INTRODUCTION DEFINITION OF FLUID plate solid F at t = 0 t > 0 = F/A plate U p F fluid t 0 t 1 t 2 t 3 FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION THE PROCESS OF CONTINUOUS DEFORMATION IS KNOWN AS FLOW OF FLUIDS Fluid Fluids cannot support tension eiter 1.01
SCOPE OF FLUID MECHANICS Civil Engineering Applications Utilitarian Water supply Energy production Transportation of fluids, of material, as waterways Mecanical Pipelines Hydraulic structures Fluvial ydraulics Coastal ydraulics Groundwater flow Wind forces on structures Sips, Cars, Fast Trains, Aeroplanes Macines, Industrial Plants Te circulatory system of uman body 1.02
CONCEPT OF CONTINUUM Actual molecular structure A ypotetical medium lim d m d Continuum assumption is te continuous distribution of matter in te flow field witout any discontinuity. A fluid particle is defined as te mass contained in te smallest fluid volume for wic te continuum assumption is not violated. 1.03
DESCRIBING PHYSICAL ENTITIES Qualitative description Quantitative description DIMENSIONS UNITS Quantity MLT FLT SI units PRIMARY Mass M FL -1 T 2 kg or Force MLT -2 F N Lengt L L m Time T T s Temperature C F ma DERIVED Area L 2 L 2 m 2 Velocity LT -1 LT -1 m/s Acceleration LT -2 LT -2 m/s 2 Force MLT -2 F N Pressure ML -1 T -2 FL -2 Pa Energy ML 2 T -2 FL Joule Power ML 2 T -3 FLT -1 Watt Angle 1 1 radian 1.04
PHYSICAL PROPERTIES OF FLUIDS Density, : Mass per unit volume, Specific Weigt, : Weigt per unit volume, = m/ []=ML -3 = W/ []=FL -3 Specific Gravity, SG: Te ratio of te density of te fluid to te density of water (or air) at standard conditions. (SG) liquid w (SG) gas air Density and Specific Weigts of some fluids (g=9.81m/s 2 ) Liquids Gases Fluid Temperature C Density kg/m 3 Specific Weigt N/m 3 Water 4.0 1000. 9810. Mercury 20.0 13600. 133416. Gasoline 15.6 680. 6671. Alcool 20.0 789. 7740. Air 15.0 1.23 12.0 Oxygen 20.0 1.33 13.0 Hydrogen 20.0 0.0838 0.822 Metane 20.0 0.667 6.54 Note tat = g 1.05
Viscosity: S B B Δθ A y U p u(y) F Deformation of fluid for a sort time interval t U p F A or U p U p S lim t0 t lim t0 t lim t0 t d dt Tus d dt Sear stress is proportional to te rate of angular deformation 1.06
For te linear velocity profile U p u(y) y u(y) U p y du U d p dy dt Terefore du dy or du dy Newton s Law of viscosity Te proportionality constant is known as dynamic viscosity of te fluid. 2 FL 2 1 1 du dy LT L 1 FL T ML T Viscosity can be made independent of fluid density; kinematic viscosity is defined as te ratio ML 1 ML T 3 1 2 L T 1 Fluid Temperature (C) (Ns/m 2 ) (m 2 /s) Water 20 1.00E-03 1.01E-06 Air 20 1.80E-05 1.51E-05 1.07
In general K du dy n=1 Newtonian fluids n1 Non-Newtonian fluids n>1 Sear tickening n<1 Sear tinning n n<1 1 = 0 ideal fluid n>1 A typical variation of sear stress y U max dy (y) u(y) u(y) du w w Wall sear stress w du dy y0 Frictional drag force 1.08
Dynamic (absolute) viscosity of some common fluids as a function of temperature 1.09
Surface tension, Intermolecular Attraction Forces Coesive Forces (C) Liquid to liquid Gas to gas Adesive Forces (A) Liquid to solid Gas to liquid A>C Solid Gas Liquid Te intensity of te molecular attraction per unit lengt along any line on an interface is called te surface tension. 1 FL A>C Capillary effects C>A Capillary rise (wetting fluid) Capillary drop (non-wetting fluid) P atm z 2R 2cos R P atm 1.10
Vapor pressure, p v Vapor Water p Boiling occurs wen pp v Heat Vapor pressure for water Temperature C p v (kpa) 0 0.61 10 1.23 25 3.17 60 19.92 100 101.33=p atm 1 2 3 p 3 Vapor pockets p 1 >p v p 2 p v p 3 >p v Cavitation 1.11
Compressibility of Fluids p=f/a A F dp 0 d/ 0 1 Bulk Modulus of Elasticity E v dp d/ 0 dp dρ /ρ Compressibility K d/ dp 0 dρ /ρ dp (E v ) water =2.15x10 9 Pa (STP) (E v ) air =1.42x10 5 Pa (STP) E steel =2.00x10 11 Pa 1.12
EXAMPLES Example 1.1 Calculate te velocity gradient and te sear stress for y=0, 0.1, and 0.5 m if te velocity profile of te flow is a parabola given by u = 50 (2y-y 2 ), 0 y 1m were u is in (m/s) and y is in (m). Draw te sear stress distribution. Also calculate te frictional drag force of te fluid on te bottom boundary on an area of 10 m 2. Use dynamic viscosity =0.001 Pas. 1.E01
Example 1.2 A space =25 mm wide between two plane surfaces is filled wit crude oil at 20C for wic oil =7.18x10-3 Pas. Wat force is required to drag a very tin plate of 0.5 m 2 area between te surfaces at a speed of v=0.15 m/s. Assume linear velocity profile. a) If te plate remains equidistant from te two surfaces? b) If it is at a distance of 10 mm from one of te surfaces. Upper stationary plate F, V Lower stationary plate 1.E02
Example 1.3 Wen a torque T is applied to te saft, te disk A rotates wit a constant angular velocity Te fluid in between transmits tis torque T to te disk B. Wat will be te angular velocity 2 for te disk B? ω 1 Disk A ω 2 Disk B R 0 1.E03
Example 1.4 Two capillary tubes of different diameter are submerged into water as seen in te figure. Find te elevation difference of water between te two tubes. D 1 σ θ x D 2 σ θ 1 2 1.E04
HOMEWORK PROBLEMS 1.1 If F=QU/g, were Q is discarge, is specific weigt, U is velocity and g is te gravitational acceleration, wat are te dimensions of F? 1.2 An expression for te volume rate of flow, Q flowing over a dam of lengt, B, is given by te equation Q=3.09 BH 3/2 were H is te dept of te water above te top of te dam (called as ead). Tis formula gives Q in ft 3 /s wen B and H are in feet. Is te constant, 3.09, dimensionless? Would tis equation be valid if units oter tan feet and seconds were used? 1.3 A liquid wen poured into a graduated cylinder is found to weig 6 N wen occupying a volume of 500 ml (milliliters). Determine its specific weigt, density and specific gravity. 1.4 A gas is compressed. Te measured volume and absolute pressure before compression are 0.30 m 3 and 50.7 kpa, respectively. After compression te volume and te pressure becomes 0.111 m 3 and 202.8 kpa, respectively. Wat is te compressibility and bulk modulus of elasticity of tis gas? 1.5 Develop an expression for te pressure variation in a liquid in wic te specific weigt increases wit dept,, as =K+o, were K is constant, o is te specific weigt at te free surface. 1.6 An 8-kg flat block of metal slides down a = 20 inclined plane wile lubricated by a tin film of oil. Te contact area, A, is 0.2 m 2. Wat is te terminal velocity of te block? oil=0.29 Pa.s, t=2 mm. Contact area, A t 1.H01
1.7 Calculate te sear stress for y= 0, 3 and 6mm. If te velocity profile of te flow in an open cannel is given as, y u UmaxSin ( ) 2 were u is in (m/s) and y in (mm). Draw te sear stress distribution. =1.8*10-5 kg/m.s, δ=6 mm, Umax=10 m/s. y δ U max u=u maxsin( ) 1.8 A triangular saft is pulled in a triangular bearing ousing (see figure) at a constant velocity of 0.3 m/s. Find te force required to pull te saft, if te lengt of te saft is 2 m. Te viscosity of te lubricating oil filling te clearing between te saft and te ousing is =1x10-1 Ns/m 2. t1=t2=t3=1 mm, l =10 cm. t 2 60 t 1 60 Saft t 3 l oil 1.9 A 25 mm-diameter saft is pulled troug a cylindrical bearing as sown in te figure. Te lubricant tat fills te 0.3 mm gap between te saft and bearing is an oil aving a kinematic viscosity of 8x10-4 m 2 /s and a specific gravity of 0.91. Determine te force P required to pull te saft at a velocity of 3 m/s. Assume te velocity distribution in te gap is linear. L=0.5 m. Bearing Saft Lubricant P L 1.H02
1.10 A torque of T=4 Nm is required to rotate te intermediate cylinder at =30 rad/min. Calculate te viscosity of te oil. All cylinders are 450 mm long. Neglect te end effects. R=0.15 m, t=0.003 m. R t t 1.11. Te device sown consists of a disk tat is rotated by a saft. Te disk is positioned very close to a solid boundary. Between te disk and boundary tere is viscous oil. a) If te disk is rotated at a rate of 1 rad/s, wat will be te ratio of te sear stress in te oil at r=2 cm to te sear stress at r =3 cm? b) If te rate of rotation is 2 rad/s, wat is te tangential velocity of te oil in contact wit te disk at r=3 cm? c) If te oil viscosity is 0.01 N.s/m 2 and te spacing y is 2 mm, wat is te sear stress for te conditions noted in (b)? Disk r Oil y D 1.12. A conical body is made to rotate at a constant speed of =42 rad/sec. A film of oil aving a viscosity of 0.5 poise (gr/cm.s) separates te cone from te container. Te film tickness, t, is 0.025 cm. Wat torque is required to maintain tis motion? Te cone radius at te base, R, is 10 cm and cone as a lengt of =30 cm. t t R t oil 1.H03
1.13. Compute te torque T required to rotate a conical object at a constant angular speed. Te clearance between te object and te casing is constant in tickness () and filled wit oil of. ( =30), D oil 1.14. Small droplets of carbon tetracloride at 68F are formed wit spray nozzle. If te average diameter of te droplets is 200 m wat is te difference in pressure between te inside and outside of te droplets? (=2.69x10-2 N/m for carbon tetracloride at 68F) 1.15 Water is filled between two parallel plates of infinite lengt, a distance d apart. Find te capillary rise between tese two plates, were surface tension angle of contact ddistance between plates unit weigt of water : capillary rise d d W 1.H04