross because, if they did, a fluid particle arriving at the intersection would have to assume two different velocities simultaneously, an impossibility. streamline fluid element 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.

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

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

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

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

We need be concerned only with chan- ges that take place at the input and out- put ends Bernoulli's Equation the principle of con- servation of energy initial state (Fig.(a)) final state (Fig.(b)). 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)). Proof of Bernoulli's Equation Apply energy conservation in the form of the work-kinetic energy theorem (10-31) (10-32)

10-10 Bernoulli's Equation Proof of Bernoulli'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 done at the input end to push the entering fluid into the tube and by the system at the output end to push forward the fluid ahead of the emerging fluid.

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

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