Important Definitions: MECHANICAL PROPERTIES OF FLUIDS: Fluid: A substance that can flow is called Fluid Both liquids and gases are fluids Pressure: The normal force acting per unit area of a surface is called pressure 3 Density: The density of any material is defined as its mass per unit its volume Its SI unit is Kgm -3 4 Relative Density: The ratio of density of a substance to the density of the water at 4 0 C is called the relative density of the substance It is a unit less quantity 5 Gauge Pressure: The difference between the pressures at any two points in a fluid is called gauge pressure Mathematically it is given by gh ρ = density of the fluid, h = difference in the vertical levels of the two points, g = acceleration due to gravity 6 Atmospheric pressure: The atmospheric pressure at a point is defined as the weight of the column of air of unit cross-sectional area extending from that point up to the top of the atmosphere Its value is 03x0 5 Pa at sea level 7 Barometer: A device used to measure the atmospheric pressure using a column of mercury in an inverted tube is called a Barometer 8 Open-tube Manometer: It is a device used to measure the pressure of an enclosed gas or liquid 9 Stream line flow: The flow of a liquid is said to be stream lined If each particle of the liquid passing a given point moves along the same path and has the same velocity as its predecessor had at that point 0 Streamline: The path followed by a fluid particle in a steady flow is called a streamline Two stream lines never cross each other Speed of efflux: The speed of outflow of fluid through a small orifice in a tank, is called the speed of efflux V eff orifice V eff gh for an open container, where h is the height of the column above Venturi meter: The venture meter is a device used to measure the rate of flow of a liquid through a pipe It is an application of Bernoulli s principle CMadan Kumar kalachandran@gmailcom Page
3 Dynamic Lift: The upward force experienced by a body when it moves through a fluid is called dynamic lift 4 Magnus effect: When a ball moving in air gets deviated from its path due to its spinning action, this effect is called Magnus effect 5 Aerofoil: A streamlined shaped, solid body that is capable of generating a dynamic lift as it moves through a fluid is called aerofoil 6 Viscosity: The property of a fluid due to which its different layers offer resistance to their relative motion is called viscosity 7 Coefficient of viscosity: It is defined as the ratio of shearing stress and the strain rate produced in the fluid 8 Terminal velocity: The maximum constant velocity acquired by a body while falling through a viscous medium is called its terminal velocity 9 Turbulence: The nature of flow of a fluid when its speed at any point in space varies rapidly and randomly with time is called turbulence 0 Critical Reynold s number: The exact value of Reynold s number at which turbulence sets in a fluid is called critical Reynold s number Surface tension: The property of liquids due to which their free space behaves like an elastic stretched membrane is called surface tension Surface tension in terms of Surface energy: Surface tension is defined as the surface energy per unit area or the force per unit length acting in the plane of the interface between the plane of the liquid and any other substance 3 Surface Energy: The excess energy possessed by the molecules present at the surface of a liquid as compared to those present in the interiors is called surface energy 4 Angle of contact: It is defined as the angle that the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid 5 Capillary action: The phenomenon of a liquid rising or falling through a tube of very fine bore (capillary), is called capillary action -oo0oo- CMadan Kumar kalachandran@gmailcom Page
Formula Chart: thrust( F) Pressure, p area( A) Mass( m) Density, Volume( V ) 3 Relative density of a substance substan ce rel 0 4 C waterat 4 Pressure at a point at depth h below a liquid = gh Where, ρ = density of the liquid g = acceleration due to gravity 5 Gauge pressure, gh P g 6 Apparent weight of a body of density σ in a fluid of density ρ, W ' W ( ), W = weight of the body in air 7 Weight of the fluid displaced = Vρg V = volume of the body inside the fluid Ρ = density of the fluid W (a) Relative density of a solid = W W W (b) Relative density of a liquid = W W W 3 W = weight of solid in air W = weight of solid inside water W 3 = weight of solid inside liquid 8 Equation of continuity Av = constant A = cross-sectional area of pipe v = fluid velocity 9 Bernoulli s equation: At any point in a streamline flow P gh v = constant CMadan Kumar kalachandran@gmailcom Page 3 P a
P = pressure v = fluid velocity 0 Speed of efflux: v ( P P agh) P = pressure above the free surface of the liquid Torricelli s Law: V eff gh Fl Coefficient of viscosity, va F = viscous force l = separation between two lamina A = Area of each lamina v = Relative velocity of two lamina 3 Stoke s Law: F 6av 4 Terminal velocity a = radius of ball or drop v = velocity of ball or drop v T a ( ) 9 g ρ = density of falling body σ = density of fluid 4 Pr 5 Hagen Poiseuille equation, V 8 L 6 Reynold s number: (a) Vd R e d = diameter of the pipe Inetial force (b) R e Viscous force CMadan Kumar kalachandran@gmailcom Page 4
7 Surface tension: F S l F = force required to change area of a film l = length of the edge of the film 8 Excess pressure inside a liquid drop or a cavity of radius R, S P P0 R 4S 9 Excess pressure inside air bubble, P P0 R 0 (a) Surface energy = Surface tension X area of film (b) Work done = Surface tension X change in film area S cos Rise and fall of a liquid in a capillary tube, h rg Quick Recap: = angle of contact ρ = density of liquid g = acceleration due to gravity -oo0oo- A substance that flow is called fluid Both liquids and gas are fluids They have no definite shop Fluids have a very small shear modulus of rigidity Liquids are treated as incompressible while gases are highly compressible 3 The normal force acting on any surface is called thrust The thrust acting per unit area is called pressure Pressure is a scalar quantity Its SI unit is Nm - or Pascal 4 Fluids also exert pressure on the objects immersed in them 5 Pascal s Law: This law states that pressure in a fluid at rest is same at all points which are at the same height A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel Pascal s law has many practical applications like hydraulic lift and hydraulic brakes etc 6 The pressure at a point located at a depth h below the surface of a incompressible fluid CMadan Kumar kalachandran@gmailcom Page 5
is given as, P Pa gh P a = pressure above the fluid surface which usually is the atmospheric pressure 7 A consequence of Pascal s law is hydrostatic paradox It means that to express the pressure at a point, the height of the fluid column is important and not the cross-sectional or base area or the shape of the container 8 Air around us also exerts pressure, it is known as atmospheric pressure Its value is 03X0 5 Pa at sea level 9 Streamline flow is shown by incompressible and non-viscous fluids at low flow speeds The volume of the incompressible fluid passing through a point in a unit time remains constant in a steady flow ie, Av = constant This relation is known as equation of continuity and is a consequence of conservation of mass 0 Another principle, which is based on conservation of energy for incompressible fluids in steady flow, is Bernoulli s principle It says that for the steady flow of an ideal fluid, total energy per unit volume remain constant P v gh = constant P = Pressure v = Kinetic energy per unit volume gh = Potential energy per unit volume Though shear strain in a fluid does not require shear stress, when a shear stress is applied to a fluid, the motion is generated which causes a shear strain grow with time The ratio of shear stress to the shearing strain rate is called coefficient of viscosity η Viscous force acting on a body can also be given by Stoke s law According to it viscous drag F, F 6av (-) negative sign here shows that drag acts opposite to the direction of motion of the body Thus viscous drag increases with increase in the velocity of moving body in opposite direction CMadan Kumar kalachandran@gmailcom Page 6
3 For liquids, viscosity decreases with increase in temperature while for gases, viscosity increases with increase in temperature 4 For a body falling freely through a fluid, its velocity increases due to acceleration due to gravity acting along on the body As a result viscous drag increases according to Stoke s law As the acceleration due to gravity and viscous drag act in opposite direction, a certain stage is reached when total force on the body remains zero The body then falls down with a constant maximum velocity called terminal velocity V T a Mathematically, V T g 9 5 The molecules present at the surface of a liquid at rest possess extra potential energy as compared to those present well inside the liquid This energy is called surface energy This surface energy makes a liquid surface behave like an elastic stretched membrane which tends to possess a minimum surface area This property of liquid is called surface tension Surface tension is measured as the force acting per unit length of an imaginary line drawn on the liquid surface, the direction of force being perpendicular to this line and tangential to the liquid surface 6 Surface tension give rises to many interesting phenomena like spherical shape of liquid drops, excess pressure inside an air bubble or a drop, capillary action etc, 7 The shape of meniscus of a liquid in a container depends on the angle of contact for the given pair of liquid and solid surfaces -oo0oo- CMadan Kumar kalachandran@gmailcom Page 7