fluid mechanics 2 (updated 2020-3-10).pdf - Chapter 2...

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Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei ( 李雨 ) Chapter 2 : Physical and Mathematical formulations of Fluid Mechanics Continuum fluid mechanics Conservation of mass, momentum (Newton’s 2 nd law) and energy (1 st law of thermodynamics) Use the Lagrangian description to derive the equations, and the Eulerian description to solve the problems Recall the Reynolds transport theorem, which expresses the time rate of change of properties within a volume with fixed mass
Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei ( 李雨 ) Reynolds Transport Theorem (three useful expressions as follows are employed) ( ) ( , ) ( ) V t d dF F F t dV F dV F dV dt dt t = + ∇ ⋅ = + ∇ ⋅    r u u V(t) V(t) ( ) ( ) V t S t F dV t F dS = +   n u (1-27) (Controlled mass system) (Lagrangian description) (Eulerian description) if V ( t ) is chosen as a fixed volume with naterials in/out
Divergence theorem for g = g ( x , y , z ) = any vector function = D D S V d d n g g D n D domain boundary
Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei ( 李雨 ) Conservation of mass - Continuity equation (1) The mass of material inside the material volume, V ( t ),  = ) ( ) , ( t V dV t m r ρ Conservation of mass implies that m = constant, 0. dm dt = Applying the Reynolds transport theorem, ( ) ( ) ( , ) 0 V t V t dm d d t dV dV dt dt dt ρ ρ ρ = = + ∇ ⋅ =   r u (Lagrangian frame)
Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei ( 李雨 ) Conservation of mass - Continuity equation (2) Since V ( t ) is arbitrary, it is required that 0, = + u ρ ρ dt d ( t ρ ρ + ∇ ⋅ = u) 0 or (Eulerian frame)
Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei ( 李雨 ) Another form of the Reynolds transport theorem Replacing F ( r , t) in the Reynolds transport theorem in (1-27) by ρ ( r , t) F ( r , t), ( ) ( ) ( ) ( ) ( ) ( ) V t V t V t d d F dF d FdV F dV F F dV dt dt dt dt ρ ρ ρ ρ ρ ρ = + ∇ ⋅ = + + ∇ ⋅    u u The sum of the last two terms in the integral is zero according to the continuity equation. Thus ( ) ( ) V t V t d dF FdV dV dt dt ρ ρ =  