Fluid Flow And Bernoulli

Fluid Mechanics Flow Rate & Bernoulli’s Equation

Volumetric Flow Rate • volumetric flow rate: the volume of fluid that passes a particular point per unit of time – example: liters/minute coming out of a faucet – metric units: m3/s – note: not the same as flow velocity (m/s)

Flow Rate

S

volumetric flow rate = Q = Sv

Continuity Equation S

• If the fluid in the pipe is incompressible (density remains constant) then the flow rate must be the same everywhere in the pipe. • Therefore: Q1 = Q2 • …and S1v1 = S2v2  This is known as the

Sample Problem 1 A pipe of non-uniform diameter carries water. At one point on the pipe, the radius is 2 cm and the flow speed is 6 m/s. c.What is the volumetric flow rate? e.What is the flow velocity at a point where the pipe constricts to a radius of 1 cm?

Sample Problem 2 If the diameter of a pipe increases from 4 cm to 12 cm, what will happen to the flow velocity?

Bernoulli’s Equation • One of the most important idea in fluid mechanics. • It is the statement of the law of conservation of energy for ideal fluid flow (mechanical energy balance equation).

Conditions for Ideal Fluid Flow 1. The fluid is incompressible. This works well for liquids and also applies to gases if the pressure changes are small.

Conditions for Ideal Fluid Flow 1. The fluid’s viscosity is negligible or zero. Viscosity is the force of cohesion between the molecules of a fluid. It can be thought of as internal friction. Syrup has a higher viscosity than water – there’s more resistance to the flow of syrup. Bernoulli’s equation gives good results when

Conditions for Ideal Fluid Flow 1. The flow is streamline (laminar). The fluid moves smoothly through the tube. The opposite is turbulent flow.

Turbulent Flow

Bernoulli’s Equation • If the conditions for ideal fluid flow are met and the volumetric flow rate, Q, is steady, Bernoulli’s equation can be applied to any pair of points along a streamline with the flow.

Bernoulli’s Equation z2 z1

• P1 & P2 = pressure at points 1 and 2 • v1 & v2 = flow velocity • z1 & z2 = elevation above a reference level Bernoulli’s Equation (each term has a unit of energy per unit mass):

Bernoulli’s Equation

Bernoulli’s Equation: or…

Implications of Bernoulli’s Equation Bernoulli’s Equation:

• Where the flow speed is high the pressure is low, and where flow speed is low, pressure is high.

Applications of Bernoulli’s Equation

• Close streamlines above wing indicate high velocity (continuity equation). Therefore the pressure above the wing is lower which creates a loft force that balances that of gravity.

Applications of Bernoulli’s Equation A person with constricted arteries will find that they may experience a temporary lack of blood to the brain (TIA – transient ischemic attack) as blood speeds up to get past the constriction, thereby reducing the pressure.

Torricelli’s Theorem

Bernoulli’s equation can be used to determine the efflux speed (how fast the liquid flows out of the hole)

Torricelli’s Theorem z2

z2-z1 z1

Torricelli’s Theorem z2

z2-z1 z1

Points 1 and 2 are open to air, so P1 = P2 = Patm Also,

Torricelli’s Theorem z2

z2-z1 z1

Torricelli’s Theorem z2

z2-z1 z1

Torricelli’s Theorem z2

z2-z1 z1

v2 ≅ 0 (when compared to v1)

Torricelli’s Theorem z2

z2-z1 z1

v2 ≅ 0 (when compared to v1)

Torricelli’s Theorem z2

z2-z1 z1

Torricelli’s Theorem z2

z2-z1 z1

Solving for v1, we have

v1 = 2g(z 2 − z1 )α = 2ghα

Siphoning The figure below shows a siphon that is used to draw water from a swimming pool. The pipe that makes up the siphon has an inside diameter of 40mm and terminates with a 25mm diameter nozzle. Assuming that there are no energy losses in the system, calculate the volume flow rate through the siphon. Calculate also the pressure at points b, c c, d and e. 1.2m

a

b

d

1.8m

40-mm inside diameter

1.2m

e

25-mm inside diameter

f

For Your Information

Other Examples

a c 6m h b

How high will the jet of water shown at the left be? (neglecting energy losses)