058 160 Intermediate Mechanics of Fluids Class Notes Fall 2006 Prepared by Professor Fred Stern Typed by Derek Schnabel Fall 2004 Nobuaki Sakamoto Fall 2006 Hamid Sadat Hosseini Fall 2006 Maysam Mousaviraad Fall 2006 Corrected by Jun Shao Fall 2004 Mani Kandasamy Fall 2005 Tao Xing Hyun Se Yoon Fall 2006 058 0160 Professor Fred Stern Chapter 1 1 Fall 2006 Chapter 1 Introduction Definition of a fluid A fluid cannot resist any applied shear stress and remain at rest whereas a non fluid i e solid can Solids resist shear by static deformation up to an elastic limit of the material after which they undergo fracture Fluids deform continuously undergo motion when subjected to shear stress Consider a fluid between two parallel plates with the lower one fixed and the upper moving at speed U which is an example of Couette flow i e wall shear driven flows y V u y u U h u y x u 0 1 D flow velocity profile 058 0160 Professor Fred Stern Chapter 1 2 Fall 2006 No slip condition Length scale of molecular mean free path length scale of fluid motion therefore macroscopically there is no relative motion or temperature between the solid and fluid in contact Knudsen number Kn 1 Exceptions are rarefied gases and gas liquid contact line Newtonian fluids Rate of Strain u uy dy dt y y u uydy dy dy u d tan 1 uydt x x u dt Fluid element with sides parallel to the coordinate axes at time t 0 Fluid element deformation at time t dt 058 0160 Professor Fred Stern Chapter 1 3 Fall 2006 tan d d uy dt u y dydt dy du dy rate of strain velocity gradient For 3D flow rate of strain is a second order symmetric tensor ij 1 2 ui u j xi x j ji Diagonal terms are elongation contraction in x y z and off diagonal terms are shear in x y x z and y z Liquids vs Gases Liquids Gases Closely spaced with large Widely spaced with small intermolecular cohesive intermolecular cohesive forces forces Retain volume but take Take volume and shape of shape of container container 1 1 constant p T Where coefficient of compressibility change in volume density with external pressure 1 1 p p 058 0160 Professor Fred Stern Fall 2006 Chapter 1 4 p p 1 K Bulk modulus Recall p v T diagram from thermodynamics Single phase two phase triple point point at which solid liquid and vapor are all in equilibrium critical point maximum pressure at which liquid and vapor are both in equilibrium Liquid gases and two phase liquid vapor behave as fluids 058 0160 Professor Fred Stern Chapter 1 5 Fall 2006 Continuum Hypothesis Fluids are composed of molecules in constant motion and collosion However in most cases molecular motion can be disregarded and the assumption is made that the fluid behaves as a continuum i e the number of molecules within the smallest region of interest a point are sufficient that all fluid properties are point functions single valued at a point For example Consider definition of density of a fluid lim M x t V V V x position vector xi yj zk t time V limiting volume below which molecular variations may be important and above which macroscopic variations may be important V 10 9 mm3 for all liquids and for gases at atmospheric pressure 10 9 mm3 air at standard conditions 20 C and 1 atm contains 3x107 molecules such that M V constant 058 0160 Professor Fred Stern Fall 2006 Chapter 1 6 Exception rarefied gas flow Note that typical smallest measurement volumes are about 10 3 100 mm3 V and that the scale of macroscopic variations are very problem dependent A point in a fluid is equantionally used to define a fluid particle or infinitesimal material element used in defining the governing differential equations of fluid dynamics At a more advanced level the Knudsen number is used to equantify the separation of molecular and fluid motion length scales Kn l molecular length scale l fluid motion length scale 058 0160 Professor Fred Stern Chapter 1 7 Fall 2006 Molecular scales Air atmosphere conditions 6 10 8 m mean free path t 10 10 s time between collisions Smallest fluid motion scales l 0 1mm 10 4 m Vmax 100m s incompressible flow Ma 0 3 t 10 6 s 3 K 10 Thus n 1 and l scales larger than 3 order of magnitude scales An intermediate scale is used to define a fluid particle l l And continuum fluid properties are an average over V l 3 V 10 9 mm3 i e l 10 3 mm 10 6 m Previsouly given smallest fluid motion scales are rough estimates for incompressible flow Estimates are VERY conservative for laminar flow since for laminar flow l is usually taken as smallest characteristic length of the flow 058 0160 Professor Fred Stern Chapter 1 8 Fall 2006 domain and Vmax can not exceed Re restriction imposed by transition from laminar to turbulent flow For turbulent flow the smallest fluid motion scales are estimated by the Kolmogorov scales which define the length velocity and time scales at which viscos dissipation takes place i e at which turbulent kinetic energy is destroyed u kinematic viscosity dissipation rate 3 14 12 14 Which can also be written l0 Re 3 4 l0 L u u0 Re 1 4 Re0 u0l0 UL 0 Re 1 2 Which even for large Re of interest given l For example 100 watt mixer in 1 kg water 100 watt kg 100 m 2 s 3 10 6 m2 s for water 10 2 mm l 058 0160 Professor Fred Stern Fall 2006 Chapter 1 9 Fluid Properties 1 Kinematic linear V angular 2 velocity rate of strain ij vorticity and acceleration a 2 Transport viscosity thermal conductivity k and mass diffusivity D 3 Thermodynamic pressure p density temperature T internal energy enthalpy h p entropy s specific heat Cv Cp Cp Cv etc 4 Miscellaneous surface tension vapor pressure pv etc 1 Kinematic Properties Kinematics refers to the description of the flow pattern without consideration of forces and moments whereas dynamics refers to descriptions of F and M Lagrangian vs Eulerian description of velocity and accelaration 058 0160 Professor Fred Stern Fall 2006 Chapter 1 10 a Lagrangian approach focuses on tracking individual fixed particles b Eulerian approach focuses on fixed points in space u v w f x t are velocity components in x y z directions a d V x t V V x V y V z dt t x t y t z t V DV V V V u v w Dt t x y z 058 0160 Professor Fred Stern Fall 2006 Chapter 1 11 DV V j k V V gradient i x y z Dt t D V Dt t substantial material derivative DV Dt Lagrangian time rate of change of velocity V V V local convective acceleration in terms of t Eulerian derivatives D Dt derivative following motion of particle a a xi a y j …
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