Chapter1 - 57:020 Fluid Mechanics Class Notes Fall 2009...

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57:020 Fluid Mechanics Class Notes Fall 2009 Prepared by: Professor Fred Stern Typed by: Stephanie Schrader (Fall 1999) Corrected by: Jun Shao (Fall 2003, Fall 2005) Corrected by: Jun Shao, Tao Xing (Fall 2006) Corrected by: Hyunse Yoon (Fall 2007 Fall 2009)
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57:020 Fluid Mechanics Chapter 1 Professor Fred Stern Fall 2009 1 CHAPTER 1: INTRODUCTION AND BA- SIC CONCEPTS Fluids and the no-slip condition Fluid mechanics is the science and technology of flu- ids either at rest (fluid statics) or in motion (fluid dynamics) and their effects on boundaries such as solid surfaces or in- terfaces with other fluids. Definition of a fluid: A substance that deforms continuous- ly when subjected to a shear stress Consider a fluid between two parallel plates, which is subjected to a shear stress due to the impulsive motion of the upper plate No slip condition: no relative motion between fluid and boundary, i.e., fluid in contact with lower plate is stationary, whereas fluid in contact with upper plate moves at speed U . Fluid deforms, i.e., un- dergoes rate of strain θ & due to shear stress τ Fluid Element & u=U u =0 t =0 t= Δ t
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57:020 Fluid Mechanics Chapter 1 Professor Fred Stern Fall 2009 2 τ τ Solid γ t =0 t= Δ t Newtonian fluid: strain of rate = θ & τ θ μ τ & = μ = coefficient of viscosity Such behavior is different from solids, which resist shear by static deformation (up to elastic limit of material) Elastic solid: γ = strain = G γ G = shear modulus Both liquids and gases behave as fluids Liquids: Closely spaced molecules with large intermolecular forces Retain volume and take shape of container Gases: Widely spaced molecules with small intermolecular forces Take volume and shape of container gas container liquid
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57:020 Fluid Mechanics Chapter 1 Professor Fred Stern Fall 2009 3 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). Liquids, gases, and two-phase liquid-vapor behave as flu- ids.
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57:020 Fluid Mechanics Chapter 1 Professor Fred Stern Fall 2009 4 Continuum Hypothesis In this course, the assumption is made that the fluid be- haves 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 ρ of a fluid () V V V lim , * δ ρ m t x = δ V * = limiting volume below which molecular variations may be important and above which macroscopic variations may be important δ V * 10 -9 mm 3 for all liquids and for gases at atmospheric pressure 10 -9 mm 3 air (at standard conditions, 20 ° C and 1 atm) con- tains 3x10 7 molecules such that δ M/ δ V = constant = ρ Note that typical “smallest” measurement volumes are about 10 -3 – 10 0 mm 3 >> δ V * and that the “scale” of ma- croscopic variations are very problem dependent x = position vector xyz =++ ij k t = time
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57:020 Fluid Mechanics Chapter 1 Professor Fred Stern Fall 2009 5 Exception: rarefied gas flow
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Chapter1 - 57:020 Fluid Mechanics Class Notes Fall 2009...

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