北京航空航天大学：《工程流体力学》课程PPT教学课件（双语版）Chapter 4 Differential Relations For Viscous Flow

4.4 Differential Equation of linear momentum Newton’ s second law F=ma=pdxdydz Rodxdvdz Elemental volume =(Xi+yj+Zk)pdxdydz What are the surface forces Fs on the elemental volume?

11 Newton’s second law dV dxdydz dt =   b s dV dxdydz F F dt   = + F R dxdydz b =  = + + ( ) Xi Yj Zk dxdydz  4.4 Differential Equation of Linear Momentum dx dz dy Elemental volume F ma = What are the surface forces Fs on the elemental volume?

Surface force on an elemental volume: for the two surfaces⊥x aP Prydz OX P Surface stress P aP d (P+-dx)dydz dx ax Vector sum aP aP aP for the two surfaces⊥y, ydxdvdz z ap aP aP Net Surface force: F=(x++一) dxdy ay a 12

12 Surface force on an elemental volume: for the two surfaces x ⊥ P dydz x ( ) x x P P dx dydz x  +  dx dz dy P x x x P P dx x  +  Vector Sum P x dxdydz x   P x Surface stress for the two surfaces y z , ⊥ P y dxdydz y   P z dxdydz z   Net Surface Force: ( ) x y z s P P P F dxdydz x y z    = + +   

It is not these stresses but their gradient, which cause a net force on the differential volume ap ap. aP p-dxdydz=pRdxdydz+(x++)dxdydz dt OX aP R+ Momentum equation dt x=Xxi+TJ+tk In the like manner P=τ,i+rj+τ,k P=t i+t,j+tk 13

13 ( ) x y z dV P P P dxdydz Rdxdydz dxdydz dt x y z      = + + +    1 [ ]i i dV P R dt x   = +  Momentum equation P i j k x xx xy xz = + +    P i j k y yx yy yz = + +    P i j k z zx zy zz = + +    In the like manner xx  xy  xz  P x It is not these stresses but their gradient, which cause a net force on the differential volume

XXx Tensor张量 T ≠ 6 0r0 F=C++)+( ax a atat at ax ay a au 7,0r =X+-( ax ay az dt

14 [( ) ( ) xx zx yx xy yy zy F i j s x y z x y z             = + + + + +       ( ) ] xz yz zz k dxdydz x y z    + + +    ij ji   = i j  xx xy xz yx yy yz zx zy zz                    ij      6 Tensor 张量 1 ( ) du xx zx yx X dt x y z        = + + +    ... ... dv dw dt dt = =

l aC =X1+-() Momentum equation(角标表示法) Constitutive Relation本构 Newton'sLaw(广义牛顿内摩擦定律) u(vv)si-P8 ou + 2 ax. a 0 i≠J 15

15 Constitutive Relation 本构 ~ i ij j u x                w du dy   = Newton’s Law (广义牛顿内摩擦定律) 2 2 ( ) 3      ij ij ij ij = −  − S V P 1 ( ) 2 j i ij j i u u S x x   = +   ij  = 1 0 i j  i j = 1 ( ) ji i i j du X dt x    = +  Momentum equation(角标表示法) xy u y    = 