Published on February 12, 2008
Solids and Fluids: Solids and Fluids Four known states of matter; solids, liquids, gases, and plasmas Plasmas, systems of charged particles, are the most common state of matter in the universe Understanding properties of these states is important in the physical and health sciences Solids: Solids Have a definite volume and shape Is play-dough a solid or liquid? Silly putty? Molecules of a solid vibrate about specific equilibrium positions due to electric forces The molecular structure can be modeled as springs connecting spheres Crystalline Solid: Crystalline Solid The atoms have an ordered structure This example is salt Gray spheres represent Na+ ions Green spheres represent Cl- ions Amorphous Solid: Amorphous Solid The atoms are arranged almost randomly Examples include glass Liquids: Liquids Have a definite volume but no definite shape They exist at a higher temperature than when in their solid state The molecules “wander” through the liquid in a random fashion The intermolecular forces are not strong enough to keep the molecules in a fixed position Gases: Gases Have no definite volume or shape The molecules are in constant random motion and exert only weak forces on each other Average distance between molecules is large compared to the size of the molecules Plasma: Plasma When matter is heated to a very high temperature many of the electrons are freed from the nucleus The result is a collection of free ions Plasmas mostly exist inside stars Deformation of Solids: Deformation of Solids All objects are deformable It is possible to change the shape or size (or both) of an object through the application of external forces Internal resistive forces of molecules cause materials to return to their original shape This is a deformation that exhibits elastic behavior Bouncing a basketball is an example Deformation of Solids: Deformation of Solids Materials can be deformed in three basic ways Stretched (tension) Squeezed (compression) Twisted (shear) We measure a material’s elasticity in terms of stress and strain Stretch a rubber band and then fold it in two and apply the same force Does it stretch as much? Why not? Stress: Stress The stress in the single rubber band was twice the stress in the doubled because stress is the force per unit area causing the deformation So stress is σ = F/A, where F is the internal force and A is the cross-sectional area of the internal surface What would the metric and English units be? Units for Stress: Units for Stress N/m2 is a Pascal (Pa) named for Blaise Pascal the French physicist, philosopher, and mathematician lb/in2 is a PSI as in tire pressure (stress applied to the inner surface of tires) Pressure is compressive stress and we usually use the term when dealing with fluids (it is also 3-dimensional) Compressive Stress: Compressive Stress If a 10 N force compresses a bone of radius 1 cm, what is the stress? What if the radius were 2 cm? Why do lumbar vertebrae and discs have larger cross-sectional areas than cervical? Compressive stress is the normal stress that occurs at the cross-sectional plane due to a load that tends to push molecules together Shear Stress: Shear Stress Shear stress is a transverse stress acting parallel to the cross-sectional plane due to forces acting parallel to this plane Most loading situations are not so simple A cross-sectional plane can experience multiple stresses or varying stresses across the plane Tensile Stress: Tensile Stress Tensile stress (σt) is due to a load that pulls apart the molecules σt = F/A Tensile because the bar is under tension More Stress: More Stress Bend a pencil. What stresses are exerted on the pencil? Where are the stresses the greatest? Do bones experiences similar stresses? What part of a bone is the strongest? The greatest stress a material can withstand without breaking is the ultimate strength of the material Slide16: Tendon 6.89 x 107 Bone shear strength Muscle 5.5 x 105 2.75 x 107 Shear 2.75 x 107 Stressful Problems?: Stressful Problems? If the ulna and radius in the forearm have a combined cross sectional area of 2.1 cm 2, an effective mass of 2.6 kg, an ultimate compressive strength of 1.5 x 108 Pa (N/m2) and comes to rest at constant deceleration in 2 ms, what is the maximum speed before impact without breaking a bone? Stress Problem: Stress Problem A driver’s head has a mass of 5 kg and the area head which would hit a windshield in an accident is 25cm2. The compressive strength of bone is 1.5 x 108 Pa. How much force is required to fracture the skull? What acceleration would that require? If the head came to rest in 3 ms, what could have been the maximum velocity of the head? More Stress Problems: More Stress Problems From what height can a 70.0 kg person land straight legged without breaking their tibia if the impact distance is 1.0 cm? The cross-sectional area of the tibia is 3.4 x 10-4 m2. What other kinds of damage could be done? What if you landed in 1 m of snow? What if you did not land straight legged and were able to absorb some of that energy? Strain: Strain A measure of the amount of relative deformation defined as ε = Δl/l0, where Δl is the change in length and l0 is the original length. Note it is dimensionless A 20 cm bone is compressed 0.2 cm, what is the strain? For shear strain the formula is ε = Δx/h Poisson Effect: Poisson Effect What happens to the cross-sectional area of a rubber band when it is stretched? compressed? The Poisson effect is named for S. D. Poisson, a French scientist investigating this phenomena in the 1820s Poisson’s ratio is that of axial strain to transverse strain, typically .25 to .35 The compressive load on intervertebral discs shorten them vertically but laterally they bulge out Too much bulge can cause a ruptured disc Creep: Creep Certain materials deform over time when subjected to a constant load The deformation-time curve approaches a maximum value assymptotically An individual loses a certain amount of height after standing all day due to the constant force of gravity Stress vs. Strain: Stress vs. Strain Compare two graphs of stress vs. strain Which material is most rigid? The slope of the line is Young’s modulus Y = Δσ/Δε = FL0/AΔL Y is a constant for small stresses The elastic modulus can be thought of as the stiffness of the material, like a spring constant Young’s Modulus: Active figure: Young’s Modulus: Active figure The elastic limit is the point at which deformation is permanent The ultimate strength is the maximum stress a material can handle before breaking. In terms of strain this is its extensibility The area under the line is the energy absorbed and represents toughness Breaking Point: Breaking Point For a brittle material, the breaking point is just beyond its ultimate strength For a ductile material, after passing the ultimate strength the material thins and stretches at a lower stress level before breaking Shear Modulus:Elasticity of Shape : Shear Modulus: Elasticity of Shape Forces may be parallel to one of the object’s faces Whiplash injuries are due to shear stresses Wearing a seatbelt too high can can cause a broken back due to shear S = (F/A)/(Δx/h) S is the shear modulus Active Figure Whiplash Problem: Whiplash Problem In a rear-end collision, if impact time is 0.010 sec, what is the minimum velocity at which there is danger from neck fracture? Assume the area of the cervical vertebra is 1.0 cm2 and the mass of the head is 5.0 kg Pressure active figure: Pressure active figure When an object is immersed in fluid it is compressed on all sides (uniform squeezing) The forces act perpendicular to all the surfaces There is no Poisson effect Volume stress, or pressure (P) = F/A The volume strain = ∆V/V0 There is a change in volume but no change in shape Bulk Modulus: Bulk Modulus The ratio of this volume stress to volume strain is the bulk modulus of the material It measures the compressibility of a material Both solids and fluids have bulk moduli The compressibility is the reciprocal of the bulk modulus Solids have Young’s, Bulk, and Shear moduli Liquids have only bulk moduli, they will not undergo a shearing or tensile stress The liquid would flow instead Post and Beam Arches: Post and Beam Arches A horizontal beam is supported by two columns Used in Greek temples Columns are closely spaced Limited length of available stones Low ultimate tensile strength of sagging stone beams Semicircular Arch: Semicircular Arch Developed by the Romans Allows a wide roof span on narrow columns Stability depends on the compression of the wedge-shaped stones Compare paper’s ability to support loads while flat and folded like an accordion Why is the vertebral column curved in the lateral view? Gothic Arch: Gothic Arch Extremely high arches first used in Europe in the 12th century The flying buttresses are needed to prevent the spreading of the arch supported by the tall, narrow columns Density: What are some physical similarities and differences between a golf ball and a ping pong ball? The density of a substance of uniform composition is defined as its mass per unit volume: ρ = m/V Units are kg/m3 (SI) or g/cm3 (cgs) or g/ml 1 g/cm3 = 1000 kg/m3 The densities of most liquids and solids vary slightly with changes in temperature and pressure Densities of gases vary greatly with changes in temperature and pressure. How so? Density Specific Gravity: Specific Gravity The specific gravity of a substance is the ratio of its density to the density of water at 4° C The density of H2O at 4° C is 1000 kg/m3 or 1 g/cm3 Specific gravity is a unitless ratio So a substance with a density of 12.3 g/cm3 has a specific gravity of 12.3 A substance with a specific gravity of 130 kg/m3 has a specific gravity of 0.13 Rank in order by mass, 1 m3 of gold, 2 m3 of silver, 6 m3 of aluminum Fluid Pressure: Pushed or Pulled?: Fluid Pressure: Pushed or Pulled? Fluids cannot be sheared or tensiled nor can they exert shear or tension Fluids can only compress or be compressed. We refer to this as pressure Measuring Pressure in a Fluid: Measuring Pressure in a Fluid This device is submerged in a fluid The spring is calibrated by a known force The force the fluid exerts on the piston is then measured Benefits of Air Bags: Benefits of Air Bags What is the advantage of snow shoes? Wider tires? How can someone lie on a bed of nails? Why are metal cleats hard on a vinyl floor? What is an advantage of a small contact area? Air bags can spread impulse over a larger time interval and energy over a longer distance thereby reducing average impact force Air bags also spread impact force over a larger area reducing the pressure on any given point There is less chance for puncture, bone fracture, or damage to internal organs and tissues Variation of Pressure with Depth: Variation of Pressure with Depth If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium Why is still water always level? All points at the same depth must be at the same pressure otherwise the fluid would flow from the higher pressure region to the lower pressure region creating equilibrium Variation of Pressure with Depth: Variation of Pressure with Depth Why is there more pressure at greater depths? P2 = P1 + ρgh, where ρ is the density of the fluid, and h is the depth Deep sea exploration vehicles are made of very strong material Pressure and Depth: Pressure and Depth The pressure does not depend on the shape of the container What is wrong with this statement? The syringe draws blood Normal atmospheric pressure is 1.013 x 105 Pa or 14.7 lb/in2 Pressure and Depth: Pressure and Depth Both dams are equally long and deep Which dam has the most pressure? Pressure and Depth: Pressure and Depth Why are the bands around silos closer together at the bottom? Why are blood pressure cuffs placed around the upper arm? How does the explain the reason for elevating injured extremities that are bleeding or swelling? Atmospheric Pressure: Atmospheric Pressure The atmosphere is also a fluid with pressure Hitters love to play in Colorado but not curveball pitchers. Why? What is wrong with these statements? He drew in a breath A vacuum sucks up dirt Meteorologists multiply measured pressure by a factor determined by elevation to report normal atmospheric pressure as 29.92 in Hg This way reported differences in pressure between two nearby cities is due to weather conditions and not differences in elevation Intravenous Solution: Intravenous Solution Intravenous fluids are introduced from a certain height in order to attain a certain pressure at the point of insertion Why not intra-arterial fluid? Does the length or diameter of the tube matter? Why are these fluids in a bag and not a bottle? What is the key to siphoning fluids from a container? Atmospheric Pressure: Atmospheric Pressure Using P = P0 + ρgh the pressure 10.3m deep in water is twice standard atmospheric pressure. What is the pressure 10.3m above sea level? (ρair = 1.29 kg/m3) At what elevation is atmospheric pressure half of standard atmospheric pressure? What factor would meteorologists use for this elevation? Pascal’s Principle: 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 Recognized by Blaise Pascal, French scientist (1623 – 1662) Pascal’s Principle: Pascal’s Principle The Hydraulic Press has a MA because F1< F2 The same elevations have the same pressure, so force is multiplied over the larger area, MA = (A2/A1) But Δx1> Δx2 because energy is conserved As the vehicle is lifted would F1 increase or decrease? P = F1/A1 = F2/A2 Pascal’s Principle (Applications): Pascal’s Principle (Applications) How could a water or air mattress prevent decubitis ulcers (bed sores)? A brain tumor which protrudes into the cerebrospinal fluid may cause a measurable increase in pressure in all parts of the fluid A spinal tap made between L3 and L4 measures cerebrospinal fluid pressure in mm or cm of water Pascal’s Principle (Applications): Pascal’s Principle (Applications) The Queckenstedt test involves squeezing the jugular veins which increases intracranial pressure This pressure is transmitted by Pascal’s Principle to the spinal tap manometer If water level does not rise it indicates an obstruction in the cerebrospinal fluid Pascal’s Principle (Applications): Pascal’s Principle (Applications) Amniotic fluid protects the fetus by distributing force to all parts of the amniotic sac as well as the fetus Wearing tight clothing causes continuous pressure on the fetus which might produce fetal deformity The eye contains enclosed fluid so a blow to the eye may not produce noticeable damage to the front but damage to the retina or optic nerve due to transmitted pressure Abnormal collection of fluid in the pericardial and pleural cavities may cause abnormal pressure on the heart and lungs Torricelli’s Barometer: Torricelli’s Barometer Torricelli’s (1608 – 1647) barometer uses Pascal’s principle to measure atmospheric pressure Why is the height of Hg in a column related to atmospheric pressure? Standard atmospheric pressure is the pressure equivalent of 76 cm of Hg in a column at 00C P = ρgh = (13,595 kg/m3)(9.80665 m/s2)(0.7600 m) = 1.013 x 105 Pa or 1 atm or 14.70 lb/in2 Ear Popping Stuff: Ear Popping Stuff How much force does the atmosphere exert on a person whose surface area is 2000 in2? Why do our ears pop when ascending or descending from great heights? If we are exposed to the vacuum of space we would explode do to internal pressure We must leave windows open during tornadoes to prevent glass from exploding outward Blood Pressure: Blood Pressure A manometer uses Pascal’s principle to measure blood pressure with a special type of manometer called a sphygmomanometer It actually measures the air pressure in the cuff in mm of Hg Would the pressure readings change if the cuff were wrapped around the thigh? How so? Why or why not? The pressure P is called the absolute pressure, P = Po + rgh P – Po = rgh is the gauge pressure Archimedes Principle: Archimedes Principle 287 – 212 BC, Greek mathematician, physicist, inventor, and engineer Recognized buoyant force Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object Buoyant Force: Buoyant Force The upward force exerted by a fluid on a submerged or partially submerged object The cause is the pressure difference between the top and the bottom of the object The magnitude is the weight of the displaced fluid It is the same for a totally submerged object of any size, shape, or density What’s “atmospheric” buoyancy? Totally Submerged Object: Totally Submerged Object Why do diet soda cans float and regular soda cans sink in water? The upward buoyant force is B = ρfluidgVobj = mfluidg = Wfluid The downward gravitational force is W = mg = ρobjgVobj The net force is ρfluidgVobj – ρobjgVobj = (ρfluid – ρobj)gVobj Totally Submerged Object: Totally Submerged Object If the net force is positive the object will accelerate up, if negative it will accelerate down If an object is less dense than the fluid the object experiences a net upward force Also it can be shown that the specific gravity of an object, SG = W/B for totally submerged objects Totally Submerged Object: Totally Submerged Object The object is more dense than the fluid The net force is downward The object accelerates downward The net force can be considered an objects submerged weight, Ws So B = W – Ws, the weight of the object in air minus the weight submerged Active figure Body Density: Body Density We can find body density this way to determine a percentage of body fat What is the density of a 90.0 kg man with an apparent mass of 3.7 kg submerged? It is therapeutic for persons with motor difficulty to exercise partially submerged in water Buoyancy force allows greater range of movement of the extremities in order to prevent disuse atrophy If an object is at equilibrium 95% submerged then the buoyancy force is only 95% what it would be if it were totally submerged. Where does the other 5% come from to keep it afloat? Archimedes’ Principle:Floating Object active figure: Archimedes’ Principle: Floating Object active figure The buoyant force is balanced by the weight Volume of the fluid displaced is the same as the volume of the object beneath the fluid level Ice floats higher in water or alcohol? As ice melts in a glass the water level will not go up Which will cause ocean levels to rise, melting ice bergs or melting glaciers? Buoyancy Problems: Buoyancy Problems Two submerged objects of the same volume but different densities will or will not experience the same buoyancy force? Prove mathematically that if an object has a specific gravity of 0.8, then it will float 80% submerged and if 0.7, 70% submerged Human Brain: Human Brain The human brain is immersed in cerebral spinal fluid of density 1007 kg/m3 which is only slightly less than the density of the brain, 1040 kg/m3 It will not float, but 1007/1040 or 96.8% of the brain’s weight is supported by fluid buoyancy If some of the fluid is removed for diagnostic purposes the nerves and blood vessels are placed under great strain and can cause extreme discomfort Great care must be exercised during these procedures until the initial fluid volume is restored Streamline Fluid Flow: Streamline Fluid Flow Every particle that passes a particular point moves exactly along the same smooth path as previous particles that passed that point Also called laminar flow The path is called a streamline Different streamlines cannot cross each other The streamline at any point coincides with the direction of fluid velocity at that point Streamline Flow: Streamline Flow Streamline flow shown in a wind tunnel Notice the smooth lines (no jagged lines or abrupt corners) Turbulent Fluid Flow: Turbulent Fluid Flow The flow becomes irregular exceeds a certain velocity any condition that causes abrupt changes in velocity Eddy currents are a characteristic of turbulent flow Characteristics of an Ideal Fluid: Characteristics of an Ideal Fluid The fluid is non-viscous, there is no internal friction between adjacent layers The fluid is incompressible, its density is constant The fluid motion is steady, its velocity, density, and pressure do not change in time The fluid moves without turbulence No eddy currents are present Its elements have zero angular velocity about their center Equation of Continuityfor Ideal Fluids: Equation of Continuity for Ideal Fluids A1v1 = A2v2 or Vol1/t = Vol2/t The product of the cross-sectional area, A, of a pipe and the fluid speed, v, is a constant Fluid speed is high where the pipe is narrow and fluid speed is low where the pipe has a large diameter Equation of Continuity: Equation of Continuity Av is called the volume flow rate and is constant The volume of fluid that enters one end of the tube in a given time interval is equal to the volume of fluid leaving the tube in the same interval If v1 = 12 cm/s, A1 = 3.1 m2, and A2 = 12.5 m2 what is v2? What is the volume flow rate? If ρ = 1.3 g/cm3 what is the mass flow rate? Equation of Continuity: Equation of Continuity Water flow from a faucet narrows as it accelerates downward. Why? How do you spray long distance with a hose? Daniel Bernoulli: Daniel Bernoulli 1700 – 1782, Swiss physicist and mathematician, did work that was the beginning of the kinetic theory of gases The sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline Bernoulli’s Equation: Bernoulli’s Equation The sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline Or W/V + KE/V + PE/V = constant because of the conservation of energy for ideal non-viscous fluids Or P/ρg + h + ½v2/g = constant which is read: Pressure head + elevation head + velocity head = constant Application - The Venturi Tube: Application - The Venturi Tube Elevation is constant so we ignore ρgy P + 1/2ρv2 = constant so v and P are inversely related Speed changes as diameter changes because of the equation of continuity The higher the velocity the lower the fluid pressure The Venturi tube can measure the speed of the fluid Problem: Problem When inhaling air moves through the bronchus at 15 cm/s. What is the pressure drop in the bronchus? (ρair = 1.29 kg/m3) P1 + ½ρv12 + ρgy1 = P2 + ½ρv22 + ρgy2 Assume no elevation difference P1 + ½ρv12 = P2 + ½ρv22 P1 – P2 = ½ρv22 - ½ρv12 Since v1 = 0 P1 – P2 = ½ρv22 P1 – P2 = ½(1.29)(.15)2 P1 – P2 = .015 Pa or 0.22 PSI Application – Golf Ball: Application – Golf Ball The dimples in the golf ball catch and help move air along its surface With backspin the ball pushes the air down Newton’s Third Law tells us the air must push up on the ball The air over the top has more velocity creating an area of low pressure above and higher pressure below so the ball is buoyed up by the air Application – Golf Ball: Application – Golf Ball The backward spinning ball travels farther The opposite is top spin in which the ball is pushed down Ever hear of a sinking line drive? Application – Airplane Wing: Application – Airplane Wing Air speed above the wing is greater than the air speed below So the air pressure above the wing is less than air pressure below (Bernoulli’s principle) This creates a net upward force called lift Vascular Flutter: Vascular Flutter Blood travels faster through a constricted artery due to the equation of continuity Due to Bernoulli’s principle this causes an area of low pressure and a collapse of the artery The artery then reopens under arterial pressure and the process repeats (flutters) Aneurysms: Aneurysms Blood flows more slowly through an aneurysm where the walls have ballooned This results in higher pressure from the blood that could cause a rupture of the aneurysm Other Applications: Other Applications Sailing up wind Other Applications: Other Applications Tailgating or standing to close to a passing train or truck Atomizers in sprayers Other Applications: Other Applications Plumbing vent pipes Opening windows during a tornado Carburetors Other Applications: Other Applications Ground effect on race cars Airflow underneath is faster creating lower pressure The car is pushed down which increases friction and aids in turns Drafting can negate this effect by reducing air flow over vehicle creating a dangerous situation in turns Surface Tension: Surface Tension Net force on molecule A is zero Net force on B is toward the center of the fluid This pull creates tension at the surface Spheres are the forms that have the lowest surface area for a particular volume so droplets are spherical (liquids in anti-gravity) What forces cause a needle to sit on the surface of water? Surface Tension on a Needle: Surface Tension on a Needle Surface tension allows the needle to float, even though the density of the steel in the needle is much higher than the density of the water The needle actually rests in a small depression in the liquid surface The vertical components of the force balance the weight Surface Tension: Surface Tension The surface tension is defined as the ratio of the magnitude of the surface tension force to the length along which the force acts SI units are N/m Slide88: Note how soap and increased temperature reduces surface tension Surface Tension and Respiration: Surface Tension and Respiration Surface tension can be decreased by adding surfactants to a liquid, like detergents and disinfectants (ST-37) Oxygen exchange from air to lungs occurs in small bubble-like air sacs called alveoli If membrane tension on the alveoli is too high it is hard to inhale and the bubble will collapse If too low it will expand and rupture Surface Tension and Respiration: Surface Tension and Respiration A surfactant made by the body coats the membrane and lowers surface tension gradually during inhalation to give an optimal radius for the alveoli Newborns struggle for their first breath much like we struggle blowing up a balloon until the alveoli are coated by this surfactant The Hay diagnostic test for jaundice is to sprinkle powdered sulfur onto urine If it sinks it indicates the presence of bile which reduces surface tension Adhesive and Cohesive Forces: Adhesive and Cohesive Forces Cohesive forces are forces between like molecules Adhesive forces are forces between unlike molecules Adhesive and Cohesive Forces: Case 1: Adhesive and Cohesive Forces: Case 1 The shape of the surface depends upon the relative size of the cohesive and adhesive forces Here adhesive forces are greater than cohesive forces so the liquid climbs the dry walls of the container The liquid “wets” the surface Adhesive and Cohesive Forces: Case 2: Adhesive and Cohesive Forces: Case 2 If cohesive forces are greater than the adhesive forces The liquid stays to itself and curves downward The liquid does not “wet” the surface Contact Angle: Contact Angle In a, Φ > 90° and cohesive forces are greater than adhesive forces In b, Φ < 90° and adhesive forces are greater than cohesive forces Φ can be changed by using wetting agents or sealants Capillary Action: Case 1: Capillary Action: Case 1 A tube in which the diameter is very small, like 0.01 cm, is called a capillary, which means “hair-like” Capillary action (wicking) is the result of surface tension and adhesive forces (like climbing a door jamb) The liquid rises in the tube when adhesive forces are greater than cohesive forces until the weight of the fluid is Fcosφ Capillary Action: Case 2: Capillary Action: Case 2 Here the cohesive forces are greater than the adhesive forces The level of the fluid in the tube will be below the surface of the surrounding fluid The height at which the fluid is drawn above or depressed below the surface of the surrounding liquid is given by: Applied Capillary Action: Applied Capillary Action Blotters, napkins, filters, and towels all use capillary action A nylon hose hanging over a sink of water can be used to water plants while on vacation The heart could not get blood to all cells of the body if it were not for the capillary action of the capillaries Viscous Fluid Flow: Viscous Fluid Flow Viscosity refers to friction between the layers of a fluid Layers in a viscous fluid have different velocities The velocity is greatest at the center Adhesive forces between the fluid and the walls slow down the fluid on the outside High concentration of red blood cells increases the viscosity of blood and requires more pumping action from the heart Slide99: How is viscosity affected by temperature? Viscosity of water at 200 C is 1 centipoise (N·s/cm2) Poiseuille’s Law for Laminar Flow and Insignificant Wall Friction: Poiseuille’s Law for Laminar Flow and Insignificant Wall Friction Gives the rate of flow of a fluid in a tube with pressure differences Frictionless flow is an idealization, real fluids have some viscosity Which quantity effects flow rate the most? (P1 + P2)/L is the pressure gradient (vehicle drafting) 8ηL/πR4 is the resistance Poiseuille’s Law: Poiseuille’s Law A lower blood flow rate means cell demands are not met The circulatory system is signaled and increases pulse rate and stroke volume Vasodilation of arterioles effectively aids stroke volume How does the formula explain the use of blood thinning, vasodilating, or vasocontricting drugs? Problem: Problem The pulmonary artery has a radius of 2.6 mm and a length of 8.4 cm. If the pressure drop is 400 Pa what is the average speed of blood in the artery? (ηblood = 2.7 x 10-3 N-s/m2) Typical Systolic Pressures: Typical Systolic Pressures 120 mm Hg Circulatory System Pressure Drops: Circulatory System Pressure Drops Compare flow rates through one vessel with A = 3.14 cm2 and through 2 vessels of combined A = 3.14 cm2 r = 1.00cm r = .707cm If flow rate is constant which will require the greater pressure gradient? However pressure drop through the arterioles is larger than through the capillaries because there is a larger number of capillaries and flow is aided by capillary action Controlling Volume Flow Rate: Controlling Volume Flow Rate Through vasodilation the body can not only control flow rate but also the distribution of blood flow (don’t eat and swim) Vasodilation is only necessary in the smaller vessels because the larger vessels offer relatively little resistance to flow If resistance is cut in half and pressure increases 15% during exercise then according to Poiseuille’s Law: ΔV/Δt = ΔP/R = 1.15ΔP/0.5R = 2.3ΔV/Δt Heart Output: Heart Output The heart must meet this demand ΔV/Δt = heart output = (heart rate)x(stroke volume) This could be accomplished by doubling the heart rate and a 15% increase in stroke volume ΔV/Δt = (heart rate)x(stroke volume) = (2 x heart rate)(1.15 x stoke volume) = 2.3ΔV/Δt It could also be accomplished by a 90% increase in heart rate and a 21% increase in stroke volume Departures from Poiseuille’s Law: Departures from Poiseuille’s Law For some fluids such as blood or synovial fluid, viscosity decreases with increased pressure So doubling pressure will more than double flow rate Suspended red blood cells moving through smaller vessels would cause greater resistance to blood flow Turbulance What may cause turbulence in the circulatory system? Reynold’s Number: Reynold’s Number At sufficiently high velocity, fluid flow can change from streamline to turbulent and laws for ideal fluids no longer apply The onset of turbulence can be found by a factor called the Reynold’s Number, RN If RN ≤ 2000 fluid flow is streamline If 2000 < RN < 3000, fluid flow is unstable If RN ≥ 3000 the fluid flow is turbulent Turbulence reduces flow rate and can be caused by obstructions, rough siding, or crooks in the vessel Transport Phenomena: Transport Phenomena Movement of a fluid may be due to differences in concentration (the number of molecules per unit volume) rather than pressure The fluid will flow from an area of high concentration to an area of low concentration The processes are called diffusion and osmosis Diffusion: Diffusion Concentration on the left is higher than on the right of the imaginary barrier Many of the molecules on the left can pass to the right, but few can pass from right to left There is a net movement from the higher concentration to the lower concentration Diffusion and Fick’s Law: Diffusion and Fick’s Law The diffusion rate equation is Fick’s Law D is the diffusion coefficient for the particular fluid It takes an about 2.8 hrs for a light molecule to move 1 cm in water, however only .01 sec to go 10 μm, the size of a cell To move 1 cm in a gas a light molecule takes about 10 seconds. Why so much faster? Slide112: Note that oxygen moves faster through air than water or tissue The tissue must be permeable A selectively permeable membrane is one that allows passage of some molecules, but not others Uses of Diffusion: Uses of Diffusion Diffusion is vital for carrying O2 to cells and removing CO2 through the cell membrane Because cells have a large surface area to volume ratio they can rapidly receive and eliminate products Consider the surface area (6L2) and volume (L3) of a cube. The ratio is 6/L So as L decreases the ratio increases and nutrient and waste transport becomes more efficient Smaller animals (about 0.5cm, linearly, such as insects) are able to absorb enough oxygen through their skin and do not need lungs Uses of Diffusion: Uses of Diffusion During hibernation frogs absorb enough O2 from water at the bottom of a frozen pond, 40 C In lungs, O2 and CO2 are diffused between blood and air due to differences in concentration levels Lung volume is about 6 L but only about ½ L is used at rest 10.5 breathes per minute are generally sufficient at rest to supply the body with enough oxygen Dialysis: Dialysis In dialysis arterial blood is thinned and passed through a semi-permeable tube that is immersed in dialysate Dialysate is chemically like blood so waste products enter the dialysate by diffusion through the membrane Osmosis: Osmosis The diffusion of a solvent (usually water) across a selectively permeable membrane from a region of high solute concentration to a region of low solute concentration leaving behind the solute Kidney function, cell function, and the passing of nerve signals along neurons depend on osmosis As arterial blood flows through the kidneys, waste products and some essential salts and minerals are removed by diffusion Osmosis: Osmosis Most essential elements are returned to the blood by osmosis with waste products not allowed to return through the selectively permeable membrane In cell osmosis if the difference in concentration levels is too high a cell can become dehydrated through shrinkage or it can burst from too much water Intravenous solutions must be introduced to the body carefully to avoid such cell damage
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