2-323 u(vv)s-PS 1(V·V)-P x -2m(V.)-P au o x=1( (V)-P en Newton fluid, linear fluid(牛顿流体,线性流体) Substitute newton's constitutive relation into me 16
16 2 2 ( ) 3 xx u V P x = − − 2 2 ( ) 3 yy v V P y = − − 2 2 ( ) 3 zz w V P z = − − ( ) xy u v y x = + ( ) xz u w z x = + ( ) yz w v y z = + Substitute Newton’s Constitutive Relation into ME 2 2 ( ) 3 ij ij ij ij = − − S V P Newton fluid, linear fluid (牛顿流体，线性流体）
1 aP +2+yx(V.V) 5 Ox Y B rrt3r(vv) N-S Equation I aP 2+y(V) P az dv R--VP+2+(V·7) a2a2 a ax Ov az 17
17 1 1 2 ( ) 3 du P X u V dt x x = − + + 1 1 2 ( ) 3 dv P Y v V dt y y = − + + 1 1 2 ( ) 3 dw P Z w V dt z z = − + + 1 1 2 ( ) 3 dV R P V V dt = − + + 2 2 2 2 2 2 2 x y z = + + N-S Equation
For incompressible flow (V·V)=0 For inviscid flow yV2+v(V7)=0 For 2-D, steady, incompressible flow I aP 1 +y( OX OX I aP l-+1 Y + ay 0 18
18 For incompressible flow 1 ( ) 0 3 = V For inviscid flow 2 1 [ ( )] 0 3 + = V V For 2-D,steady,incompressible flow 2 2 2 2 1 ( ) u P u u u u v X x y x x y + = − + + 2 2 2 2 1 ( ) v P v v v u v Y x y y x y + = − + + 0 u v x y + =
F OX F=[()+()j+()]hxvo dz ax Ti =2uSi-ou(vv-Po dv R-VP+少2+(V·V) dt 3
19 F ma = 1 ( ) ji i i j du X dt x = + [( ) ( ) ( ) ] ix iz iy s i i i F i j k dxdydz x x x = + + 1 1 2 ( ) 3 dV R P V V dt = − + + 2 2 ( ) 3 ij ij ij ij = − − S V P
4.5 The Differential Equation of Energy Infinitesimal fluid element The first thermodynamic law de de 2-w O dz dx +u L+1+w u+ 2 20
20 4.5 The Differential Equation of Energy Infinitesimal fluid element dx dz The first thermodynamic law dy . . dE de Q W dxdydz dt dt − = = 2 2 V e u = + 2 2 2 2 u v w u + + = +