FLUIDS IN MOTION The equations that follow are applied when a moving fluid exhibits streamline flow. Streamline flow assumes that as each particle in the fluid passes a certain point it follows the same path as the particles that preceded it. There is no loss of energy due to internal friction (viscosity) in the fluid. In reality, particles in a fluid exhibit turbulent flow, which is the irregular movement of particles in a fluid and results in loss of energy due to internal friction in the fluid. Turbulent flow tends to increase as the velocity of a fluid increases.

FLOW RATE Consider a fluid flowing through a tapered pipe:

 The flow rate is the mass of fluid that passes a point per unit time: m/t = ρ A v Units: kg/s   Where ρ is the density of the fluid, A is the cross-sectional area of the tube of fluid at the particular point in question, and v is the velocity of the fluid at the point in question.

Since fluid cannot accumulate at any point, the flow rate is constant Since fluid cannot accumulate at any point, the flow rate is constant. This is expressed as the equation of continuity. ρ A v = constant In streamline flow, the fluid is considered to be incompressible and the density is the same throughout. The equation of continuity can then be written in terms of the volume rate of flow (R) that is constant throughout the fluid:   R = Av = constant Units: m3/s or: ρ1 A1 v1 = ρ2 A2 v2

A = πr2 = π(0.04)2 = 0.005 m2 R = Av = 0.005(4) = 0.02 m3/s 10.11 Oil flows through a pipe 8.0 cm in diameter, at an average speed of 4 m/s. What is the flow in m3/s and m3/h? r = 8/2 = 0.04 m v = 4 m/s A = πr2 = π(0.04)2 = 0.005 m2 R = Av = 0.005(4) = 0.02 m3/s R = 0.02 m3/s (3600 s/h) = 72.4 m3/h

BERNOULLI’S EQUATION In the absence of friction or other non- conservative forces,the total mechanical energy of a system remains constant, that is, PE1 + K1 = PE2 + K2   There is a similar law in the study of fluid flow, called Bernoulli’s principle, which states that the total pressure of a fluid along any tube of flow remains constant.

Assume incompressible fluid in a laminar flow with no viscosity. Bernoulli’s equation is a form of the energy conservation law.

Bernoulli’s principle: Where the velocity of fluid is high the pressure is low. Where the velocity of fluid is low the pressure is high.

Find the velocity of a liquid flowing out of a spigot: The pressure is the same P1=P2. The velocity v2 » 0. Bernoulli’s equation is: This is called Torricelli’s Theorem.

Fluid Pressure in a Pipe of Varying Elevation

Another special case: Fluid flow but no change in height: y1 = y2. Bernoulli’s equation: When pressure is low (high), velocity is high (low).

= 9.9 m/s R = v2A2 = 9.9 (π(1.5x10-2)2) = 7x10-3 m3/s 10.12 What volume of water will escape per second from an open top tank through an opening 3.0 cm in diameter that is 5.0 m below the water level in the tank? r = 1.5x10-2 m Top (1) Bottom (2) h1 = 5 m h2 = 0 If tank is large v1 = 0 = 9.9 m/s (Torricelli’s equation) R = v2A2 = 9.9 (π(1.5x10-2)2) = 7x10-3 m3/s