Ice cube in a glass of water After the piece of ice melts: Water level, h ? Barge with steel beams:

Steady flow in a river. Velocity in each point is shown by a vector with the length proportional to the velocity. Velocity gets higher, where the river gets narrower. Flow represented by streamlines, that are everywhere tangent to flow direction. Higher density of the streamlines corresponds to higher flow velocity. In a steady flow there are no variations in velocity and pattern of flow in time. Nevertheless, the actual fluid elements flowing past any particular point at different times are always different. The fluid elements also get accelerated and decelerated as they move along the streamlines.

Motion of fluids obeys the standard laws of mechanics. Newton’s second law: Becomes Navier-Stokes equation: Newton’s second law is actually a complicated differential equation! Any way to make our life easier?! Let’s try to use the laws of conservation!!

Motion of fluids obeys the standard laws of mechanics. Conservation of mass: Conservation of momentum: Conservation of energy: Using the laws of conservation means doing appropriate bookkeeping and doing algebra instead of solving differential equations!

Flow tube - A small tubelike region bounded on its sides by a continuous set of streamlines and on its ends by small areas at right angles to the streamlines. Cross-section areas on the left and right ends are: A 1 and A 2. Densities and velocities are:  1,  2 and v 1, v 2 Steady flow

Mass of fluid entering the tube from the left over the time interval  t By mass conservation, over the time interval  t, the same mass is exiting the tube from the right Therefore everywhere along a flow tube If the fluid is incompressible and its density, , is constant, we have Steady flow

Does it work for traffic? Once you pass the spot of accident there are more lanes available (larger A ) and the traffic speeds up (higher v ). What is the matter? Traffic is highly compressible. You have got to use

How does the total energy of a small fluid element change, as it moves inside the flow tube from cross-section 1 to cross-section 2? Kinetic energy: Potential energy:

How does this change in the total energy become possible? There are external forces originating from pressure of the liquid outside the tube, which do work on the fluid element! The total energy balance Positive work as it enters from the left Negative work as it exits from the right

The total energy balance Incompressible fluids – constant density and volume Bernoulli’s equation

Inertial jet The fire truck pump is generating a pressure P. What is the maximal velocity at the nozzle? What is the maximal height the jet can reach?

The venturi flowmeter is a practical instrument which makes use of the Bernoulli effect and a manometer pressure gauge.Bernoulli effectmanometer pressure gauge Venturi Flowmeter