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Transcript
Fluid Dynamics AP Physics 2 Fluid Flow Up till now, we have pretty much focused on fluids at rest. Now let's look at fluids in motion Characteristics of an Ideal Fluid The fluid is nonviscous The fluid is incompressible Its density is constant The fluid motion is steady There is no internal friction between adjacent layers Its velocity, density, and pressure do not change in time The fluid moves without turbulence No eddy currents are present The elements have zero angular velocity about its center Fluids in Motion: Streamline Flow A fluid’s motion can be said to be streamline, or laminar. The path itself is called streamline. By laminar we mean that every particle moves exactly along the smooth path as every other particle that follows it. If the fluid does not have laminar flow, it has turbulent flow. Fluids in Motion: Streamline Flow Streamline flow Every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier Also called laminar flow Streamline is the path Different streamlines cannot cross each other The streamline at any point coincides with the direction of fluid velocity at that point Streamline Flow, Example Streamline flow shown around an auto in a wind tunnel Fluid Flow: Viscosity Viscosity is the degree of internal friction in the fluid The internal friction is associated with the resistance between two adjacent layers of the fluid moving relative to each other Laminar Flow Laminar Flow ESSENTIALLY: Laminar flow, type of fluid (gas or liquid) flow in which the fluid travels smoothly or in regular paths Fluids in Motion: Turbulent 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 Turbulent Flow, Example The rotating blade (dark area) forms a vortex in heated air The wick of the burner is at the bottom Turbulent air flow occurs on both sides of the blade Flow Rate Flow Rate (ƒ): Volume of fluid that passes a particular point in a given time Units used to measure Flow Rate = m³/sec Equation for: Flow Rate ƒ = Aν = (m2)(m/s) (A = cross sectional area) (ν = velocity of fluid) Rate of Flow V Avt A vt Volume = A(vt) R Avt vA t Rate of flow = velocity x area Since A1 > A2… For an incompressible, frictionless fluid, the velocity increases when the cross-section decreases: R v1 A1 v2 A2 v1 < v2 Continuity Equation Flow rates are the same at all points along a closed pipe Continuity Equation: ƒ₁ = ƒ₂ A₁ν₁ = A₂ν₂ Reminder: the equation for Area of a circle: A = πr² Question: Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out through a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle? ANS: 19 m/s Bernoulli's Principle The Swiss Physicist Daniel Bernoulli, was interested in how the velocity changes as the fluid moves through a pipe of different area. He especially wanted to incorporate pressure into his idea as well. Conceptually, his principle is stated as: " If the velocity of a fluid increases, the pressure decreases and vice versa." The velocity can be increased by pushing the air over or through a CONSTRICTION A change in pressure results in a NET FORCE towards the low pressure region. Bernoulli's Principle in Action The constriction in the Subclavian artery causes the blood in the region to speed up and thus produces low pressure. The blood moving UP the LVA is then pushed DOWN instead of up causing a lack of blood flow to the brain. This condition is called TIA (transient ischemic attack) or “Subclavian Steal Syndrome. One end of a gopher hole is higher than the other causing a constriction and low pressure region. Thus the air is constantly sucked out of the higher hole by the wind. The air enters the lower hole providing a sort of air re-circulating system effect to prevent suffocation. Bernoulli’s Equation Relates pressure to fluid speed and elevation Bernoulli’s equation is a consequence of Conservation of Energy applied to an ideal fluid Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steadystate manner Bernoulli’s Equation, cont. States that 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 1 2 P v gy constant 2 Bernoulli's Equation Derivation X=L F1 on 2 Let’s look at this principle mathematically. -F2 on 1 Work is done by a section of water applying a force on a second section in front of it over a displacement. According to Newton’s 3rd law, the second section of water applies an equal and opposite force back on the first. Thus is does negative work as the water still moves FORWARD. Pressure*Area is substituted for Force. Bernoulli's Equation Derivation v2 A2 y2 L1=v1t L2=v2t v1 y1 A1 ground Work is also done by GRAVITY as the water travels a vertical displacement UPWARD. As the water moves UP the force due to gravity is DOWN. So the work is NEGATIVE. Bernoulli's Equation Derivation Now let’s find the NET WORK done by gravity and the water acting on itself. WHAT DOES THE NET WORK EQUAL TO? A CHANGE IN KINETIC ENERGY! Bernoulli's Equation Derivation Consider that Density = Mass per unit Volume AND that VOLUME is equal to AREA time LENGTH Bernoulli's Equation Derviation We can now cancel out the AREA and LENGTH Leaving: Bernoulli's Equation Derivation Moving everything related to one side results in: What this basically shows is that Conservation of Energy holds true within a fluid and that if you add the PRESSURE, the KINETIC ENERGY (in terms of density) and POTENTIAL ENERGY (in terms of density) you get the SAME VALUE anywhere along a streamline. An Object Moving Through a Fluid Many common phenomena can be explained by Bernoulli’s equation At least partially In general, an object moving through a fluid is acted upon by a net upward force as the result of any effect that causes the fluid to change its direction as it flows past the object Example Water circulates throughout the house in a hot-water heating system. If the water is pumped at a speed of 0.50 m/s through a 4.0 cm diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6 cm-diameter pipe on the second floor 5.0 m above? A1v1 A1v2 1 atm = 1x105 Pa r12 v1 r22 v2 (0.04) 2 0.50 (0.026) 2 v2 v2 1.183 m/s 1 2 1 vo gho P v 2 gh 2 2 1 1 3x105 (1000)(0.50) 2 (1000)(9.8)(0) P (1000)(1.183) 2 (1000)(9.8)(5) 2 2 P 2.5x105 Pa(N/m2) or 2.5 atm Po Application – Golf Ball The dimples in the golf ball help move air along its surface The ball pushes the air down Newton’s Third Law tells us the air must push up on the ball The spinning ball travels farther than if it were not spinning Airplane Wings - Application Application – Airplane Wing The air speed above the wing is greater than the speed below The air pressure above the wing is less than the air pressure below There is a net upward force Called lift Other factors are also involved Applications of Bernoulli’s Principle: Venturi Tube Shows fluid flowing through a horizontal constricted pipe Speed changes as diameter changes Can be used to measure the speed of the fluid flow Venturi Meter The higher the velocity in the constriction at Region-2, the lower the pressure... Wait why? Venturi Effect Venturi Effect Law of Conservation of Energy ~ Bernoulli’s Equation! Energy due to pressure gets converted into energy due to velocity (kinetic energy) So higher the velocity the lower the pressure