Fluid Dynamics PDF

Unit 2 FLUID DYNAMICS CONTENTS FUNDAMENTAL CONCEPTS IDEAL FLUIDS • Continuity Equation • Bernouilli’s Theorem • Bernouilli’s Theorem Applications REAL FLUIDS • • • • Dynamics of a Real Fluid. Viscosity. Poiseuille’s Law Movement Regimes. Reynolds Number Viscosity measurement. Viscosimeters. HE...

Unit 2 FLUID DYNAMICS CONTENTS FUNDAMENTAL CONCEPTS IDEAL FLUIDS • Continuity Equation • Bernouilli’s Theorem • Bernouilli’s Theorem Applications REAL FLUIDS • • • • Dynamics of a Real Fluid. Viscosity. Poiseuille’s Law Movement Regimes. Reynolds Number Viscosity measurement. Viscosimeters. HEMODYNAMICS. CIRCULATORY SYSTEM OF A LIVING BEING • • • • • Blood Circuit Model Blood speed Pressure losses in the blood circuit Arterial pressure measurement Power developed by the heart FUNDAMENTAL CONCEPTS CURRENT LINE Curve tangent to the speed of a particle of the moving fluid (matches the trajectory). CURRENT TUBE A set of lines passing through a given closed curve. VISCOSITY (𝜼) A measurement of the resistance of a fluid to deformations produced by tangent traction tensions. “Ability to Flow”. STATIONARY MOVEMENT OF A FLUID Stationary regime can be achieved when, at any point of the fluid, speed, pressure and density are constant along time. FUNDAMENTAL CONCEPTS Fluid types according to viscosity Ideal Fluids Zero Viscosity 𝜼=𝟎 Real Fluids 𝜼≠𝟎 Newtonian Fluids NON Newtonian Fluids Constant Viscosity NON constant Viscosity Part I IDEAL FLUIDS CONTINUITY EQUATION  Let us consider an incompressible fluid flowing in steady state.  Flow:  Is the volumen of a fluid that passes through a given section during the time unit.  Speed of the fluid that passes through a surface (current line) 𝑉 𝐿 𝑄 = = 𝑡 𝑇 𝑉 𝑄= 𝑡 𝑄 = 𝑣 𝑄=𝑣 𝑆  Units: ⁄ (SI), 𝑆 = 𝐿 𝑇 ⁄ (C.G.S.), L/s, L/h,  FLOW REMAINS CONSTANT 𝑣 𝑆 =𝑣 𝑆 = 𝑐𝑡𝑒 CONTINUITY EQUATION CONTINUITY EQUATION CONSEQUENCES OF CONTINUITY EQUATION Flow remains constant: 𝑣 𝑆 =𝑣 𝑆 =𝑄 At a narrowing → speed increases. If 𝑆 < 𝑆 ⇒ 𝑣 >𝑣 At a widening → speed decreases. If 𝑆 > 𝑆 ⇒ 𝑣 <𝑣 A change in a Conduit section gives place to a change in speeds. The volumen of fluid that crosses a section in the unit time remains constant. BERNOUILLI’S THEOREM Let us consider the dots along a current line 𝑝 𝑝 𝑣 𝑣 Punto 2 Punto 1 𝑦 BERNOUILI’S EQUATION 1 1 𝑝 +𝜌𝑔𝑦 + 𝜌𝑣 = 𝑝 + 𝜌𝑔𝑦 + 𝜌𝑣 2 2 What happens if the fluid is at rest? 𝑦 BERNOUILLI’S 1 1 𝑝 +𝜌𝑔𝑦 + 𝜌𝑣 = 𝑝 + 𝜌𝑔𝑦 + 𝜌𝑣 2 2 Changes in the SHAPE of conduits Results in Changes in the HEIGHT of conduits Changes in the SPEED of the fluid Result in Results in Changes in the PRESSURE of the fluid APPLICATIONS OF BERNOUILLI’S THEOREM TORRICELLI’S THEOREM At points 1 and 2 it is fulfilled: 1 𝑆 ≫𝑆 2 y1 y2 𝑝 =𝑝 Exit speed Using Bernuilli’s Equation y 𝑣 = 2𝑔𝑦 APPLICATIONS OF BERNOUILLI’S THEOREM VENTURI EFFECT If displayed horizontally, points 1 and 2 fulfill that: 𝑦 =𝑦 Using Bernouilli’s Equation 1 𝑝 −𝑝 = 𝜌 𝑣 − 𝑣 2 Horizontally Changes in the SHAPE of conduits Results in Changes in the SPEED of the fluid Results in Changes in the PRESSURE of the fluid APPLICATIONS OF BERNOUILLI’S THEOREM VENTURI EFFECT For instance: at a narrowing 𝑆 <𝑆 ⇒ 𝑝 <𝑝 Horizontally Changes in the SHAPE of conduits Results in Changes in the SPEED of the fluid Results in Changes in the PRESSURE of the fluid Part II REAL FLUID DYNAMICS DYNAMICS OF A REAL FLUID. VISCOSITY  Real fluids exert tangential forces when they are in motion.  When moving, real fluids experience certain effects due to frictional forces or viscous forces.  Imagine a fluid placed between two parallel plates. In the first case one of the plates is in motion while in the second both are static.  In the first case:  To move the upper layer, a force is needed to counteract the viscous forces that appear between the molecular layers of the liquid and oppose their sliding over each other.  In the second case:  The speed of the fluid is greater in the central part where friction occurs between layers of fluid. Viscous forces can be considered as frictional forces that occur between the layers of fluid when we try to move one over the other. DYNAMICS OF A REAL FLUID. VISCOSITY  The fluid moves with different speeds depending on the distance to the moving surface.  The viscous force does not originate from the sliding of the fluid along a surface, but from the sliding of some layers over others. d FRICTION FORCE. VISCOUS FORCE.     Proportional to the area of the plates (S) and the speed (v) Inversely proportional to the distance between the plates (d) Proportionality coefficient: viscosity (η) (viscosity coefficient). Velocity gradient in the direction of d: ⁄ 𝑆 𝑣 Δ𝑣 𝐹∝ ⟶ 𝐹 = 𝜂𝑆 𝑑 Δ𝑑 Newton’s Viscosity Law VISCOSITY COEFFICIENT 𝐹 𝜂= Δ𝑣 [𝐹] 𝑆 Δ𝑑 𝜂 = [𝑆] [Δ𝑣] [Δ𝑑] = 𝐹 [Δ𝑡] [𝑆] 𝜂 = 𝜂 = 𝑁 𝑠 = 𝑃𝑎 𝑠 𝑚 (𝐼𝑆) 1𝑃𝑎 𝑠 = 10𝑃 = 1000𝑐𝑃 𝑑𝑖𝑛𝑒 𝑠 = 𝑃 (𝑃𝑜𝑖𝑠𝑒) (𝑐𝑔𝑠) 𝑚 DYNAMICS OF A REAL FLUID. VISCOSITY DEPENDENCY OF η WITH OTHER QUANTITIES     It is almost independent of pressure In gases ↑ 𝜂 when ↑ 𝑇: viscosity is due to collisions between molecules In liquids ↓ 𝜂 when ↑ 𝑇: viscosity is due to cohesive forces between molecules Depends on the size, shape, and nature of the fluid molecules. DYNAMIC, KINEMATIC AND RELATIVE VISCOSITY 𝐹 DIYNAMIC: RELATIVE: 𝜂= 𝜂 Δ𝑣 𝑆 KINEMATIC: Δ𝑑 = 𝜂 𝜂 𝜇= 𝜂 𝜌 ¿Units? FLUID TYPES NEWTONIAN: constant viscosity (depends only on T) and complies with Newton’s Law for η NON NEWTONIAN: viscosity not constant and does not comply with Newton’s Law for η POISEUILLE’S LAW  Let us consider a real fluid that moves in a laminar and stationary regime through a horizontal cylindrical pipe of radius r.  Let us consider two cross sections 1 and 2 that are distant by a length l  The variation of the speed of the different layers of fluid is symmetrical with respect to the axis of symmetry of the cylindrical pipe and shows a profile like the one in the figures. l r  The flow (or flow rate, Q) of a fluid through a pipe depends on the viscosity of the fluid, the pressure difference and the dimensions of the pipe itself. 8𝜂𝑙 𝜋𝑟 (𝑝 − 𝑝 ) Governs the pressure loss experienced by a real Δ𝑝 = 𝑄 𝑄= 𝜋𝑟 fluid flowing through a cylindrical pipe of radius 8𝜂𝑙 r due to the viscosity between two points that are a distance l apart. POISEUILLE’S LAW FLOW REGIMES. ℜ Ideal Flow Bernouilli’s Regime Same speed in one section Laminar Flow Poiseuille’s Regime Turbulent Flow Venturi’s Regime Parallel stream lines. Parallel stream lines. Different speed in one section (v=vmax/2) Eddy formation Chaotic movement FLOW REGIMES. ℜ  Reynolds number (ℜ) gives us the relationship between the convective forces and the viscous forces in a fluid.  It is related to the density, viscosity, speed and dimension of the fluid (r, radius) 2𝜌𝑣𝑟 ℜ= 𝜂 Units?  V = vaverage = vmax / 2  The value of the Reynolds number indicates the rate of movement of our fluid. RANGES • If ℜ < 2400 laminar flow • If ℜ > 2400 turbulent flow RANGES • If ℜ < 2000 laminar flow • If ℜ > 3000 turbulent flow • If2000 < ℜ < 3000 transitional flow Part III HEMODYNAMICS. CIRCULATORY SYSTEM OF A LIVING BEING HEMODYNAMICS BLOOD CIRCUIT MODEL Drive motor Ducts Transporting Blood Veins Ducts Transporting Blood Heart Blood 𝜂 = 2.08 · 10 𝑃𝑎 · 𝑠 Tissues Oxygen Exchange Aorta Arteries Capillaries Cylindrical shape Radius r Ellastic Muscle fibers HEMODYNAMICS BLOOD SPEED 𝑟 = 9 𝑚𝑚 Aorta 𝑄 =𝑆 ·𝑣 𝑣 = 33 𝑐𝑚⁄𝑠 𝑆 = 20 𝑐𝑚 𝑟 = 4 𝑚𝑚 Arteries 𝑄 = 83 𝑐𝑚 𝑠 𝑄 𝑣= 𝑆 𝑣 = 4.1 𝑐𝑚⁄𝑠 𝑆 = 2500 𝑐𝑚 𝑟 = 2 · 10 Capillaries 𝑄 𝑣= 𝑆 𝑚 𝑣 = 0.33 𝑚𝑚⁄𝑠 HEMODYNAMICS PRESSURE DROP ALONG THE BLOOD CIRCUIT 𝜌 = 1.06 𝑔 𝑐𝑚 𝜂 = 2.08 · 10 𝑃𝑎 · 𝑠 Blood Poiseuille’s Law Pressure Drop Reynolds Number 8𝜂𝑙 Δ𝑝 = 𝑄 𝜋𝑟 2𝜌𝑣𝑟 ℜ= 𝜂 Aorta ℜ = 3027 Turbulent flow Arteries ℜ = 167 Laminar flow Capillaries ℜ = 6.7 · 10 Laminar flow HEMODYNAMICS PRESSURE DROP ALONG THE BLOOD CIRCUIT Aorta Experimental value 8𝜂𝑙𝑣 Δ𝑝 = 𝑟 𝑙 Arteries Capillaries Veins 𝑙 Return Circuit ≫𝑙 ≫𝑙 0 0 0 0 0 0 0 Δ𝑝 = 3 𝑇𝑜𝑟𝑟 Δ𝑝 maximum in the arteries Δ𝑝 ≃ 10 𝑇𝑜𝑟𝑟 HEMODYNAMICS PRESSURE DROP ALONG THE BLOOD CIRCUIT Aorta Δ𝑝 = 3 𝑇𝑜𝑟𝑟 Arteries Arterioles Δ𝑝 ≃ 67 𝑇𝑜𝑟𝑟 Capillaries Δ𝑝 ≃ 20 𝑇𝑜𝑟𝑟 Veins Δ𝑝 ≃ 10 𝑇𝑜𝑟𝑟 HEMODYNAMICS PRESSURE DROP ALONG THE BLOOD CIRCUIT Hemodynamic Resistance Poiseuille’s Law Δ𝑝 𝑄= 𝑅 𝑄 constant If 𝑅 ↓ Δ𝑝 ↓ 𝑄 constant If Paralell Scheme 𝑅 ↑ 𝑅 ↓ Δ𝑝 ↑ 8𝜂𝑙 𝑅 = 𝜋𝑟 HEMODYNAMICS PRESSURE DROP ALONG THE BLOOD CIRCUIT Hemodynamic Resistance Complete Circuit Δ𝑝 = 100 𝑇𝑜𝑟𝑟 𝑅 = 𝑄 = 83 𝑐𝑚 𝑠 Arteries + Capillaries Capillary Capilary Net (paralel) Veins Δ𝑝 𝑄 · 𝑅 = 1.61 · 10 𝑅 = 90 𝑇𝑜𝑟𝑟 80 𝑐𝑚 ⁄𝑠 𝑅 = 6.6 · 10 · 𝑅 = 3.2 · 10 10 𝑇𝑜𝑟𝑟 𝑅 = 80 𝑐𝑚 ⁄𝑠 · HEMODYNAMICS PRESSURE DROP ALONG THE BLOOD CIRCUIT Effects relative to a change in 𝑅 Hypertension 𝑅 ↑ 𝑄 constant 8𝜂𝑙 𝑅 = 𝜋𝑟 Arteries hemorrhages surrounded by muscle fibers Blood Loss Δ𝑝 ↓ leads to Δ𝑝 ↑ 𝑟↓ 𝜂↑ vary 𝑅 ↓ Arteriosclerosis CO2 Excess 𝑟 regulates for 𝑄 Closure of blood vessels HEMODYNAMICS BLOOD PRESSURE Blood pressure (commonly known as "blood tension") is the force or pressure that carries blood to all parts of the body. When measuring blood pressure, the result of the pressure exerted by the blood against the walls of the arteries is known. The result of the blood pressure reading is given in 2 figures. One of them is the systolic that is on top or the first number in the reading. The other one is called diastolic which is on the bottom and is the second number in the reading. Traditionally, the following values ​have been considered ideal blood pressure: <120 mmHg systolic and <80 mmHg diastolic. Accepted as high blood pressure (hypertension = HA)) when the systolic values ​are over 140 and/or the diastolic over 90. An example of a blood pressure reading is 120/80 (120 over 80) in which 120 is the systolic number and 80 is the diastolic number. Sphygmomanometer Maximum value of blood pressure in systole (when the heart contracts). It refers to the pressure effect exerted by blood ejected from the heart on the vessel wall. HEMODYNAMICS Which is then lost through the vessels, arteries,... POWER DEVELOPED BY THE HEART 𝑊 ℘= 𝑡 Power Low pressure High pressure Blood Circuit Fluids ℘ = Δ𝑝 · 𝑄 ℘=𝑅 ·𝑄 Δ𝑝 = 100 𝑇𝑜𝑟𝑟 𝑄 = 83 𝑐𝑚 ℘ = 1.1 Total Power developed by the heart ℘=℘ +℘ +℘ ℘ ℘ ℘ = 3𝑊 ó = 1.4 𝑊 = 1.1 𝑊 ℘ = 5.5 𝑊