Tiêu đề 8. Fluid Mechanics.pdf ✅

Fluid Mechanics 155 8 Fluid Mechanics QUICK LOOK The force of attraction or repulsion acting between the molecules are known as intermolecular force. The nature of intermolecular force is electromagnetic. They include Figure: 8.1 Intermolecular Force The force in between two molecules F F F = + attraction repulsion 7 9 a b F r r − = + The potential energy can be expressed as 12 6 A B U r r = − Figure: 8.2 Table 8.1: Types of Intermolecular Forces of Attraction Cohesive Force Adhesive Force The force of attraction between molecules of same substance is called the force of cohesion. This force is lesser in liquids and least in gases. The force of attraction between the molecules of the different substances is called the force of adhesion. (i) Two drops of a liquid coalesce into one when brought in mutual contact. (i) Adhesive force enables us to write on the blackboard with a chalk. (ii) It is difficult to separate two sticky plates of glass welded with water. (iii) It is difficult to break a drop of mercury into small droplets because of large cohesive force between the mercury molecules. (ii) A piece of paper sticks to another due to large force of adhesion between the paper and gum molecules. (iii) Water wets the glass surface due to force of adhesion. Note Cohesive or adhesive forces are inversely proportional to the eighth power of distance between the molecules. Density Pressure and Flotation Density mass volume M V ρ = = Relative density = specific gravity density of body density of water = Relative density weight of body in air up thrust of water = air air water W W W = − With rise of temperature, density of liquid decrease, so up thrust decreases. Relative density of liquid fraction of solid immersed in water fraction of same solid immersed in liquid = air liquid air water W W W W − = − Up thrust on a solid body within liquid =V g in l σ where σ l = density of liquid Vin = volume inside liquid. Figure: 8.3 When ice floating in water in a beaker melts, there is no change in level of water. If ice containing metal floating in water melts completely, level of waterfalls. If ice containing cork floating in water metals, there is no change in level. Density of Mixture If masses m1 and m2 and densities ρ1 and ρ ρ 2 are mixed, Sinker in air Wair Sinker in water Wwater Sinker in liquid Wliquid Repulsive force Sum of forces: net force Separation lf atoms Attractive force Force between pair of atoms r0 attractive force Repulsive force Energy 0 (r) 0 dE dr > 0 dE dr < Dipole dipole Dipole charges Intermolecular Ion-dipole H-bond Ion-induced dipole Dipole-induced dipole Dispersion (London) Ion charge-dipole charge Polar bond to H-dipole charge (high EN of N, O, F) Ion charge- polarizable e– cloud Dipole charge polarizable e– cloud Polarizable e– clouds A H B – ......... δ δ δ − + − − ⋅ − – ......... δ δ δ − + −
156 Quick Revision NCERT-PHYSICS then density of mixture 1 2 1 2 1 2 m m d m m ρ ρ + = + If volume V1 and V2 and densities ρ1 and ρ2 are mixed, then density 1 1 2 2 1 2 ρ ρ V V ρ ρ ρ + = + Pressure Force Area F P A = = Pressure is a scalar quantity. Hydrostatic pressure at a depth h below the surface of a liquid P P hdg = +0 where P0 = atmospheric pressure Gauge Pressure: The difference of hydrostatic pressure and atmospheric pressure is called the gauge pressure i.e., P P h g − =0 ρ Figure: 8.4 It is independent of shape of vessel in which liquid is filled. When g varies barometric height remains unchanged For floating equilibrium Weight of body = weight of liquid displaced. Apparent weight of a floating body is a always zero. The condition of equilibrium for a floating body can be expressed in terms of Meta centre (M) and the centre of gravity of body (GM) height as follows: GM > 0 M is above G Stable equilibrium GM = 0 M coinciding with G Neutral equilibrium GM < 0 M is below G Unstable equilibrium Pascal's Law Figure: 8.5 Pressure of Liquid in Equilibrium It states that if gravity effect is neglected, the pressure at every point of liquid in equilibrium of rest is same or The increase in pressure at one point of the enclosed liquid in equilibrium of rest is transmitted equally to all other points of the liquid and also to the walls of the container, provided the effect of gravity is neglected. Example: Hydraulic lift, hydraulic press and hydraulic brakes Surface Tension Surface tension / F T N m L = ; Dimensions of surface tension 2 [MT ] − = Surface tension of liquids generally decreases with rise of temperature. At critical temperature surface tension is zero. By mixing detergents surface tension of liquid decreased. Due to surface tension liquid drop assumes a spherical shape. Figure: 8.6 Angle of contact: A contact angle can be measured by producing a drop of liquid on a solid. The angle between the tangent to the liquid surface and the tangent to the solid surface at the point of contact (inside the liquid) is known as angle of contact. Figure: 8.7 If angle of contact is less than 90° then the liquid wets the surface, the liquid spreads on the surface, there is a capillary rise and the liquid meniscus is concave up. If the angle of contact is greater than 90° then the liquid does not wet the surface, the liquid does not spread on the surface, there is a capillary dip and the liquid meniscus is convex up. For mercury and glass, the angle of contact is 135°. For ordinary water and glass, the angle of contact is nearly 8°. Hydrophobic surface Hydrophilic Surface θ θ high poor poor low low poor poor high Contact angle adhesiveness wet ability solid surface free energy Surface Tension Liquid Load F f C D p y x A P p p0 > ptm ptm p0 Q B
Fluid Mechanics 157 When the free surface of a liquid comes in contact with a solid, it becomes curved near the place of contact. The angle of contact is different for different pairs of solids and liquids. It increases on increasing the temperature and decrease on adding soluble impurity in liquid. Angle of contact does not depend upon the inclination of the liquid. A water proofing agent increases the angle of contact from an acute angle to an obtuse angle. Young's Equation is used to describe the interactions between the forces of cohesion and adhesion and measure what is referred to as surface energy. Figure: 8.8 Young’s Equation 1 1 cos sv s v γ γ γ θ = + θ is the contact angle s1 γ is the solid/liquid interfacial free energy sv γ is the solid surface free energy 1v γ is the liquid surface free energy Figure: 8.9 Total surface energy, W TA = Surface energy per unit surface area W T A = ∆ For a spherical drop: 2 W T R = ⋅ 4π Figure: 8.10 Excess pressure inside a spherical drop 2T R = Inside a soap bubble 4T R = inside a cylindrical film T R = For a soap bubble: W = T⋅2 (4π R2 ) Work done in breaking a bid drop of radius R into n drops of equal radius r ⇒ 2 1/ 3 W R T n = − 4 1 π     Decrease in temperature 3 1 1 T JSd r R θ   ∆ = −     Capillarity Figure: 8.11 Phenomenon of capillarity depends on the nature of liquid and solid both, i.e., T, θ and d. If θ > 90°, i.e., meniscus is convex, h will be negative i.e., the liquid will descend in the capillary tube as actually found in case of mercury in a glass capillary. θ = 90°, i.e., meniscus is plane, h = 0, so no phenomenon of capillarity. If θ < 90°, i.e., meniscus is concave towards air, h will be +ve i.e., the liquid will rise in the capillary. Height of liquid rise in capillary 2 2 cos T T h R g r g θ ρ ρ = = where R = radius of meniscus, r = radius of capillary. Jerlin’s law, hR = constant; or cos hr θ = constant Figure: 8.12 If capillary is tilted, liquid rises to a length cos hr l θ = In case of capillary of insufficient length, i.e., L h > , the liquid will neither overflow from the upper end like a h p < patm p = patm p > patm Water table θ < 90°; H > 0 θ p = patm d P There are two interfaces! R + P Soap bubble 0 ( ) F p p R downward i = − π 2 2 2 F R upward = σ π + R θ lv γ sl γ sv γ u Liquid Vapour
158 Quick Revision NCERT-PHYSICS fountain nor will trickle along the vertical sides of the tube. The liquid after reaching the upper end will increase the radius of the meniscus without changing its nature such that: hr > Lr'. When two bubbles of different radii 1 r and r r r 2 2 1 ( > ) come into contact the radius of common surface 1 2 2 1 r r R r r = − When two bubbles of radii r1 and r2 coalesce in vacuum under isothermal condition, then 2 2 R r r = + 1 2 When two bubbles of different sizes are in communication two with each other, air passes from smaller one to larger one and larger one grows at the expense of smaller one. This happens due to pressure inside the smaller bubble being higher than that inside the larger bubble. Figure: 8.13 If two soap bubbles of different radii r1 and r2 (r1> r2) coalesce to form a single double bubble having a common surface, then the radius of curvature of the interface is given by ( ) 1 2 1 2 r r r r r = − The interface will be concave towards smaller bubble and convex towards larger bubble. If two spherical soap bubbles of radii r1and r2 coalesce in vacuum to form a bigger bubble of radius R, then there is no change in temperature and surface energy. This implies that surface area remains unchanged i.e., 2 2 2 1 2 4 4 4 π π π r r R + = or 2 2 R r r = + 1 2 If a small drop of water is squeezed between two parallel glass plates so that a very thin layer of large area is formed then the pressure inside the water layer is less than the pressure on the plates by 2T d       (where d is the distance between the plates). When a bigger drop splits into smaller drops, energy is required to break if but then smaller drop coalesce to form a bigger drop energy is released. The excess pressure in case of a drop or bubble in a liquid is 2T R       and is directed form inside to outside, i.e., from concave to convex side. This result is also valid for meniscus of liquid, i.e., in case of concave meniscus, 2 A B T P P R − = , i.e., 0 2 A T P P R = − i.e., pressure below the meniscus is lower than above it by 2T R       . Where R is the radius of meniscus. Similarly, in case of convex meniscus, 2 , A B T P P R − = i.e., 0 2 2 B A T T P P P R R = + = + i.e., pressure below the meniscus is more than above it by 2T R       . Viscosity The graphic shows laminar flow of fluid between two plates of area A. The bottom plate is fixed. When the top plate is pushed to the right, it drags the fluid along with it. Each successive layer from the top down exerts a force on the one below it, trying to drag it along, producing a continuous variation in speed from v to 0 as shown. Figure: 8.14 Viscous Force v F A x η ∆ = − ∆ Dimension of viscosity 1 1 η ML T . − − =     The SI unit of viscosity is ( ) 2 2 / N m N s Pa s m s m m ⋅   = = ⋅     0 For streamiline flow For turbulent flow c v v u v <   >  where c K v D η ρ = Reynolds number v v K ρ η = If K < 2000, the flow is streamlined and laminar; in between 2000 to 3000 the flow of liquid is unstable and changing from streamline to turbulentand above 3000 the flow of liquid is definitely turbulent. A F L v v = 0 Turbulent Laminar P2 > P1 P1 P2