FluidAny substance that can flow is a fluid. A fluid is a substance that deforms continually under the action of an external force.

A fluid flows under the action of a force or a pressure gradient.

Properties of ideal fluid1. Incompressible2. Density is constant.2. Flow is irrotational and smooth4. No turbulences in the flow.5. It is nonviscous i.e. there is no internal friction in the flow, i.e., the fluid has no viscosity.6. Its flow is steady: its velocity at each point is constant in time.7. Fluid do not oppose deformation, they get permanently deformed.8. They have ability to flow.9. They have ability to take the shape of the container.10. Fluid cannot oppose a shear stress when in static equilibrium.11. Ideal fluids, can only be subjected to normal, compressive stress (called pressure).

HydrostaticsThe branch of physics which deals with the properties of fluids at rest is called hydrostatics.

PressureP=πgh$P=\mathrm{\pi \x9d\x9c\x8c}gh$

Absolute PressureIt is total pressure at depth h below the surface of liquid which includes pressure due to atomosphere and liquid.

Gauge PressureThe difference between the absolute pressure and the atmospheric pressure is called the gauge pressure.

Pascalβs LawPascalβs law states that the pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every point of the fluid and also on the walls of the container.

Application of Pascal Law1. Hydraulic lift:2. Hydraulic brakes

Measurement of Pressure1. Mercury Barometer2. Open tube manometer

Surface Tension1. Extra energy is associated with surface of liquids.2. Liquids have no definite shape but have a definite volume. Liquid acquire a free surface when poured in a container. These surfaces possess some additional energy. This phenomenon is known as surface tension.3. Surface tension T is defined as, the tangential force acting per unit length on both sides of an imaginary line drawn on the free surface of liquid.4. Surface tension is also equal to the surface energy per unit area.

Cohesive forces result in the phenomenon of surface tension.

Molecular Theory of Surface Tension1. Intermolecular forceIt is force between molecules.2. Range of molecular forceThe maximum distance from a molecule up to which the molecular force is effective is called the range of molecular force.3. Sphere of influenceAn imaginary sphere with a molecule at its center and radius equal to the molecular range is called the sphere of influence of the molecule.4. Surface filmThe surface layer of a liquid with thickness equal to the range of intermolecular force is called the surface film.5. Free surface of a liquidIt is the surface of a fluid which does not experience any shear stress. For example, the interface between liquid water and the air above.6. Surface tension on the basis of molecular theory

Intermolecular force

Cohesive forceThe force of attraction between the molecules of the same substance is called cohesive force.

Adhesive forceThe force of attraction between the molecules of different substances is called adhesive force.

The force due to surface tension acts tangential to the free surface of a liquid.

Pressure inside a spherical drop is more than outside.

Pressure on the concave side is always greater than that on the convex side

Angle of ContactWhen a liquid surface comes in contact with a solid surface, it forms a meniscus, which can be either convex (mercury-glass) or concave (water glass).

Shape of meniscusi) Concave meniscus - acute angle of contact.ii) Convex meniscus - obtuse angle of contact.

Conditions for concavityAdhesive force >Cohesive Force

A fluid will stick to solid surface if the surface energy between fluid and solid is smaller than sum of surface energy between solid-air and fluid-air.

Shape of liquid drops on a solid surface1. T1$T}_{1$: Force due to surface tension at the liquid-solid interface2. T2$T}_{2$: Force due to surface tension at the air-solid interface3. T3$T}_{3$: Force due to surface tension at the air-liquid interface4. π$\mathrm{\pi \x9d\x9c\x83}$ : Angle of contactcosπ=T2-T1

Factors affecting the angle of contact1. Any increase in the temperature of a liquid decreases its angle of contact. 2. Impurities present in the liquid change the angle of contact.

Effect of impurities on Surface tension1. When soluble substance such as common salt (i.e., sodium chloride) is dissolved in water, the surface tension of water increases.2. When a sparingly soluble substance such as phenol or a detergent is mixed with water, surface tension of water decreases. For example, a detergent powder is mixed3. When insoluble impurity is added into water, surface tension of water decreases.

Effect of temperature on Surface tension1. For most liquid as temperature increases surface tension decreases.2. The surface tension of a liquid becomes zero at critical temperature.3. In the case of molten copper or molten cadmium, the surface tension increases with increase in its temperature.The critical temperature of a substance is the temperature at and above which vapor of the substance cannot be liquefied, no matter how much pressure is applied. Every substance has a critical temperature.

Excess pressure across the free surface of a liquid(a) Plane liquid surface(b) Convex liquid surface(c) Concave liquid surface

Laplace equationThe Laplace law gives the relation between the pressure difference across an interface and the surface tensionFor spherical surfaceπ₯P=2T

R$\mathrm{\pi \x9d\x9b\u20af}P=\frac{2T}{R}$where R=curvature of the surfaceFor cylindrical surfaceπ₯P=T

R$\mathrm{\pi \x9d\x9b\u20af}P=\frac{T}{R}$where R=curvature of the surface

Capillary riseh=2Tcosπ

rπg$h=\frac{2Tcos\mathrm{\pi \x9d\x9c\x83}}{r\mathrm{\pi \x9d\x9c\x8c}g}$where , r is the radius of tube

Capillary fallh=2Tcosπ

rπg$h=\frac{2Tcos\mathrm{\pi \x9d\x9c\x83}}{r\mathrm{\pi \x9d\x9c\x8c}g}$where , r is the radius of tube

DropsIf the drop is in equilibrium this energy cost is balanced by the energy gain due to expansion under the pressure difference (π₯P=Pi-Po$\mathrm{\pi \x9d\x9b\u20af}P={P}_{i}-{P}_{o}$) between the inside of the bubble and the outside.Work doneπ₯W=π₯P.(4πr2)π₯r$\mathrm{\pi \x9d\x9b\u20af}W=\mathrm{\pi \x9d\x9b\u20af}P.\left(4\mathrm{\pi \x9d\x9c\x8b}{r}^{2}\right)\mathrm{\pi \x9d\x9b\u20af}r$Extra Surface energyπ₯U=T{4π(r+π₯r)2-r2}$\mathrm{\pi \x9d\x9b\u20af}U=T\{4\mathrm{\pi \x9d\x9c\x8b}(r+\mathrm{\pi \x9d\x9b\u20af}r{)}^{2}-{r}^{2}\}$π₯U=8πrπ₯r T$\mathrm{\pi \x9d\x9b\u20af}U=8\mathrm{\pi \x9d\x9c\x8b}r\mathrm{\pi \x9d\x9b\u20af}r\phantom{\rule{0.22em}{0ex}}T$π₯W=π₯U$\mathrm{\pi \x9d\x9b\u20af}W=\mathrm{\pi \x9d\x9b\u20af}U$βΉ π₯P.(4πr2)π₯r=8πrπ₯r T$\mathrm{\pi \x9d\x9b\u20af}P.\left(4\mathrm{\pi \x9d\x9c\x8b}{r}^{2}\right)\mathrm{\pi \x9d\x9b\u20af}r=8\mathrm{\pi \x9d\x9c\x8b}r\mathrm{\pi \x9d\x9b\u20af}r\phantom{\rule{0.22em}{0ex}}T$βΉ π₯P=2 T

Steady FlowStreamlines and flow lines are identical for a steady flow.The flow of the fluid is said to be steady if at any given point, the velocity of each passing fluid particle remains constant in time.

Flow LineIt is the path of an individual particle in a moving fluid.

Streamline1) Two streamlines can never intersect, i.e., they are always parallel.2) It is a curve whose tangent at any point in the flow is in the direction of the velocity of the flow at that point. 3. The path taken by a fluid particle under a steady flow is a streamline. It is defined as a curve whose tangent at any point is in the direction of the fluid velocity at that point.

Flow tube1. It is an imaginary bundle of flow lines bound by an imaginary wall.2. For a steady flow, the fluid cannot cross the walls of a flow tube. 3. Fluids in adjacent flow tubes cannot mix.

Laminar Flow or Streamline flow

Turbulent Flow

Equation of continuityThe volume rate of flow of an incompressible fluid for a steady flow is the same throughout the flow.Av = constant, where A: cross-sectional area and v: velocity

Critical VelocityThe velocity beyond which a streamline flow becomes turbulent is called critical velocity.

ViscosityViscosity is that property of fluid, by virtue of which, the relative motion between different layers of a fluid experience a dragging force. This force is called the viscous drag.

Velocity gradientThe rate of change of velocity (dv) with distance (dx) measured from a stationary layer is called velocity gradient (dv/dx).

Coefficient of viscosityIt defined as the viscous force per unit area per unit velocity gradient.The coefficient of viscosity (π$\mathrm{\pi \x9d\x9c\x82}$) for a fluid is defined as the ratio of shearing stress to the strain rate.π=F/A

v/l=F l

v A$\mathrm{\pi \x9d\x9c\x82}=\frac{F/A}{v/l}=\frac{F\phantom{\rule{0.22em}{0ex}}l}{v\phantom{\rule{0.22em}{0ex}}A}$SI unit of viscosity is poiseiulle.

Reynolds numberReynolds observed that turbulent flow is less likely for viscous fluid flowing at low rates. He defined a dimensionless number, whose value gives otells us whether the flow would be turbulent . This number is called the Reynolds R .Re=πvcd

Stokes LawThe viscous force acting on a small sphere falling through a viscous medium is directly proportional to the radius of the sphere (r), its velocity (v) through the fluid, and the coefficient ofviscosity (Ξ·) of the fluid.Fv=6ππrv${F}_{v}=6\mathrm{\pi \x9d\x9c\x8b}\mathrm{\pi \x9d\x9c\x82}rv$

Terminal VelocityConsider a spherical object falling through a viscous fluid.The constant downward velocity is called terminal velocity. v=(2

Applications of Bernoulliβs equationa) Speed of effluxb) Ventury tubec) Working of an atomizerd) Blowing off of roofs by stormy winde) Lifting up of an aeroplane

Speed of effluxv=2gh$v=\sqrt{2gh}$

Ventury tubeA ventury tube is used to measure the speed of flow of a fluid in a tube. p1-p2=1