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Physics 1029A/B
Martin Zinke- Allmang

Chapter 11: Liquid water and aqueous solutions, static fluids Chapter 12: Cardiovascular System: Fluid flow - p. 368 – - Fluid = deformable so evolve toward mechanical eq’m in given - flow no longer stationary fluid model because dynamic process .: not in mechanical eq’m space. Intermolecular forces dominate = liquid fluid. Thermal - ideal stationary fluid AND dynamic fluid = incompressible/deformable energy of system dominates = gas fluid. Ideal fluid model: fluid is - ideal dynamic fluid: (1) no turbulences (.: no air through trachea), (2) no sound waves (b/c density fluctuations, automatic if incompressible, but incompressible. If in mechanical eq’m, ideal stationary fluid. If in physiologically bun dis), (3) no friction with walls or objects that move @ diff speeds .: interaction of fluid particles = elastic collisions only condensed liquid state, Pascal’s law: pressure increase linearly - laminar flow is fluid flow satisfying (1)(2); established when (i) flow lines in the fluid never cross each other, and (ii) flow tubes never penetrate with depth below surface (Pressure at liquid surface = pressure in each other adjacent gas space) - conservation of fluid mass: ∆m /∆t =1∆m /∆t; 2 D * ∆V 1∆t = D * ∆V /∆2 ; mass flow rate vs volume flow rate ; rewrite volume --> D * - Surface of condensed ideal stationary fluid has properties A 1 ∆l 1∆t = D * A *2∆l /∆2 ; using velocity --> A *1v | 1 A *|2 | .2 A *|v| = const (= ∆V/∆t) = equation of continuity distinct from those of bulk material due to excess energy required to - Equation of continuity = expression of conservation of mass or volume of an incompressible fluid. It states that the volume fluid rate is form surface (surface tension/energy), which causes pressure diff constant along a tube. The fluid flows faster when it passes through a section of tube with smaller cross-section. Apples to laminar flow whether ideal across bubble / droplet surface inversely proportional to radius dynamic fluid or not (b/c type interactions irrelevant). - Model system: ideal stationary fluid = systems that yield to any force that attempts to alter their shape, causing system to flow until - venturi meter measures changes of fluid speed based on pressure (pressure ~ 1/speed) reaches mechanical eq’m in which fluid then conforms to shape of - Bernoulli’s law: expression of conservation of (total) energy for closed system. States that increase in speed of ideal dynamic fluid in tube is container accompanied by drop in pressure. Correct pressure locally due to tube diameter variations. Only applies to ideal dynamic fluid (conservation of E)  - Liquids and gases, but not solids (don’t change shape for const = p + ½ D * v 2 ; p = pressure in fluid ; Work = – (p – p )∆V 2 1 container) - Liquids: molecules in condensed state, maintain - Newtonian fluid model: describes fluid movement by removing restrictive assumption, since must have friction interactions constant intermolecular distance, forms surface in container of larger - Viscosity: interaction between neighbouring layers of moving fluid in direction perpendicular to flow lines (F ext= Resistance volume than liquid occupying volume - F ext= n * A |∆v| / ∆y ; n = viscosity coefficient (N * s/m ) ; ∆y = distance between the 2 layers ; v = velocity ; A = area ; F = force required ; n for - Gases: adjust intermolecular distance and fill any gases INCREASE with temp (more interaction with layers); n for liquids DECREASE with temp (molecules have to squeeze past) provided space uniformly, have no natural surfaces - viscosity is dynamic effect requires velocity gradient perpendicular to direction of flow lines in fluid ∆v/∆y .: no role in stationary (v = 0) or ideal - Fluids in mechanical eq’m = stationary fluid (all “essential dynamic (∆v/∆y = 0) fluids .: all fluids even ideal gas, behave as Newtonian fluid once velocity gradient introduced. In Newtonian fluid, inelastic parameters” time-independent) interaction with fluid-confining walls causes velocity gradients. Viscosity replaces assumption of elastic collisions required between ideal dynamic fluid - Liquid: no obvious changes occur while observe system and its confining walls. In Newtonian fluids, the flow is laminar and viscous. - Gas: microscopic level =lot of motion of particles -.: macroscopic eq’m excludes microscopic - Newtonian assumes steady-state .: only 2 forces = pressure diff force pushing forward (1), and viscosity force pushing back (2). (1) dominates in - Properties of ideal stationary fluid: middle, (2) dominates close to walls 2 2  = incompressible: V (volume) and p (density) = m/V are constant - velocity profile: v = ( r tube– r / 4 * n ) * ( ∆p / l ) ;tube= radius of tube, r = distance to container, ∆p = pressure diff along segment of tube length l, (pressure n = coefficient (always > 0), ∆p/l = constant pressure gradient along tube independent) -----Good for liquids, doesn’t apply to gases (retain 4 ideal gas model) - Poiseuille’s law: ∆V / ∆t = ( π / 8 * n ) * r tube* ( ∆p / l ), pi/8 = factor for cylindrical tube = states that volume flow rate of Newtonian fluid through th  = deformable under forces and seeks mechanical eq’m, and only cylindrical tube is proportional to 4 power of radius of tube. Actual flow through the tube. .: narrow tube reduces flow severely i.e. reduce diameter 2 then = flow reduced by 16 does fluid become stationary----- Good for both liquids and gases - Ohm’s law: ∆p = R * ( ∆V / ∆t ) ; R = flow resistance (Pa * s / m ) ; Ohm’s law states that volume flow rate of Newtonian fluid ~ pressure diff along 4 - No condition for interactions with fluid molecules or container tube, and that proportionality constant = flow resistance . .: R = n * (8 * l / π * r tube) walls, no elastic collision limitation - Pascal’s law: p 2 p =1–density * g (y – y2) 1 - Q. water flow in sink: volume flow rate constant, gravity accelerates it .: jet narrows as it falls - difference between pressures at 2 diff positions in fluid is - Q. In a decorative fountain in a garden, water is shot vertically from a pipe. The uprising jet of water A) broadens as it rises 4 proportional to vertical distance between 2 positions. Proportionality - Q. diver: pressure @ depth = D g h ; too wide breathing tube = too much dead space; too narrow = too much resistance (1/r ) = exhausting factor is product of density of fluid and gravitational acceleration - Q. cardboard/pin = Bernoulli preventing cardboard from falling - in systems with identifiable surface of fluid, y = 1, p = 1 - Q. vascular flutter: constriction (lower area)  lower pressure  collapse  build up of pressure  forced open p atm, make downward direction positive .: p = p atm+ density * g * - Q. prairie dog mound / hurricane and ROOF: mound causes increase speed = lower pressure = air sucked in to tunnels depth = p atm+ p gauge - Q. WRONG: poiseuille’s law applies as derived only to ideal dynamic fluids; RIGHT: Poseuille’s law applies as derived only to laminar flow, - doesn’t apply to gases, since gases are compressible, density incompressible fluids, to Newtonian fluids; Poseuille’s law and the equation of continuity can be used together for the same system depends on pressure - Q. water tank exposed to air, hole __ m below, leaks at rate ___, speed?(a) Diameter?(b) (a) around hole: p + ½ D * v = p + ½ D * v 2 ; inside, v = 2 2 - doesn’t contain any info on shape of container .: regardless 0, p = p.atm + Dgh; outside p = p.atm .:  p.atm + ½ D*g*h = p.atm + ½ D *v  gh = ½ v (b) A * v = V/t of shape, pressure increases below surface at fixed values - Q. hypodermic needle, velocity out ? p = F / A + p  Bernouille  v = 0, p = 1 atm ; v = = 1 ext 1 atm 1 2 2 - gauge pressure: p gauge= p absolutep air can be positive or - Q. hypodermic needle, length = , radius = , flow =  pressure? Poseuille negative, density * g * d = gauge pressure - Q. blood vessel splits into 2  take new area, because splits into 2 multiply area by 2 before calculate velocity o 5 - p atm= 1 atm @ 20 C = 1.01325 x 10 Pa = 760 torr = 760 - Q. volume flow rate in = volume flow rate out, if nothing else open and affecting mmHg - Q. suction, range of y it’ll work @? Y max = p atm(D*g) - 1.0 g / cm = 1.0 kg/L = 1 x 10 kg/m 3 3 - Q. pipe underground. Faucet. Change in pressure using Bernouille + Dgh 2 - Pressure = force/area = N/m - Q. aorta volume = total cap volume .: given # cap, A cap, cap, Aaorta solve for v aortahen multiply by #cap - 1 Pa = 1 N / m = 1 kg / m*s = J / m 3 - Q. venture-meter / tapering: v = ; = = when R >> r (in thick part, with Area A) - Q. Streamlined race cars as the one at right usually have “spoilers”, which are basically upside–down wings. Due to the Equation of Continuity and - Diastolic blood pressure: 10.7 kPa, 80 mmHg  veins, Bernoulli’s law in what direction will an additional force act on the spoiler at high speed? DOWN pulmonary circulation - Q. A viscous fluid is forced through a pipe to obtain a certain volume flow rate (experiment 1). If the same fluid is then forced through - Systolic blood pressure: 16.0 kPa, 120 mmHg a pipe of same length with double the cross–sectional area (experiment 2), by how much has the pressure difference )p2 along the - Supine position: species that person is lying down, and blood pipe dropped from the previous value )p1 to obtain the same volume flow rate? /\p = ¼ /\p 2 1 pressure has maximum variation of 15% atmospheric pressure - Q. Bird doesn’t benefit from Bernoulli when tailwind pushes forward - Q. fish can float in water, .: their density = density of water - For box in fluid… - Q. Laplace’s law describes pressure in hollow bubble (e.g. alveoli in lungs) in form p inside poutside 4o/r, in which o is surface tension and r is radius of - Fnet= F up– Fdown – W B W Fluid– WBlock curvature of bubble. In healthy alveoli, surfactant is used to reduce surface tension by coating parts of surface. WRONG = surfactant must coat - Fnet> 0. Weight of block < weight of displaced liquid, block particularly areas with large radius of curvature. RIGHT: surfactant must coat areas with SMALL radius curvature, does not change pressure in bubble B rises to surface - Fnet= 0 .lock floats at current depth below fluid surface, - Q. We introduced Pascal’s law in 2 forms, one is , can NOT be used when fluid is compressible weight of block = weight of displaced fluid - Q. Turtle of common shape, if swims fast horizontally under water, in what direction will force act on body due to Equation of Continuity and - Fnet< 0 .eight of block > weight of displaced fluid, block Bernoulli’s law? Note: turtle has a flat bottom and domed top. UP sinks to bottom - Q. Lady floating effortless on surface of water, what is true is water is extremely salty, increasing density to above that of lady’s body - Archimedes principle: when object is immersed in a fluid, - Q. how does mass flow rate m/t of an incompressible, viscous fluid flowing through a cylindrical pipe depend on pipe’s diameter d? m/t ~ d 4 fluid exerts an upward force on object equal to weight of fluid - Q. A viscous fluid is forced through a pipe to obtain a certain volume flow rate (experiment 1). If the same fluid is then forced through a pipe of same displaced by object cross–sectional area but triple the length (experiment 2), how has the pressure difference along the pipe changed from the value in experiment 1?.: Δp2 - Buoyant force = magnitude of weight of displaced fluid: = 3 Δp1 F bouyant= density fluid* Vobject g - Q. The viscosity coefficient is a material’s constant Always against gravity .: up ; Always non-zero, CAN USE - Q. The sixth figure shows a neat experiment you can do at home. Stick a pin through the middle of a small postcard. Hold it under a thread spool so FORMULA IN AIR that the pin projects up into the hole. If you blow down the hole and let go of the card, it won’t fall down. How many of the following laws do you need to explain this experiment? (I) Bernoulli’s law, (II) Pascal’s law, (III) Equation of continuity (IV) Poiseuille’s law, (V) Laplace’s law. 1 or 2 - Surface tension = fluid property associated with presence of - Q. ideal dynamical fluid is flowing in a circular pipe which maintains a fixed diameter and remains level while it is routed around a large external surface toward air Chapter 16: Elastic Tissue: Elasticity and Vibrations SPRINGS Chemical Bonds - Interfacial tension used to describe analogous phenomena for p = W / A (pressure = weight / area) ; SPHERE: - Hooke’s law applies in elastic regime of material & - HCl  Cl is a lot bigger .: acts like a fluids having interfaces with solids or other liquids SA = 4πr2 ; V = (4/3)πr3 states that stress and strain of material = linear wall, H is like an object on a spring - spring makes obect move to eq’m in response to attached to it (model dies at high - surface/interface don’t have symmetry of interactions of molecules with immediate neighbours broken ELASTICITY displacement temps and high separation) + - - sufficiently far below surface, water molecule attractive interactions - stress = force exerted per unit area of surface on - HOOKES LAW for object attached to horizontal - Na Cl  when too close, repulsion; in all directions, .: eq’m, but within 1 nm of surface sphere of extended object, leads to strain = relative change in spring: (note F.elast = opposite F.external) -F elastic otherwise obviously attract size of object (tensile stress  stretching for length, Fext= k(x – x eqm) ; k = spring constant (N/m) - Eattract - (1 / 4 π ε0) * (e / r) ; neighbours not complete .: net force acts pulling molecule down away from surface shearing stress  twisting for angle, hydraulic stress - .: elastic -k(x – x eqm) = m · a Erepulsive b · e -a · (a and b are - force pulls molecules away from surface, molecules can’t leave  compression/expansion for volume) - an object on an ideal springs that undergoes a experimental constants, ε 0 surface as surface area can’t shrink .: brought to surface AGAINST small displacement from its eqm position = Hooke’s permittivity, e = elementary charge) net force down - stretching strain = length change caused by tensile law - .: energy associated with formation of surface .: surface stress = F/A (Pa) ; A = area of object on which force - W = area under F*displacement  W = ½ F extx final 2 energy σ = ∆E/∆A ; Energy needed to increase surface of fluid by acts W = ½ k * (x – x eq) = elastic potential energy - Q. Cable and shit  Steel sphere, 2 2 - elastic response = strain/stress are proportional diameter = 1.5 m, thickness 5 cm. area A .: = energy required to form 1 m of new surface (J/m ) - W = ∆E = σ *∆A ; W = work required to increase surface by A; (linear)  F/A = Y · Δl/l ; Y = Young’s modulus (Pa), - amplitude (A) = maximum displacement of Weight = 4900 N. STEEL Rope = 3 cm 3 W = F * d = F *∆l y ∆A = l x∆ y materials constant (~ temp) vibrating object .: kinetic 0 = E total E spring E total diameter, density = 7.5 g/cm . σ = 2 2 - σ = F / lx ; units = N/m (.: = surface tension) = J/m .: is - σ (stress) = F/A ; ε (strain) = Δl/l .: σ = Y · ½ k A 6.87E8 Pa. SA sphere * thickness = energy to increase surface or force to stretch film ε  strong materials = HIGH Y (large force = little - At minimum displacement, all E = kinetic .: E total ½ volume .: 4 π r d * ρ g = weight - wire example for soap with high cohesiveness; water change e.g. metal); soft = LOW Y m v 2 ; v = MAXIMUM speed  .: v = sqrt (k / m) * A sphere 25980. Buoyant = 4/3 π r ρ 3 measured by measuring energy to lift solid off fluid surface - elastic limit = threshold above which permanent g (volume obj*densityfluid*g) = deformation 17320. .: sum force down – Fb = - sphere has least surface : volume, .: raindrops release E when 17320 N needed to be supported by reshaping to sphere, or need E to change from sphere Plastic Deformations - position of object on spring: x(t) = A · cos( sqrt(k/ rope. Check max weight  F = σ A = 2 - when droplet on flat surface, shape complex --> flat frame = - equation doesn’t always hold  irreversible because m) · t ) σ r π = 4.86E5. .: that – 17 = flat layer film, drops on surface = partial sphere (alveoli) strain doesn’t return to 0 when stress removed (e.g.  x(t) = A · cos(ω · t) (simple harmonic motion); 4.72E5 of length. Translates to W = π playdough) ω = angular frequency (1/s, rad/s) r l (ρsteel ρfluidg (to integrate Fb into - bubbles have increased pressure inside them. Internal p > - stress increases beyond linearity  microscopic - ω = sqrt(k/m) (1/s, rad/s) - period (T) = full it)  10480 m = l structural alterations  significant strain increase  cycle - Q. Muscle has no elastic regime @ external p - Laplace’s Law: pressure difference between inside and outside of permanent plastic deformation  material no longer  ω = 2 · π / T  f = 1/T .: ω = 2πf resting length fluid with curved surface is inversely proportional to the radius of the withstand stress, pass through ultimate stress point - ***note (lectures): may have to include phase curvature of the curved surface. This means that smaller bubble,  beyond which material approaches rupture point angle in cos( ) if initial displacement*** quickly, without need for further increase in stress droplet, or cylinder has a larger pressure difference ∆p. *negative if curve is U  material = elastic when responds to external forces - bubbles: ∆p = p insidep outside 4*σ / r = (stress) with linear deformation (strain). Elastic transmural pressure deformations are reversible. A material is called - droplet / hollow cylinder: ∆p = p inside poutside 2*σ / r plastic when it responds to stress non-linearly - homogenous cylinde
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