Types of fluid flow Steady (or unsteady) - velocity at any point is constant. Turbulent flow - the velocity at any particular point changes erratically from moment to moment.

Compressible (or incompressible) – Most liquids are incompressible, gases are highly compressible.

Viscous (or nonviscous) Does not flow readily. An incompressible, non-viscous fluid is an ideal fluid.

Rotational (or irrotational) – A part of the fluid has rotational as well as translational motion.

Steady flow is called streamline flow.

The equation of continuity expresses the idea that if a fluid enters one end of a pipe at a certain rate of flow, it must exit at the same rate. The mass of fluid per second is called the mass flow rate.

The mass Δm of a fluid that passes given point in a given amount of time is: Δm = ρAvΔtThe mass flow rate Δm/Δt = ρAv. The mass flow rate is the same throughout a conducting tube.

This is the equation of continuity.

ρ1A1v1 = ρ2A2v2

ρ = fluid density (kg/m3)A = cross-sectional area of

tube (m2)v = fluid speed (m/s)The unit of mass flow rate is kg/s.

If we are dealing with an incompressible fluid, ρ1 = ρ2, and the equation of continuity reduces to A1v1 = A 2v2. Av is the volume flow rate Q.

Ex. 11 - A garden hose has an unobstructed opening with a cross-sectional area of 2.85 x 10-4 m2, from which water fills a bucket in 30.0 s. The volume of the bucket is 8.00 x 10-3 m3. Find the speed of the water that leaves the hose through (a) the unobstructed opening and (b) an obstructed opening that has only half as much area.

Ex. 12 - In the carotid artery, blood flows three time faster through a partially blocked region than it does through an unobstructed region. Determine the ratio of the effective radii of the artery at the two places.

Bernoulli’s Principle - a moving fluid exerts less pressure than a stationary fluid. Bernoulli’s equation also includes elevation, the pressure is greater at a lower level than at higher elevation.

Bernoulli’s Equation:

P1 + ½ρv12 + ρgy1 =

P2 + ½ρv22 + ρgy2

y is the elevation.This applies to the steady, irrotational flow of a nonviscous, incompressible fluid.

P + ½ρv2 + ρgy has a constant value at all points.

If a moving fluid is in a horizontal pipe, y remains constant and the equation simplifies to: P1 + ½ρv1

2 = P2 + ½ρv22

P1 + ½ρv12 = P2 + ½ρv2

2

P + ½ρv2 remains constant throughout the pipe.

Ex. 14 - Because of an aneurysm, the cross-sectional area A1 of the aorta increases to a value A2 = 1.7A1. The speed of the blood (ρ = 1060 kg/m3) through a normal portion of the aorta is v1 = 0.40 m/s. Assuming the aorta is horizontal, determine the amount by which the pressure P2 in the enlarged region exceeds the pressure P1 in the normal region.

An airplane wing is an example of how fluid flow affects pressure. Ski jumpers also use this principle to provide lift.

Ex. 15 - A tank is open to the atmosphere at the top. A pipe allowing the fluid to exit is a height h below the top of the fluid. Find an expression for the speed of the liquid leaving the pipe. (This is called efflux speed).

We find that the speed v1 = √2gh is the same as if the fluid had freely fallen through height h. (vf

2 = v02 +2ax)

This is known as Torricelli’s theorem. If the pipe were directed upward, the water would rise to a height equal to the fluid level.

But this only occurs if the fluid is ideal. A non-ideal fluid has viscosity that will decrease the height to which the liquid will rise.

Fluids with any viscosity will not flow uniformly across the area of a tube; they will move faster at the center and slower at the edges.

The force needed to move a viscous fluid is thus greater at the edges than at the center. The force is also proportional to the area of contact and to the velocity.

These relationships are expressed using the coefficient of viscosity (or the viscosity) represented by Ɲ.

F = ƝAv/y

The unit of viscosity Ɲ is the the Pa•s, 0.1 Pa•s is a unit called the Poise, P (after Jean Poiseuille). Ideal fluids have a viscosity Ɲ of zero.

Poiseuille’s Law relates flow rate to pressure difference, h, and length and radius of the pipe. Q = πR4(P2 -P1)/8ƝL

Ex. 16 - A hypodermic syringe is filled with a solution whose viscosity is 1.5 x 10-3 Pa•s. The plunger area of the syringe is 8.0 x 10-5 m2, and the length of the needle is 0.025 m. The internal radius of the needle is 4.0 x 10-4 m. The gauge pressure in a vein is 1900 Pa. What force must be applied to the plunger, so that 1.0 x 10-6 m3 of solution can be injected in 3.0 s?