Reading 1: Review of basic principles of fluid mechanics13.42 Spring 2004c°A.H. Techet & M.S. Triantafyllou1. Description of Fluid Motion1.1. Lagrangian Description. In rigid body mechanics the motion of a body is described interms of the body’s position in time. This body can be translating and possibly rotating, but cannot deform. This description, following a particle in time, is a Lagrangian description.1.2. Eulerian Description. A fluid is a continuum of particles, and unlike rigid bodies, parcels offluid tend to deform continuously as they move. Thus it is useful to use the Eulerian description, orcontrol volume approach, and describe the flow at every fixed point in space (x, y, z) as a function oftime, t. In general this is less costly than tracking each individual fluid particle as in a Lagrangianapproach.2. Governing LawsThe governing laws of fluid motion can be derived in multiple forms (e.g. differential or integralform) using a simple control volume approach. The control volume (CV) is equivalent to a “fluidicblack box” where all we know is what is going in and coming out of the volume (mass, momentum,energy, work, etc). The CV can be fixed or move with the fluid. For simplicity it is often idealto fix the CV. In an ideal situation you can pick the control volume that makes your life easiermathematically.When analyzing a control volume problem there are three laws that MUST be followed:(1) Conservation of Mass(2) Conservation of Momentum(3) Conservation of Energy2.1. Conservation of Mass: For an incompressible fluid, with constant density, the total massentering a control volume must equal the total mass exiting the control volume. We can write a12mass conservation equation, in 2-D, balancing the fluid entering and exiting an infinitesimal controlvolume, δxδz (figure 1).δzuu+∂u∂xδxu+∂u∂xδx∂u∂zδz+∂u∂zδzu+δx Figure 1. Infinitesimal Control Volume, δxδz.In the x-direction we have,(2.1) ρ{u +12∂u∂zδz} δz − ρ{u +∂u∂xδx +12∂u∂zδz} δz = −ρ∂u∂xδxδz.The first term, on the LHS of equation ( 2.1), represents the mass entering the control volume andthe second term represents the fluid exiting the control volume. The RHS is the net mass inflowinto the CV. Similarly, the net vertical fluid flux in the z-direction can be written as(2.2) −ρ∂w∂zδxδz.For a fluid with constant density, net mass flux must be zero in order for mass to be conserved.(2.3) −ρ(∂u∂x+∂w∂z)δxδz = 0.3This equation is valid for all δx and δz and can be simplified to arrive at the two-dimensionalequation for conservation of mass:(2.4)∂u∂x+∂w∂z= 0.Similarly in three-dimensions, the equation for mass conservation can be written as:(2.5)∂u∂x+∂v∂y+∂w∂z= 0.Recalling from vector calculus, ∇ = (∂∂x,∂∂y,∂∂z), we can abbreviate equation 2.5 as(2.6) ∇ · V = 0.2.2. Newton’s Second Law: Newton’s second law is simply the law of conservation of momentum.It states that the time rate of change of momentum of a particle is equal to the sum of external forcesacting on that particle.tt+δtδxδzδxp=u δtδzp=w δtFigure 2. Motion of a fluid particle, δxδz, in time δt.In an infinitesimal time interval, δt, a particle of size δxδz moves a distance (δxp, δzp) = (uδt, wδt),where u and w are the horizontal and vertical velocities of the particle. The sum of the externalforces, F , acting on the particle equal the time rate of change of momentum of the particle as follows,4(2.7) ΣF =ddt{MV}pwhere M = ρδxδz is the mass of the fluid particle (per unit length) and MV is the linearmomentum (V is the velocity vector). Since the fluid density is constant, the time rate-of-change oflinear momentum can be written as(2.8)ddt{MV}p= ρ δxδzdVdtp.The time rate of change of velocity of the fluid particle is found, for small δt, as(2.9)dVdtp=limδt→01δt{V(x + δxp, z + δzp, t + δtp) − V(x, z, t)}.We can substituteδxp= u δt and δzp= w δtinto equation 2.9 and cancelling terms we arrive at(2.10)dVdtp=limδt→01δt{»»»»»V(x, z, t) +∂V∂xu · δt +∂V∂zw · δt +∂V∂tδt −»»»»»V(x, z, t)}.The RHS of equation (2.10) is the familiar form of total derivative of the velocity for a particle,(2.11)DVDt=∂V∂t+∂V∂xu +∂V∂zw,which can be simplified, using the vector identity,(2.12) V · ∇ = u∂∂x+ v∂∂y+ w∂∂z.5The total (material) derivative of the velocity is the sum of the acceleration,∂V∂t, and the convectiveterm, (V · ∇)V:(2.13)DVDt=∂V∂t+ (V · ∇)V.Finally, the momentum equation from 2.7 can be rewritten as(2.14) ΣF = ρDVDtδxδz2.3. External Body and Surface Forces. The LHS of equation 2.7 is the sum of the forces actingon the control volume. Contributions from gravity and pressure both play a role in this term as wellas any applied external forces.2.3.1. Gravity: Body force acting on a fluid volume due to gravity:(2.15) Fg= −(ρg δxδz)ˆk2.3.2. Pressure: Force acting normal to the surface of a fluid particle, or volume, due to pressure isFP= P · A.Pressure force in x-direction:(2.16) FP x= {(p +12∂p∂zδz)δz − (p +12∂p∂zδz +∂p∂xδx)δz = −∂p∂xδxδzPressure force in z-direction:(2.17) FP z= −∂p∂zδxδz6Total pressure force in two dimensions:(2.18) FP= −(∂p∂x,∂p∂z)δxδz = −∇p δxδz.2.3.3. Viscosity: Viscous stresses act tangentially on the surface of a fluid particle, or volume. Unlessotherwise specified, viscous forces will be ignored herein and the fluid will be considered to be inviscid.3. Irrotational flowVorticity of a fluid is defined as the curl of the flow velocity field. For flow to be consideredirrotational, the curl of the velocity must be zero,(3.1)−→ω = ∇ × V =ˆiˆjˆk∂∂x∂∂y∂∂zu v wsuch that(3.2) ω =ˆi(∂w∂y−∂v∂z) +ˆj(∂u∂z−∂w∂x) +ˆk(∂v∂x−∂u∂y) = 0.For two-dimensional flows this reduces to(3.3) ωy=∂u∂z−∂w∂x= 0.It is interesting to note here discuss in further detail the differences between the motion of asimple rigid body and that of a fluid body. We can classify types of motion as (1) pure translation,(2) pure strain, (3) angular deformation, (4) pure rotation.7tt+δtV = (u,v)V = (u,v)tt+δtV = (-u,0)V = (0,v)V = (0,0)V = (u1,v1)tt+δtV = (0,0)V = (u2,v2)δz = (dw/dx) δtδx = (du/dy) δt{{δzδxtt+δtV = (u,v)V = (-u,0)θV = (-u,0)a. b. c. d.Figure 3.