流体力学+第三讲

2:12451219H~Pn
~VdS=Rr
(P
~V)dV;~Pn=~n
Px1.10ox1x2x3ox01x02x03ijij=cos(~e0i;~ej)=~e0i
~ejox1x2x3;~ej;j=1;2;3ox01x02x03;~e0i;i=1;2;3~e0i=ij~ej;~ei=ji~e0jO~ej~e0iij~e0iij~e0i=ij~ej=cos(~e0i;~e1)~e1+cos(~e0i;~e2)~e2+cos(~e0i;~e3)~e3(1)~ei~e0j~ei=ji~e0j;~ej=kj~e0k(2)(2)(1)(ki+ijkj)~e0k=0)ijkj=ik;jijk=ik~e0i~ej;j=1;2;3()ij=~e0i
~ej=cos'ji=~e0j
~ei~e0i=ij~ej=i1~e1+i2~e2+i3~e3~e0i
~e0k=ij~ej
km~em=ijkmjm=ijkj=ik~ei
~ek=ji~e0j
mk~e0m=jimkjm=jijk=ikijik=jk;jiki=jk1'(x01;x02;x03;t)='(x1;x2;x3;t)a0i=ijaja1;a2;a3~aox1x2x3~a=ai~eiox01x02x03~a=a0j~e0j2~a=ai~ei=aiji~e0j)a0j=jiai0BBBBBBBBBBB@1112132122233132331CCCCCCCCCCCAai=jia0jjiNnox1x2x332PmnP0ij=imjnPmn32PmnP=Pmn=0BBBBBBBBBBB@P11P12P13P21P22P23P31P32P331CCCCCCCCCCCAnox1x2x33nPj1j2


jnox01x02x033nP0i1i2


in=i1j1i2j2


injnPj1j2


jn23nnn01P0=P131P0i=ijPj232P0ij=ikjmPkm333P0ij`=ikjm`nPkmn:::::::::n3nP0j1j2


jn=j1k1j2k2


jnknPk1k2


knijijij=0ij*ikjk=ij)0ij=ikj`k`ij21P=Pj1j2


jnQ=Qj1j2


jnnPQTT=Tj1j2


jn=Pj1j2


jnQj1j2


jnnPQP=QPQP=QPQ=0P0Q0=00)P0=Q0PQ2P=Pi1i2


immQ=Qj1j2


jnnTi1i2


imj1j2


jn=Pi1i2


imQj1j2


jn3Tm+nT=PQP''Qn~a~baibj~a~b=aibjnPP=Pj1j2


jn~ej1~ej2


~ejn3P=Pj1j2


jnn2Qn2P0i1i2


in2kk=i1j1i2j2


in2jn2krksPj1j2


jn2rs=i1j1i2j2


in2jn2rsPj1j2


jn2rs=i1j1i2j2


in2jn2Pj1j2


jn2rrQ=Pi1i2


in2kkn2P4PQmPnQm+n2PQP
QP
~a=Pijaj~a
P=aiPijP
~a,~a
PP;QP
Q=PikQkjP;QP:Q=PijQjix**P=Pi1i2


imj1j2


jnnQ=Qj1j2:::jnPi1i2


imj1j2


jnQj1j2:::jnmPm+nmTP0i1i2


imj1j2


jnQ0j1j2:::jn=T0i1i2


im=i1k1i2k2


imkmTk1k2


km=3i1k1i2k2


imkmPk1k2


kmr1r2


rnQr1r2


rn=Pk1k2


kmr1r2


rni1k1i2k2


imkmj1r1j2r2


jnrnQ0j1j2


jn(P0i1i2


imj1j2


jni1k1i2k2


imkmj1r1j2r2


jnrnPk1k2


kmr1r2


rn)Q0j1j2:::jn=0Q0nP0i1i2


imj1j2


jn=i1k1i2k2


imkmj1r1j2r2


jnrnPk1k2


kmr1r2


rn4Pm+nPi1i2


imnQPi1i2


imQj1j2


jn=Ti1i2


imj1j2


jnm+nPi1i2


immx1.111.P=PijPc=PjiPPij=PjiPPij=PjiP2.P=A+SA=12(PPc)S=12(P+Pc)!1;!2;!3A=aij=0BBBBBBBBBBB@0a12a31a120a23a31a2301CCCCCCCCCCCA=0BBBBBBBBBBB@0!3!2!30!1!2!101CCCCCCCCCCCAaij=ijk!k1.Aa0ij=imjkamka0ji=jkimakm=imjkamk=a0ij2.!1;!2;!3~!3.A~bA
~b=~!~b~b
A=~b~!A~!=(!1;!2;!3)A
~b=aijbj=ijkbj!k=~!~bx1.1251.nP=Pi1i2


inrPrP=gradP=@@xkPi1i2


inpi1i2


inkn+1r0P0=@@x0kP0i1i2


in=@x`@x0k
(i1k1i2k2


inkn@@x`Pk1k2


kn)=i1k1i2k2


inknk`@@x`Pk1k2


kn2.nP=Pi1i2


inr
Pr
P=divP=@@xkPki2


inrPn1~adiv~a=r
~a=@ai@xi~ar~a=@[e-mail]~n
PdS=ZVdivPdV(1)ISnkPkj2


jndS=ZV@@xkPkj2


jndV(1)1P'r('P)='r
P+r'
Pr('P)=@@xk('Pkj)=@'@xkPkj+'@Pkj@xk='r
P+r'
P2div(~a~b)=~b(r
~a)+(~a
r)~bdiv(~a~b)=@@xk(akbj)=@ak@xkbj+ak@bj@xk=~b(r
~a)+(~a
r)~b3r
(~a
r~a)=r~a:r~a+~a
r(r
~a)r
(~a
r~a)=@@xjai@aj@xi!=ai@2aj@xj@xi+@ai@xj@aj@xi=r~a:r~a+~a
r(r
~a)6