FLUID

Characteristics of Fluid Flow (1) Steady flow (lamina flow, streamline flow) The fluid velocity (both magnitude and direction) at any given point is constant in time The flow pattern does not change with time Non-steady flow (turbulent flow) Velocities vary irregularly with time e.g. rapids, waterfall

Rotational and irrotational flow The element of fluid at each point has a net angular velocity about that point Otherwise it is irrotational Example: whirlpools Compressible and incompressible fluid Liquids are usually considered as incompressible Gas are usually considered as highly compressible Characteristics of Fluid Flow (2)

Viscous and non-viscous fluid Viscosity in fluid motion is the analog of friction in the motion of solids It introduces tangential forces between layers of fluid in relative motion and results in dissipation of mechanical energy Characteristics of Fluid Flow (3)

Streamline A streamline is a curve whose tangent at any point is along the velocity of the fluid particle at that pointstreamline It is parallel to the velocity of the fluid particles at every point No two streamlines can cross one another In steady flow the pattern of streamlines in a flow is stationary with time

Change of speed of flow with cross-sectional area If the same mass of fluid is to pass through every section at any time, the fluid speed must be higher in the narrower region Therefore, within a constriction the streamlines must get closer togetherconstriction

Kinematics (1) Mass of fluid flowing past area A a =  a v a  tA a Mass of the fluid flowing past area A b =  b v b  tA b

In a steady flow, the total mass in the bundle must be the same   a v a A a  t=  b v b A b  t i.e.  a v a A a =  b v b A b or  vA = constant The above equation is called the continuity equation For incompressible fluids vA = constant Kinematics (2) Further reading

Static liquid pressure The pressure at a point within a liquid acts in all directions The pressure depends on the density of the liquid and the depth below the surface P =  gh Further reading

Bernoulli’s equation This states that for an incompressible, non- viscous fluid undergoing steady lamina flow, the pressure plus the kinetic energy per unit volume plus the potential energy per unit volume is constant at all points on a streamline i.e.

Derivation of Bernoulli’s equation (1) The pressure is the same at all points on the same horizontal level in a fluid at rest In a flowing fluid, a decrease of pressure accompanies an increase of velocity

In a small time interval  t, fluid XY has moved to a position X’Y’ At X, work done on the fluid XY by the pushing pressure = force  distance moved = force  velocity  time = p 1 A 1  v 1   t Derivation of Bernoulli’s equation (2) figure

At Y, work done by the fluid XY emerging from the tube against the pressure = p 2 A 2  v 2   t Net work done on the fluid W = (p 1 A 1  v 1 - p 2 A 2  v 2 )  t For incompressible fluid, A 1 v 1 = A 2 v 2  W = (p 1 - p 2 )A 1 v 1  t Derivation of Bernoulli’s equation (3) figure

Gain of p.e. when XY moves to X’Y’ = p.e. of X’Y’ - p.e. of XY = p.e. of X’Y + p.e. of YY’ - p.e. of XX’ - p.e. of X’Y = p.e. of YY’ - p.e. of XX’ = (A 2 v 2  t  )gh 2 - (A 1 v 1  t  )gh 1 = A 1 v 1  t  g(h 2 - h 1 ) Derivation of Bernoulli’s equation (4) figure

Gain of k.e. when XY moves to X’Y’ = k.e. of YY’ - k.e. of XX’ = = Derivation of Bernoulli’s equation (5) figure

For non-viscous fluid net work done on fluid = gain of p.e. + gain of k.e. (p 1 - p 2 )A 1 v 1  t = A 1 v 1  t  g(h 2 - h 1 ) + Derivation of Bernoulli’s equation (6) figure

or Derivation of Bernoulli’s equation (7) figure

Assumptions made in deriving the equation Negligible viscous force The flow is steady The fluid is incompressible There is no source of energy The pressure and velocity are uniform over any cross-section of the tube Derivation of Bernoulli’s equation (8) Further reading

Applications of Bernoulli principle (1) Jets and nozzles Bernoulli’s equation suggests that for fluid flow where the potential energy change h  g is very small or zero, as in a horizontal pipe, the pressure falls when the velocity rises The velocity increases at a constriction and this creates a pressure drop. The following devices make use of this effect in their action

Bunsen burner The coal gas is made to pass a constriction before entering the burner The decrease in cross-sectional area causes a sudden increase in flow speed The reduction in pressure causes air to be sucked in from the air hole The coal gas is well mixed with air before leaving the barrel and this enables complete combustion Applications of Bernoulli principle (2)

Carburettor Carburettor of a car engine The air first flows through a filter which removes dust and particles It then enters a narrow region where the flow velocity increases The reduced pressure sucks the fuel vapour from the fuel reservoir, and so the proper air- fuel mixture is produced for the internal combustion engine Applications of Bernoulli principle (3)

Filter pump The velocity of the running water increases at the constriction The surrounding air is dragged along by the water jet and this causes a drop in pressure Air is then sucked in from the vessel to be evacuated Applications of Bernoulli principle (4)

Spinning ball If a tennis ball is `cut’ it spins as it travels through the air and experiences a sideways force which causes it to curve in flight This is due to air being dragged round by the spinning ball, thereby increasing the air flow on one side and decreasing it on the other A pressure difference is thus created Further readingfigure

Aerofoil A device which is shaped so that the relative motion between it and a fluid produces a force perpendicular to the flowdevice Fluid flows faster over the top surface than over the bottom. It follows that the pressure underneath is increased and that above reduced. A resultant upwards force is thus created, normal to the flow e.g. aircraft wings, turbine blades, sails of a yacht

Pitot tube (1) a device for measuring flow velocity and in essence is a manometer with one limb parallel to the flow and open to the oncoming fluiddevice The pressure within a flowing fluid is measured at two points, A and B. At A, the fluid is flowing freely with velocity v a. At B where the Pitot tube is placed, the flow has been stopped

By Bernoulli’s equation: Pitot tube (2) where P 0 = atmospheric pressure

Note: In real cases, v varies across the diameter of the pipe carrying the fluid (because of the viscosity) but if the open end of the Pitot tube is offset from the axis by 0.7  radius of the pipe, then v is the average flow velocity The total pressure can be considered as the sum of two components: the static and dynamic pressures Pitot tube (3) 

A moving fluid exerts its total pressure in the direction of flow. In directions at right angles to the flow, the fluid exerts its static pressure only figures Pitot tube (4) Further readingFurther reading: paragraph of ‘Pitot Static System’ near the bottom of the page

Venturi meterVenturi meter (1) This consists of a horizontal tube with a constriction. Two vertical tubes serving as manometers are placed perpendicular to the direction of flow, one in the normal part and the other in the constriction In steady flow the liquid level in the manometer connected to the wider part of the tube is higher than that in the narrower part figure

Venturi meter (2) From Bernoulli’s principle (h 1 = h 2 ) For an incompressible fluid, A 1 v 1 = A 2 v 2 

Hence Venturi meter (3)  v 1 can be deduced

Streamline P Q

Change of speed in a constriction Streamlines are closer when the fluid flows faster

Derivation of Bernoulli’s equation X Y X’ Y’ v1v1 p1A1p1A1 p2A2p2A2 v1tv1t v2tv2t Area A 1 Area A 2 v2v2 h1h1 h2h2

Bunsen burner

Carburettor fuel air filter to engine cylinder

Filter pump

Spinning ball

Aerofoil

Pitot tube (1)

Pitot tube (2) Pitot is here

Pitot tube: fluid velocity measurement (1) Flow of air Stagnant air, higher pressure inside tube Fast moving air, lower pressure inside chamber Total tube Static tube Static pressure holes P 1 = total pressure P 2 = static pressure P 2 – P 1 = ½(  v 2 )

Pitot tube: fluid velocity measurement (2)

Ventri meter (1)

Venturi meter (2)

Venturi meter (3) A1A1 A2A2 v1v1 v2v2 Density of liquid = 