Physics 106 Lecture 13 Fluid Mechanics SJ 7th Ed.: Chap 14.1 to 14.5
• What is a fluid? • Pressure • Pressure varies with depth • Pascal’s principle • Methods for measuring pressure • Buoyant forces • Archimedes principle • Fluid dyypnamics assumptions • An ideal fluid • Continuity Equation • Bernoulli’s Equation
Goals
Ideal fl uid
Equation of continuity: Mass conservation
Bernoulli’s principle: Work-Energy theorem
1 Ideal Fluids
• non-viscous – Negligible internal friction (not sticky) • laminar (steady, streamline flow) – all particles passing through the same point have the same velocity • incompressible – the density remains constant • irrotational – no local rotation
Flow rate Mass flow rate = amount of mass crossing area A per unit time = a “current” sometimes called a “mass flux”
cross-section area A
velocity v volume of fluid in cylinder = dV = Adx dM ρdV I === mass flow rate mass dt dt dx == ρρ A Av dt length dx I ≡ mass flow rate = ρAv during time dt mass Volume flow rate = volume of fluid crossing area A per unit time
Ivol ≡ volume flow rate = Av Mass flow rate per area
Jmass ≡ mass flow/unit area = ρv
2 Equation of Continuity: conservation of mass • An ideal fluid is moving through a pipe of nonuniform diameter Æ The rate of mass entering one end equals the mass leaving at the other end
mass flow rate at "1" = ρ111A vAv==mass flow rate at "2" ρ 2 22
• Ideal fluid is incompressible so: ρ2
ρ1 = ρ2 = a cons tan t
∴ A1v1 = A2v2
Æ Equation of continuity ρ1
Where the pipe narrows, the fluid moves faster, and vice versa
Example:
Water flows through a fire hose of diameter 6.0 cm at a rate of 0.012 m^3/s. The fire hose ends in a nozzle of inner diameter 4.0 cm. What is the speed with which the water exits the nozzle?
3 Bernoulli’s Equation – Work-Energy theorem for ideal fluid
p2 pressure pressure p1 y2
y1
Question: If height and area change, how do the pressure and velocity of fluid change?
Derivation of Bernoulli’s Equation from Work-Energy theorem The fluid is incompressible. The continuity equation (no leaks) requires:
ΔV = A1Δx1 = A2Δx2 m1 = m2 ≡ m = ρΔV The net work done on fluid volume by external pressures:
ΔW = F1dx1 − F2dx2 = P1A1dx1 − P2A2dx2 = [P1 − P2 ] ΔV The change in the mechanical energy: ΔE = E − E = 1 mv2 − 1 mv2 + mgy − mgy 2 1 2 2 2 1 2 1 Equate to the net work and divide by the volume ΔV: 1 2 1 2 P − P = ρv − ρv + ρg(y − y ) 1 2 2 2 2 1 2 1 Rearrange: 1 2 1 2 P + ρv + ρgy = P + ρv + ρgy 1 2 1 1 2 2 2 2
Pressure P 1 Equal volumes flow in equal times Pressure P2 speed v 1 diiidue to continuity equation speed v2 height y 1 height y2 Volume Volume ΔV = A1Δx1 ΔV = A2Δx2
time t time t + Δt
4 Bernoulli’s Equation from analogy with Work-Energy theorem for solids
For solids,
net net ⎛⎞⎛⎞1122 Wnon−− conservative=Δ=+−+ F non conservative x⎜⎟⎜⎟ mv f mgh f mv i mgh i ⎝⎠⎝⎠22
For ideal fluids
⎛⎞⎛⎞1122 pp1222−=⎜⎟⎜⎟ρρ v + gh − ρρ v 11 + gh ⎝⎠⎝⎠22
that is, 11 p++ρρ v22 gh= p ++ ρρ v gh 11122222 1 or, pvgh++ρρ2 is conserved. 2
Bernoulli’s Equation – Work-energy theorem
1 2 1 2 P + ρv + ρgy = P + ρv + ρgy 1 2 1 1 2 2 2 2 1 2 That is, P + ρv + ρgy = Constant 2
For a fluid at rest Bernoulli’s Equation reduces to the earlier result for variation of pressure with depth:
P1 = P2 + ρg[y2 - y1]
For flow at constant altitude (y1 = y2), the pressure is lower where the fluid flows faster: 1 2 1 2 P + ρv = P + ρv 1 2 1 2 2 2 (Gases are compressible and are non-ideal fluid, but the above pressure / speed relation holds true.)
5 Bernoulli’s Equation : Principle behind airplane wing
Bernoulli’s Equation & venturi meter
6 Bernoulli’s Equation : Principle behind curve ball
Water flows in a tube that changes cross-sectional area.
A1= 2.0 m^2
A2=1.0 m^2
V1=3.0m/s Same height V2=?
P1 P2
iClicker Q
What is v2? a) 1.5 m/s b) 3.0 m/s c) 6.0 m/s
7 Water flows in a tube that changes cross-sectional area.
A1= 2.0 m^2
A2=1.0 m^2
V1=3.0m/s Same height V2=? P1 P2
Example: Find P1 - P2. Water density is 1000 kg/m^3
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