Chapter 9: Solids and Fluids

Chapter 9: Solids and Fluids • Three states of matter • Normally matter is classified into one of three (four) states: • solid, liquid, gas (, plasma). solid : crystalline solid (salt etc.) amorphous solid (glass etc.) ordered structure atoms arranged at almost at random States of Matter

Three (four) states of matter (cont’d) • Normally matter is classified into one of three (four) states: • solid, liquid, gas (, plasma). liquid : A molecule in a liquid does random-walk through a series of interactions with other molecules. • - For any given substance, the liquid • state exists at a higher temperature • than the solid state. • The inter-molecular forces in a liquid • are not strong enough to hold mole- • cules together in fixed position. • The molecules wander around in • random fashion. States of Matter

Three (four) states of matter (cont’d) • Normally matter is classified into one of three (four) states: • solid, liquid, gas (, plasma). gas : In gaseous state, molecules are in constant random motion and exert only weak forces on each other. • The average distance between the molecules of a gas is quite • large compared with the size of molecules. • Occasionally the molecules collide with each other, but most of • them move freely. • Unlike solids and liquids, gases can be easily compressed. States of Matter plasma : At high temperature, electrons of atoms are free from nucleus. Such a collection of ionized atoms with equal amounts of positive (nucleus) and negative charges (electrons) forms a state called plasma.

Stress, strain and elastic modulus • Until external force becomes strong enough to deform permanently • or break a solid object, the effect of deformation by the external • force goes back to zero when the force is removed – Elastic behavior. • Stress : the force per unit area causing a deformation • Strain : a measure of the amount of the deformation • Elastic modulus : proportionality constant, similar to a spring constant stress = elastic modulus x strain Deformation of Solids

Young’s modulus: elasticity in length • Consider a long bar of cross-sectional area A and length L0, • clamped at one end. When an external force F is applied along • the bar, perpendicular to the cross section, internal forces in the • bar resist the distortion that F tends to produce. • Eventually the bar attains an • equilibrium in which: • (1) its length is greater than L0 • (2) the external force is balanced • by internal forces. Deformation of Solids The bar is said to be stressed. Young’s modulus SI unit: dimensionless tensile stress tensile strain SI unit: Pa = 1 N/m2

Young’s modulus: elasticity in length (cont’d) • Typical values Deformation of Solids • Stress vs. strain

Shear modulus: Elasticity of shape • Another type of deformation occurs when an object is subjected to • a force F parallel to one of its faces while the opposite face is held • fixed by a second force. • The stress in this • situation is called • a shear stress. Deformation of Solids Shear modulus SI unit: dimensionless shear stress shear strain SI unit: Pa = 1 N/m2

Bulk modulus: Volume elasticity • Suppose that the external forces acting on an object are all • perpendicular to the surface on which the force acts and are • distributed uniformly. • This situation occurs when a • object is immersed in a fluid. Deformation of Solids bulk modulus SI unit: dimensionless volume stress volume strain SI unit: Pa = 1 N/m2

An example • Example 9.3 : Stressing a lead ball A solid lead sphere of volume 0.50 m3, dropped in the ocean, sinks to a depth of 2.0x103 m, where the pressure increases by 2.0x107 Pa. Lead has a bulk modulus of 4.2x1010 Pa. What is the change in volume of the sphere? Deformation of Solids

Density • The density r of an object is defined as: M: mass, V: volume SI unit: kg/m3 (cgs unit: g/cm3 ) • The specific gravity of a • substance is the ratio of • its density to the density • of water at 4oC, which is • 1.0x103 kg/m3, and it is • dimensionless. Density and Pressure

Pressure • Fluids do not sustain shearing stresses, so • the only stress that a fluid can exert on a • submerged object is one that tends to • compress it, which is a bulk stress. • The force F exerted by the fluid on the • object is always perpendicular to the • surfaces of the object. Density and Pressure • If F is the magnitude of a force exerted • perpendicular to a given surface of area A, • then the pressure P is defined as: F: force, A: area SI unit: Pa = N/m2

Variation of pressure with depth • When a fluid is at rest in a container, all portions of the fluid must • be in static equilibrium – at rest with respect to the observer. • All points at the same depth must be at the same pressure. If this • were not the case, fluid would flow from the higher pressure region • to the lower pressure region. • Consider an object at rest • with area A and height h in • a fluid. Density and Pressure • Effect of atmospheric pressure: P0 : atmospheric pressure, P: pressure at depth h

Examples • Example 9.5 : Oil and water r=0.700 g/cm3 h1=8.00 m r=1025 kg/m3 h2=5.00 m Density and Pressure

Pascal’s principle • A change in pressure applied to an enclosed fluid is transmitted • undiminished to every point of the fluid and to the walls of the • container. • Hydraulic press Density and Pressure F2 > F1 if A2 > A1

Car lift • Example 9.7 : Car lift (a) Find necessary force by compressed air at piston 1. weight=13,300 N (b) Find air pressure. Density and Pressure circular x-sec (c) Show the work done by pistons is the same. r1=5.00 cm r2=15.0 cm

Absolute and gaugepressure • An open tube manometer (Fig.(a)) P=PA=PB measures the gauge pressure P-P0 P : absolute pressure P0 : atmospheric pressure • A mercury barometer (Fig.(b)) Pressure Measurements measures the atmospheric pressure vacuum • One atmospheric pressure defined as the pressure equivalent of a column of mercury that is exactly 0.76 m in height.

Blood pressure measurement • A specialized manometer • (sphygmomanometer) • A rubber bulb forces air into a cuff wrap. • A manometer is attached under cuff and • is under pressure. • -The pressure in the cuff is increased until • the flow of blood through brachial artery • is stopped. • -Then a valve on the bulb is opened, and • measurer listens with a stethoscope to • the artery at a point just below the cuff. • -When the pressure at the cuff and the • artery is just below the max. value • produced by heart (the systolic pressure), • the artery opens momentarily on each beat. • -At this point, the velocity of the blood is high, • and the flow is noisy and can be heard… Pressure Measurements

Archimedes’s principle Any object completely or partially submerged in a fluid is buoyed up by a force with magnitude equal to the weight of the fluid displaced by the object. Upward force (buoyant force) : Buoyant Forces and Archimedes’s Principle Downward force:

Archimedes’s principle and a floating object Upward force (buoyant force) : Vobj Downward force: Vfluid Buoyant Forces and Archimedes’s Principle

Examples • Example 9.8 : A fake or pure gold crown? Is the crown made of pure gold? Tair =7.84 N Twater =6.86 N Buoyant Forces and Archimedes’s Principle rgold=19.3x103 kg/m3

Examples • Example 9.9 : Floating down the river What depth h is the bottom of the raft submerged? A=5.70 m2 rwood=6.00x102 kg/m3 Buoyant Forces and Archimedes’s Principle

Some terminology • When a fluid is in motion: • (1) if every particle that passes a particular point moves along exactly • the same smooth path followed by previous particles passing the • point, this path is called streamline. If this happens, this flow is said • to be streamline or laminar. • (2) the flow of a fluid becomes irregular, or turbulent, above a certain • velocity or under any conditions that can cause abrupt changes in • velocity. • Ideal fluid : Fluid in Motion • The fluid is non-viscous : • There is no internal friction force between adjacent layers. • The fluid is incompressible : • Its density is constant. • The fluid motion is steady : • The velocity, density, and pressure at each point in the fluid do not • change with time. • The fluid moves without turbulence : • Each element of the fluid has zero angular velocity about its center.

Equation of continuity • Consider a fluid flowing through a pipe of non-uniform size. The • particles in the fluid move along the streamlines in steady-state flow. In a small time interval Dt, the fluid entering the bottom end of the pipe moves a distance: The mass contained in the bottom blue region : Fluid in Motion From a similar argument : Since DM1=DM2(flow is steady): Equation of continuity

An example • Example 9.12 : Water garden Fluid in Motion

Bernoulli’s equation • Consider an ideal fluid flowing through a pipe of non-uniform size. Work done to the fluid at Point 1 during the time interval Dt: Work done to the fluid at Point 2 during the time interval Dt: Fluid in Motion Work done to the fluid :

Bernoulli’s equation (cont’d) If m is the mass of the fluid passing through the pipe in Dt , the change in kinetic energy is: The change in gravitational potential energy in Dt is: Fluid in Motion From conservation of energy:

Bernoulli’s equation (cont’d) From conservation of energy: Fluid in Motion Bernoulli’s equation

Venturi tube Consider a water flow through a horizontal constricted pipe. Fluid in Motion

Examples h =0.500 m y1 =3.00 m • Example 9.13 : A water tank • Consider a water tank with a hole. • Find the speed of the water • leaving through the hole. y x Fluid in Motion (b) Find where the stream hits the ground.

Examples • Example 9.14 : Fluid flow in a pipe A2=1.00 m2 A1=0.500 m2 h =5.00 m Find the speed at Point 1. Fluid in Motion

Surface tension • The net force on a molecule at A is zero • because such a molecule is completely • surrounded by other molecules. • The net force on a molecule at B is downward • because it is not completely surrounded by • other molecules. There are no molecules • above it to exert upward force. this asymmetry • makes the surface of the liquid contract and • the surface area as small as possible. Surface Tension, Capillary Action, and Viscous Fluid Flow • The surface tension is defined as : where the surface tension force F is divided by the length L along which the force acts. SI unit : N/m=(N m)/m2=J/m2

Surface tension (cont’d) • The surface tension of liquids • decreases with increasing • temperature, because the faster • moving molecules of a hot liquid • are not bound together as strongly • as are those in a cooler liquid. Surface Tension, Capillary Action, and Viscous Fluid Flow • Some ingredient called surfactants • such as detergents and soaps decrease surface tension. • The surface tissue of the air sacs in the lungs contain a fluid that has • a surface tension of about 0.050 N/m. As the lungs expand during • inhalation, the body secretes into the tissue a substance to reduce • the surface tension and it drops down to 0.005 N/m.

Surface of liquid Surface Tension, Capillary Action, and Viscous Fluid Flow • Forces between like-molecules such as between water molecules are • called cohesive forces. • Forces between unlike-molecules such as those exerted by glass on • water are called adhesive forces. • Difference in strength between cohesive and adhesive forces creates • the shape of a liquid at boundary with other materials.

Viscous fluid flow • Viscosity refers to the internal friction of a fluid. It is very difficult for • layers of a viscous fluid to slide past one another. • When an ideal non-viscous fluid flows • through a pipe, the fluid layers slide • past one another with no resistance. • If the pipe has uniform cross-section • each layer has the same velocity. Surface Tension, Capillary Action, and Viscous Fluid Flow ideal fluid, non-viscous • The layers of a viscous fluid have • different velocities. The fluid has the • greatest velocity at the center of the • pipe, whereas the layer next to the wall • does not move because of adhesive • forces between them. viscous fluid

Viscous fluid flow • Consider a layer of liquid between two solid surfaces. The lower surface • is fixed in position, and the top surface moves to the right with a velocity • v under the action of an external force F. • A portion of the liquid is distorted from • its original shape, ABCD, at one instance • to the shape AEFD a moment later. The • force required F to move the upper plate • at a fixed speed v is : • where h is the coefficient of viscosity of • the fluid, and A is the area in contact with fluid. Surface Tension, Capillary Action, and Viscous Fluid Flow 1 poise=10-1 N s/m2 1 cp (centipoise) = 10-2 poise SI unit : N s/m2 cgs unit: dyne s/cm2= poise h

Poiseuille’s law • Consider a section of tube of • length L and radius R • containing a fluid under • pressure P1 at the left end and • a pressure P2 at the right. • Poiseuille’s law describes the • flow rate of a viscous fluid • under pressure difference: Surface Tension, Capillary Action, and Viscous Fluid Flow

Reynolds number • At sufficiently high velocities, fluid flow changes from simple streamline • flow to turbulent flow, characterized by a highly irregular motion of the • fluid. Experimentally the onset of the turbulence in a tube is determined • by a dimensionless factor called Reynolds number, RN, given by: r : density of fluid v : average speed of the fluid along the direction of flow Surface Tension, Capillary Action, and Viscous Fluid Flow d : diameter of tube h : viscosity of fluid • If RN is below about 2000, the flow of fluid through a tube is streamline. • If RN is above about 3000, the flow of fluid through a tube is turbulent. • If RN is between 2000 and 3000, the flow is unstable.

Examples • Example 9.18 : A blood transfusion A patient receives a blood transfusion through a needle of radius 0.20 mm and length 2.0 cm. The density of blood is 1,050 kg/m3. The bottle supplying the blood is 0.50 m above the patient’s arm. What is the rate of the flow through the needle? Surface Tension, Capillary Action, and Viscous Fluid Flow

Examples • Example 9.19: Turbulent flow of blood Determine the speed at which blood flowing through an artery of diameter 0.20 cm will become turbulent. Surface Tension, Capillary Action, and Viscous Fluid Flow

A fluid can move place to place as a result of difference in • concentration between two points in the fluid. There are two • processes in this category : diffusion and osmosis. • Diffusion • In a diffusion process, molecules move from a region where their • concentration is high to a region where their concentration is lower. • Consider a container in which a • high concentration of molecules • has been introduced into the left • side (the dashed line is an • imaginary barrier). Transport Phenomena All the molecules move in random direction. Since there are more molecules on the left side, more molecules migrate into the right side than otherwise. Once a concentration equilibrium is reached, there will be no net movement.

Diffusion (cont’d) • Fick’s law where D is a constant of proportion called the diffusion coefficient (unit : m2/s), A is the cross-sectional area, (…) is the change in concentration per unit distance (concentration gradient), and DM/Dt is the mass transported per unit time. The concentrations, C1 and C2 are measured in unit of kg/m3. Transport Phenomena

Size of cells and osmosis • Diffusion through cell membranes is vital in supplying oxygen to the • cells of the body and in removing carbon dioxide and other waste • products from them. • A fresh supply of oxygen diffuses from the blood, where its concentration is high, into the cell, where its concentration is low. • Likewise, carbon dioxide diffuses from the cell into the blood where its concentration is lower. Transport Phenomena • A membrane that allows passage of some molecules but not others is called a selectively permeable membrane. • Osmosis is the diffusion of water across a selectively permeable • membrane from a high water concentration to a low water • concentration.

Motion through a viscous medium • The magnitude of the resistive force on a very small spherical • object of radius r moving slowly through a fluid of viscosity h with • speed v is given by: resistive frictional force Stokes’s law • Consider a small sphere of radius r falls • through a viscous medium. Transport Phenomena force of gravity buoyant force

Motion through a viscous medium (cont’d) • At the instance the sphere begins to fall, the force of friction is • zero because the velocity of the sphere is zero. • As the sphere accelerates, its speed increases • and so does Fr. resistive frictional force • When the net force goes to zero, the speed • of the sphere reaches the so-called terminal • speed vt. Transport Phenomena force of gravity buoyant force

Sedimentation and centrifugation • If an object is not spherical, the previous argument can still be • applied except for the use of Stokes’s law. In this case, we assume • that the relation Fr=kv holds where k is a coefficient. Terminal speed condition Transport Phenomena

Sedimentation and centrifugation (cont’d) • The terminal speed for particles in biological samples is usually • quite small; the terminal speed for blood cells falling through plasma • is about 5 cm/h in the gravitational field of Earth. • The speed at which materials fall through a fluid is called • sedimentation rate. The sedimentation rate in a fluid can be • increased by increasing the effective acceleration g: for example • by using radial acceleration due to rotation • (centrifuge). Transport Phenomena

Chapter 9: Solids and Fluids