# Fluid mechanics spring 2021 chapter 2 u lei 李雨

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Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei (李雨)Specific heat ratio as function of temperature(from White, “viscous fluid flows,” 1991)pvCCγ=(from White’s book)gases
Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei (李雨)Non-Newtonian fluid21pseudoplastic1newtonian ()1dilatantnxyxyCnnCnτεμ<==>(from White’s book)(nonlinear)(for simple shear flow)Equations on page 75need to be modified(different constitutive equations)
Derivation of the Continuity using control volume approach (1)xΔyΔzΔ()2uxuyzxρρΔ+Δ Δ()2wzwx yzρρΔ+Δ Δ()2vyvx zyρρΔ+Δ Δ()2uxuyzxρρΔΔ Δ()2vyvx zyρρΔΔ Δ()2wzwx yzρρΔΔ ΔVelocity at the center of the cube: (, ,)u v w(Fluxes at the surfaces of an infinitesimal cubic control volume)(Control volume: a fixed volumein space, mass may cross thesurface of the control volume).
Derivation of the Continuity using controlvolume approach (2)()()()()()2222Mass Balance: Rate of mass increase = net influx of massuuxxx y zuy zuy ztxxvvyyvx zvx zyywwρρρρρρρρρρρΔΔΔ Δ Δ =Δ Δ −+Δ ΔΔΔ+Δ Δ −+Δ Δ+()()()()()()()2200wzzx ywx yzzuvwx y zx y zx y zxyzuvwtxyztρρρρρρρρρρρΔΔΔ Δ+Δ Δ= −Δ Δ Δ −Δ Δ Δ −Δ Δ Δ+++=+ ∇ ⋅=u(similar approach for derivingmomentum and energy equations)
The following 8 pages are for the detailedequations written in Cartesian, cylindricaland spherical coordinates, which are adoptedfrom the appendix of the book by R. L.Panton, “Incompressible flow,” John Wileyand Sons, 1984.Electronic version of Panton’s book (2013)is available at NTU library.
(in the lecture notes)ijijSD=
Stokes hypothesishas been assumed2In general, replacing3in the normal stress components.μλ∇ ⋅=∇ ⋅vv
Incompressible Newtonian fluid
Incompressible Newtonian fluid
Some useful vector identities(from the appendix of the book “Fundamental mechanicsof fluids,” by I. G. Currie, McGraw-Hill, 1974._____________________________________________________________________
Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei (李雨)Boundary and Initial ConditionsSince there exist spatial derivative terms in thegoverning equations, it is required to specify certainvalues of the flow quantities on the boundaries of thefluid domain of interest.Boundary conditionsIf the flow is not steady, then it is necessary to specifythe flow quantities having time derivatives at a specified time.Initial conditions
Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei (李雨)Boundary conditions (1)(for continuum fluid mechanics)Experimentally, it is found that the fluid adheres to the boundary surface.B.C. for Navier-Stokes equationAt solid boundaries,Buu=(no-slip boundary condition)nunu=B(slip boundary condition)Inviscid fluids, including perfect and ideal fluid, slip on the wall.

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