Physics 153 Chapter 11 Notes PDF

Preview

Summary

notes from lecture 7 ...

Description

Fluids (Statics and Dynamics): Chapter 11 (sections 1-11) 11.1 Define Mass Density ● Fluids are materials that can flow; include both gases and liquids ● Mass density of a liquid or a gas is one of the important factors that determine its behavior as a fluid ○ The mass density p  is the mass m d  ivided its volume V

○ ○ Units = kg/m^3 ● Gasses have the smallest densities because gas molecules are relatively far apart and a gas contains a large fraction of empty space ○ Sensitive to changes in temperature and pressure ● Molecules are much more tightly packed in liquids and solids, and the tighter packing leads to larger densities ○ Not much sensitive to changes in temperature and pressure ● Specific gravity is its its density divided by the density of a standard reference material, usually chosen to be water at 4 degrees Celsius (1.000*10^3 kg/m^3)

○ 11.2 Pressure  xerted by a fluid is defined as the magnitude F  of the ● The pressure P e  ver which force acting perpendicular to a surface divided by the area A o the force acts ○ P=F/A ■ *area of the shape of the described thing ● i.e for circle use A=pi*r^2 ○ Units = N/m^2 or pascals (Pa) ■ 10^5 Pa = 1 barr ● In general, a static fluid cannot produce a force parallel to a surface

○ If it did, the surface would apply a reaction force to the fluid, consistent with Newton’s action-reaction law ■ Thus the fluid would not be static ● While fluid pressure can generate force, which is a vector quantity, pressure itself is not a vector ● Atmospheric pressure at sea level: 1.013 * 10^5 atmospheres = 14.7 lb/in^2 ● The force generated by the pressure of a static fluid is always perpendicular to the surface that the fluid contacts 11.3 Pressure and Depth in a Static Fluid ● External forces that act on a fluid at rest ○ Gravitational force (the weight of the fluid) ○ Collisional force responsible for fluid pressure ■ Since fluid is at rest, acceleration is zero and is in equilibrium ● Net force = 0 ● Container of fluid, more pressure generated upward because of how bottom is supporting more weight ○ ○ ■ The pressure increment pgh is affected by vertical distance h, but not any horizontal distance within the fluid ● With gases ○ The lower layers are compressed markedly by the weight of the upper layers, with the result that the density varies with vertical distance ■ The relation

can only be used when h is

small enough and any variation in p is negligible 11.4 Pressure Gauges ● One of the simplest pressure gauges is the mercury barometer used for measuring atmospheric pressure ○ Sealed at one end, filled with mercury, then inverted

○ Space above the mercury in the tube is empty and the pressure P1 is nearly zero there ○ The pressure P2 at Point A at the bottom of the column is the same as the atmospheric pressure at point B because these two points are at the same level ■ ● Another pressure gauge is the open tube manometer ○ One side of the U-tube is open to atmospheric pressure ■ Tube often contains a liquid like mercury ○ Other side is connected to the container whose pressure P2 is measured ○ When pressure in container is greater than atmospheric pressure, the liquid in the tube is pushed downward on the left side and upward on the right side ■ ■ Gauge pressure is how the height is proportional to P2-Patm ● The amount by which the container pressure differs from atmospheric pressure ■ Absolute pressure is the actual value for P2 11.5 Pascal’s Principle ● With the definition of pressure, P1=F1/A1 ○ To know the pressure P2 at any deeper place in the liquid, we just add the increment to P1 ■ The pressure P1 adds to the pressure pgh due to the depth of the liquid at any point ● If the applied pressure P1 is increased or decreased, the pressure at any other point within the confined liquid changes correspondingly ○ This is known as Pascal’s Principle: ■ Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and the enclosing walls ● P2=P1

● 11.6 Archimedes’ Principle ● Buoyant force describes how water pushes back with a strong upward force, and all fluids apply such a force to objects that are immersed in them ○ Exists because fluid pressure is larger at greater depths ● Cylinder of height h i s being held under the surface of a liquid ○ Since pressure is greater at greater depths, the upward force exceeds the downward force - consequently the liquid applies to the cylinder a new upward force, a buoyant force ■ ● FB= pghA ● Archimedes' Principle ○ Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces

■ ■ No matter what the shape is, the buoyant force pushes it upwards ● Any object that is solid throughout will float in a liquid if the density of the object is less than or equal to the density of the liquid 11.7 Fluids in Motion ● Fluid flow can be steady or unsteady ○ In steady flow the velocity of the fluid particles at any point is constant as time passes ○ Unsteady flow exists whenever the velocity at a point in the fluid changes as time passes

○ Turbulent flow is an extreme kind of unsteady flow and occurs when there are sharp obstacles ■ Velocity at a point changes erratically from moment to moment both in magnitude and direction ● Fluid flow can be compressible or incompressible ○ Most liquids are nearly incompressible ■ Density of a liquid remains almost constant as the pressure changes ○ Gases are highly compressible ■ There are instances where the density of a flowing gas remains constant enough that the flow can be incompressible ● Fluid flow can be viscous or nonviscous ○ Large viscosity = does not flow readily ○ Small viscosity = flows readily ○ Flow of a viscous fluid is an energy dissipating process ■ The viscosity hinders neighboring layers of fluid from sliding freely past one another ■ A fluid with zero viscosity flows in an unhindered manner with no dissipation of energy ● Not really seen at normal temperatures, but some may have negligible velocity ○ Ideal fluid = an incompressible, nonviscous fluid ● When flow is steady ○ Streamlines are often used to represent the trajectories of the fluid particles ■ Drawn such that a tangent to the streamline at any point is parallel to the fluid velocity at that point ■ Steady flow is often called streamline flow 11.8 The Equation of Continuity ● The equation of continuity expresses: If a fluid enters one end of a pipe at a certain rate, then the fluid must also leave at the same rate, assuming that there are no places between the entry and exit points to add or remove fluid

● ● Equation of continuity - the mass flow rate has the same value anywhere in the tube ○ The mass flow rate (pAv) has the same value at every position along a tube that has a single entry and a single exit point for fluid flow

○ ● Incompressible fluid ○ The density does not change during flow, so p1=p2 ● Volume flow rate Q:

○ 11.9 Bernoulli’s Equation ● Observations about moving fluid ○ Whenever a fluid is flowing in a horizontal pipe and encounters a region of reduced cross-sectional area, the pressure of the fluid drops, as the pressure gauges indicate ■ The accelerating fluid must be subjected to an unbalanced force ○ If the fluid moves to a higher elevation, the pressure at the lower level is greater than the pressure at the higher level ● Bernoulli’s Equation ○ In the steady flow of a nonviscous, incompressible fluid of density p, the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by

■ ● Since points 1 and 2 were selected arbitrarily, the term P+1/2pv2^2+pgy has a constant value at all positions in the flow ○ P2=P1+pgh 11.10 Applications of Bernoulli’s Equation ● When a moving fluid is contained in a horizontal pipe, all parts of it have the same elevation (y1 =y2), and Bernoulli’s equation ○ ■ If v increases, P decreases ● Torricelli’s theorem ○ When the liquid is assumed to be an ideal fluid, and the speed with which it leaves the pipe is the same as if the liquid had freely fallen through a height ○ If the outlet piper were pointed directly upward, the liquid would rise to a height h equal to the fluid level above the pipe ○ If the liquid is not an ideal fluid, its viscosity cannot be neglected ■ Then the efflux speed would be less than that given by Bernoulli’s equation, and the liquid would rise to a height less than h 11.11 Viscous Flow ● In an ideal fluid, no viscosity to hinder the fluid layers as they slide past one another ○ The adjacent fluid layer moves along with it and remains at rest relative to the moving surface ● We may imagine the fluid to be composed of many thin horizontal layers ○ The velocity of each layer is different, changing the uniformly from v at the top plate to zero at the bottom plate ■ The resulting flow is called laminar flow since a thin layer is often referred to as a lamina

● As each layer movies, it is subjected to viscous forces from its neighbors ● The purpose of the force F is to compensate for the effect of these forces, so that any layer can move with a constant velocity ● The amount of force required depends on several factors ○ Larger areas A, being in contact with more fluid requires larger forces so that the force is proportional to the contact area ○ For a given area, greater speeds require larger forces, with the result that the force is proportional to the speed ○ The force is also inversely proportional to the perpendicular distance y between the top and bottom plates ○ All three expressed as: F is proportional to Av/y ● Coefficient of Viscosity or Viscosity ○ FORCE NEEDED TO MOVE A LAYER OF VISCOUS FLUID WITH A CONSTANT VELOCITY ■ The magnitude of the tangential force F required to move a fluid layer at a constant speed v, when the layer has an area A and is located to a perpendicular distance y from an immobile surface, is given by

● ● Under ordinary conditions, the viscosities of liquids are significantly larger than those of gases ● The viscosities of either liquids or gases depend markedly on temperature ○ Viscosities of liquids decrease as the temperature is increased

○ Viscosities of gases increase as the temperature is raised ● Factors that determine the volume flow rate Q (in m^3/s) ○ A difference in pressures P2-P1 must be maintained between any two locations along the pipe for the fluid to flow ■ Q is proportional to P2-P1, a greater pressure difference leading to a larger flow rate ○ A long pipe offers greater resistance to the flow than a short pipe does and Q is inversely proportional to length L ○ High viscosity fluids flow less readily than low-viscosity fluids, and Q is inversely proportional to the viscosity n ○ The volume flow rate is larger in a pipe of a larger radius, other things being equal ● Poiseuille’s Law ○ A fluid whose viscosity is n, flowing through a pipe of radius R and length L, has a volume flow rate Q given by

○...