fluid pressure Flashcards
derive the pressure at a depth D
draw a column of cross-section A. and height h₀.
pressure is P₀ on the top surface.
height, h, is from the bottom
Force on top face is P₀A
weight of fluid above h is (h₀ -h)ρAg
∴ force at h is P₀A + (h₀ -h)ρAg
∴ pressure at h: P(h)= P₀ + (h₀ -h)ρg
D = h₀ -h
∴ P(D) = P₀ + Dρg
Buoyancy equation
F = ρVg
V: object volume
F: Buoyancy force
ρ: fluids density
Archimedes principle
Buoyancy force is equal and opposite to the weight of the fluid displaced
describe the flow of an incompressible fluid
incompressible (ρ is constant)
rate of mass flow through a surface is ρv⋅ds
rate of mass flow through a volume with surface s:
ρ∫v⋅ds = 0 otherwise the density would be changing
For a horizontal pipe that changes from cross-section A₁ to A₂ :
v₁A₁ = v₂A₂ (volumetric flow is constant)
Bernoulli’s equation
P + ½ρv² + ρgh = constant
what are laminar and turbulent flow
Laminar: steady flow (low velocity)
Turbulent: high velocity
Laminar and turbulent flow drag force equations
laminar: (for a ball)
F= -6πRvη
η: fluid viscosity
R: ball radius
v: ball’s velocity
Turbulent:
F= -½ρv²A*C
C: drag coefficient (depends on object)
A: objects cross-section
v: objects velocity
properties of laminar flow vs turbulent flow
Laminar:
smooth streamlines
F ∝ v (drag force)
drag due to shearing of the fluid
Turbulent:
chaotic streamlines
F ∝ v²
drag due to mass of liquid that must be pushed out of the way
derive Reynold’s number
for a fluid of density ρ and object of dimensions L:
inertial force ~ ρL³a (a: acceleration)
v² ~ 2aL
so a ~ v²/L
∴ inertial force ~ ρL²v²
Viscous force ~ ηLv
ratio: ρL²v²/ηLv = ρLv/η = Re (Reynolds number, dimensionless)
what does Reynolds number tell you
Re > > 1 its turbulent flow
Re < 1 its laminar flow
What’s Stokes’ law
F = -6πRvη
η: fluid viscosity
R: ball radius
v: ball’s velocity
