Fluids in motion continue…
Δm = pv
thus
Δm = pAvΔt
Viscous flow:
the magnitude of the tangential force required to move a fluid layer at a constant speed is
given by:
Equation of continuity: the mass flow rate has the same value at every position
along a tube that has a single entry and a single exit for fluid flow.
P1A1v1 = P2A2v2
F = ηAv
y
SI-unit: kg/s
η = coefficient of viscosity
SI-unit (η): Pa.s
Common unit: Poise (P)
1 Poise = 0,1 Pa.s
Bernoulli’s equation: fluid accelerates towards the low pressure regions.
According to the pressure-depth relationship, the pressure lowers at higher levels,
provided the area of the pipe does not change.
Poiseuille’s law: fluids whose viscosity is η flowing through a pipe of radius R and length L,
has a volume flow rate Q given by:
Example is real life: Atherosclerosis (when a deposit is formed on an arterial wall)
Q = πR4 (P2 - P1)
8ηL
when a fluid is accelerated because of s difference in pressures, work is being
done by nonconservative forces and this work changes the total mechanical
energy of the fluid.
total mechanical energy:
E = KE + PE
E = 1/2mv2 + mgh
total work done:
W = E1 - E2
W = (1/2mv12 + mgh1) - (1/2mv22 + mgh2)
- top surface: pressure P
- this pressure gives rise to a force of magnitude: F = PA
- bottom surface: pressure P + ΔP
- thus the force: F + ΔF = (P + ΔP)A
- magnitude of net force pushing the fluid up the pipe: ΔF = ΔP.A
- the work done is the product of the magnitude of the net force and the distance:
W = ΔF.s
W = ΔP.v
In steady flow of a non-viscous, incompressible fluid of density p, pressure P, the
fluid speed v and the elevation h at ant two points are related by:
P1 + 1/2pv12 + pgh1 = P2 + 1/2pv22 + pgh2
ΔP = 8ηLQ = 8πηLQ
πR4
A2