流体力学+第一讲

1.jiong20002.Z.U.AWarsi,FluidMechanics:TheoreticalandComputationalApproaches,19933.YihC.S.FluidMechanics158511.2.3.1234ENSOELNiNo512341t2(!)~V34P5~roxyz;orz;or21p48:13578(3-5,10-12)101-513x1.11'='(~r;t)='(x;y;z;t)~a=~a(~r;t)=~a(x;y;z;t)~r=(x;y;z)2t'(~r);~a(~r)=)@@t=0@@t,03'(t);~a(t)=)@@xi=0;i=1;2;3@@xi,0;i=1;2;3x1.21t0'(~r;t0)='0Figure1:32~ad~r~ad~r=~i~j~kdxdydzaxayaz=0dxax(x;y;z;t)=dyay(x;y;z;t)=dzaz(x;y;z;t)H1dq1a1(q1;q2;q3;t)=H2dq2a2(q1;q2;q3;t)=H3dq3a3(q1;q2;q3;t)tx1.3t0'(~r;t0)MM'(M;t0)=c~n~sMP~s@'@s=limMP!0'(P)'(M)MP*P'(Q)='(P);MQ=MPcos(~n;~s))@'@s=limMQ!0'(Q)'(M)QMcos(~n;~s)=@'@ncos(~n;~s)'grad'=@'@n~n@'@s=~s
grad';~s@'@s@'@n'~n'~n'grad'grad'=@'@x~i+@'@y~j+@'@z~k=r'1.M'2.grad''~n@'@n43.grad'~s1grad'd'=d~r
grad'd'=d~r
~a~a=grad'grad'=@'@x~i+@'@y~j+@'@z~k;d~r=dx~i+dy~j+dz~k'd~r
grad=@'@xdx+@'@ydy+@'@zdz=d'd'=d~r
~ad'=d~r
grad'd~r
(~agrad')=0d~r~a=grad'2~a=grad''~rLIL~a
d~r=0~aLIL~a
d~r0~a'~a=grad'~a=grad'1IL~a
d~r=ILgrad'
d~r=ILd''~rLILd'=0IL~a
d~r=0~aLIL~a
d~r=0M0M(~r)L1L2ZMM0(L1)~a
d~r+ZM0M(L2)~a
d~r=0ZMM0(L1)~a
d~r=ZMM0(L2)~a
d~r5M0M(~r)~r'(~r)'(~r)=ZMM0~a
d~rd'=~a
d~r1:~a=grad''~rgrad'1'r=~n=~rr;@'@n=d'dr;grad'=d'dr~rr2oxyz@'@x=@'@r@r@x=xrd'dr;


31d'='0(r)dr=d~r
'0(r)~rr!x1.4SdS~nd~S=~ndSdScos(~n;~i)=dydz;dScos(~n;~j)=dxdz;dScos(~n;~k)=dxdyan=~a
~n=axcos(~n;~i)+aycos(~n;~j)+azcos(~n;~k)SZSandS=ZS~a
~ndS=ZS~a
d~S=ZSaxcos(~n;~i)+aycos(~n;~j)+azcos(~n;~k)dS=ZSaxdydz+aydxdz+azdxdy~a(~r;t)MVSoxyzISandS=ZV@ax@x+@ay@y+@az@zdVISandS=@ax@x+@ay@y+@az@zQ
V;QVV!0QMM6div~a=limV!0ISandSV=limV!0@ax@x+@ay@y+@az@zQ=@ax@x+@ay@y+@az@zISandS=ZVdiv~adVIS~n
~adS=ZVr
~adVIS~n~adS=ZVr~adVIS~n'dS=ZVr'dV~b=(';0;0);div~b=@'@xIS~b
d~S=IS'cos(~n;~i)dS=ZVdiv~bdV=ZV@'@xdV;div~a=01.~aISandS=ZVdiv~adV=0P;P1P0S)ZPandS+ZP0andS+ZP1andS=0P0an=0ZPandS=ZP1andSFigure2:2.73.~aLSLSx1.5:~a(~r;t)LZL~a
d~r=ZLaxdx+aydy+azdzMLMLS~n;~LIL~a
d~r=ILaxdx+aydy+azdz=ISh@az@y@ay@z
cos(~n;~i)+@ax@z@az@x
cos(~n;~j)+@ay@x@ax@y
cos(~n;~k)idS=S
h@az@y@ay@z
cos(~n;~i)+@ax@z@az@x
cos(~n;~j)+@ay@x@ax@y
cos(~n;~k)iQ~arot~a~nrotn~a=limS!0IL~a
d~rS=@az@y@ay@z
cos(~n;~i)+@ax@z@az@x
cos(~n;~j)+@ay@x@ax@y
cos(~n;~k)rot~a=~i~j~k@@x@@y@@zaxayaz:IL~a
d~r=ZSrotn~adS=ZSrot~a
d~Srot~a=0~a~a=grad'~arot~a=0rot~a=0~a=grad'8x1.6p191.grad('+)=grad'+grad2.grad(')='grad+grad'3.gradF(')=F0(')grad'4.div(~a+~b)=div~a+div~b5.div('~a)='div~a+grad'
~a6.div(~a~b)=~b
rot~a~a
rot~b7.rot(~a+~b)=rot~a+rot~b8.rot('~a)='rot~a+grad'~a9.rot(~a~b)=(~b
r)~a(~a
r)~b+~adiv~b~bdiv~a10.grad(~a
~b)=(~b
r)~a+(~a
r)~b+~brot~a+~arot~b11.grada22=(~a
r)~a+~arot~a12.divgrad'=4'13.divrot~a=014.rotgrad'=015.rotrot~a=graddiv~a4~a16.div('grad)='4+grad'
grad17.4(')=4'+'4+2grad'
grad1.ZVgrad'dV=ZS'~ndS2.ZVdiv~adV=ZS~n
~adS3.ZVrot~adV=ZS~n~adS4.ZV(~V
r~a)dV=ZS(~V
~n)~adS~V5.ZV4'dV=ZS@'@ndS=ZS~n
r'dS96.ZV4~adV=ZS@~a@ndS=ZS(~n
r)~adS7.ZV('4+grad'
grad)dV=ZS'@@ndSZV(4'+grad
grad')dV=ZS@'@ndS8.ZV('44')dV=ZS'@@n@'@ndS9.ZV(grad')2dV=ZS'@'@ndS'4'=04=@2@x2+@2@y2+@2@z210