Bernoulli's Equation Flashcards
What is Archimede’s principle?
P1/ρ + gz1 = P2/ρ *gz2, where z1 is the distance between bottom and surface of water, z2 is the distance between the bottom and an object of fluid density ρ submerged in the water.
How can we generalise Archimede’s principle to any point on the object?
P(z) = P1 + ρ*g(z1-z)
What is the equation for the net upward force on the object?
F.k = - integral over surface of object of P(z) k.ds = - integral over volume of ∇.(P(z) k) dV = integral over vlume of ρg dV = Mg, where M is the mass of the water displaced by the object
What is a second example we can consider for Bernoulli’s equation?
Tank of water with exit tube at bottom right, with atmospheric pressure outside and height H.
What is Bernoullis equation for this tank problem?
P1 + 1/2 ρv1^2 + ρgH = P2 + 1/2ρv2^2
How can we simplify this version of Bernoullis equation for this problem?
P1 = P2 = P(atmos), and v1 = 0, so 1/2 *ρv2^2 = ρgH, so v2 = sqrt(2gH) = water exit velocity.
How do we find an equation for the time to empty the tank? (first steps)
- Crossectional area of pipe = Ap, cross sectional area of tank is Az
- V(0) = volume of tank = H(0)Az
- dV/dt = -Apv2 = AzdH/dt
How do we find an equation for the time to empty the tank? (second steps)
- Sub in equation for v2
- Put H on other side and dt on other side and integrate those parts from 0 to td and from H(0) to 0
- Do the integrals and rearrange for td
What is a third example to use Bernoulli’s equation?
Static tube which measures fluid velocity. Tube with bends round under surface and then comes up with right angle at top. Point zero at right angle part with ρf,vf and Pf entering the top part and z0 distance to surface, zw depth of surface and
What is Bernoulli’s equation for the third example?
Pf +1/2 ρfvf^2 + ρfgz0 = P0 +1/2ρfv0^2 + ρgz0, so P0 - Pf = 1/2 ρfvf^2
What is the equation for the velocity of the water entering the tube at 0?
vf^2 = 2g*ρm/ρf * Δz, if ρm > ρf
What are the variables in the aerofoil example?
vt and Pt above aerofoil, vb and Pb below aerofoil, vf and Pf before aerofoil.
How can we use Bernoulli’s equation for a qualitative description of an aerofoil?
- Fluid has further to travel over top, so vt > vb
- Pf + 1/2ρvf^2 = Pt + 1/2ρvt^2 = Pb + 1/2ρvb^2
- (Pb-Pt) = 1/2*(vt^2-vb^2) > 0, implies a net force upwards.
