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7/23/2019 De Motu Fluidorum from L. Euler
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AL= x Ll= y.
lm = u,
ln= v =
uu + vv =
u2 +v2
=v/u
du = Ldx+ ldy,
dv = Mdx+ mdy,
dL/dy= dl/dx
dM/dy = dm/dx dL/dy
dl/dx
/x
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dL/dy, dl/dx, dM/dy, dm/dx
L= du/dx, l= du/dy, M=dv/dx,
m= dv/dy;
du= Ldx + ldy dv= M dx+mdy,
=dx,
= dy,
=u + Ldx + ldy;
=v + Mdx + mdy. dt
=dt(u + Ldx + ldy)
=dt(v+ mdx+ mdy).
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lm = dx ln = dy;
x+dx x
y + dy.
APAL = udt,P p Ll = vdt.
AQ AM = (u+ Ldx)dt,QqMm = (v+ Mdx)dt.
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ARAL = (u + ldy)dt,Rr Ln = (v+ mdy)dt.
u = 0
= 12
dxdy,
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pqr = P prR + RrqQ PpqQ.
PprR = 1
2P R(P p + Rr),
RrqQ = 1
2RQ(Rr+ Qq),
PpqQ = 1
2P Q(P p + Qq).
pqr = 12
P Q Rr 12
RQ P p12
P R Qq.
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AQ = AP+ Q
AR = AP+ R
Qq = P p + q
Rr = P p + r
P Q= Q, P R= R, RQ= Q R.
pqr = 12
Q(P p+r) 12
(QR)P p 12
R(P p+q) pqr = 12
Qr 12
Rq.
Q = dx+ Ldxdt; q = 0 + Mdxdt;
R = 0 + ldydt; r = dy+ mdydt.
pqr = 12
dxdy(1 + Ldt)(1 + mdt)12
Mldxdydt2,
pqr = 12
dxdy(1 + Ldt + mdt + Lmdt2 Mldt2)
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= 12
dxdy,
Ldt + mdt +Lmdt2 Mldt2 = 0,
L + m +LmdtMldt = 0.
Lmdt Mldt
L
m,
L + m= 0.
du = Ldx + ldy,
dv = Mdx+ mdy
L+m = 0. L = du/dx m = dv/dy,
du/dx+ dv/dy = 0,
du/dx+dv/dy = 0.
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AL= x, Ll = y
l= z; l
lL
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.
o
= u, = v, o = w
,
x, y z.
du = Ldx+ ldy+ dz
dv = Mdx+ mdy+ dz
dw = Ndx+ ndy+ dz
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L, l, , M, m, , N, n,
x, y z
dL/dy= dl/dx; dL/dz= d/dx; dl/dz= d/dy
dM/dy= dm/dx; dM/dz= d/dx; dm/dz= d/dy
dN/dy= dn/dx; dN/dz= d/dx; dn/dz= d/dy
dt
udt vdt wdt
V,
V =
(uu + vv+ ww)
dt =V dt
dt.
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, , , o
o
x + dx x x
y y+ dy y
z z z+ dz
=lmn= 12dxdy,
o= dz, = 16
dxdydz
, , , o
dt.
u, v, w
o
u u + Ldx u+ ldy u + dz
v v+ Mdx v+ mdy v+ dz
w w+ Ndx w+ ndy w+ dz
, , , o
dt
, , ,
,,
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,
APAL = udt,
P p Ll = vdt,
p l = wdt;
AQAM = (u+ Ldx)dt,
QqMm = (v+ Mdx)dt,
q m = (w+ Ndx)dt,
ARAL = (u + ldy)dt,Rr Ln = (v+ mdy)dt,
r n = (w+ ndy)dt,
o
AS
AL = (u + dz)dt,
Ss Ll = (v+ dz)dt,
s lo = (w+ dz)dt.
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AP = x+ udt,
P p = y+ vdt,
p = z+ wdt;
AQ = x+ dx + (u+ Ldx)dt,
Qq = y+ (v+ Mdy)dt,
q = z+ (w+ Ndx)dt;
AR = x+ (u+ ldy)dt,
Rr = y+ dy+ (v+ mdy)dt,
r = z+ (w+ ndy)dt;
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AS = x + (u+ dz)dt,
Ss = y+ (v+ dz)dt,
s = z+ dz+ (w+ dz)dt.
, , , o
, , , dt
= 16
dxdydz
pqr pqr
pqr
pqr
u = 0
pqr
z
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pqs
prs
qrs
1
6dxdydz= pqrs+prs + qrspqr.
pqs = 1
3pqs(p+ q+ s),
prs = 1
3prs(p+ r + s),
qrs = 1
3qrs(q + r + s),
pqr = 1
3pqr(p+ q + r).
pqr = pqs+ prs+ qrs
1
6dxdydz= 1
3p.qrs1
3q.prs1
3r.pqs +
1
3s.pqr;
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dxdydz= 2pqr.s 2pqs.r 2prs.q 2qrs.p.
AQ= AP+ Q; Qq= P p + q; q= p+ ,
AR= AP+ R; Rr= P p + r; r= p+ ,
AS=AP+ S; Ss = P p + s; s= p+
p
dxdydz= 2pqr. 2pqs. 2prs.
pqr PpqQ
PprqQ PprR+ RrqQ
pqr =1
2P R(P p + Rr) +
1
2RQ(Rr+ Qq)1
2P Q(P p + Qq);
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P R= R, RQ= Q R, P Q= Q,
pqr = 12
R(P pQq) +12
Q(Rr P p) = 12
Qr 12
Rq.
pqs = 12
P S(P p + Ss) +1
2SQ(Ss + Qq)1
2P Q(P p + Qq),
pqs =
1
2
S(P p + Ss) +1
2
(Q
S)(Ss + Qq)
1
2
Q(P p + Qq)
pqs= 12
S(P pQq) + Q(Ss P p) =12
Qs12
Sq.
prs = 12
P R(P p + Rr) +1
2RS(Rr+ Ss)1
2P S(P p + Ss),
prs = 12
R(P p + Rr) +12
(SR)(Rr+ Ss)12
S(P p + Ss)
prs= 12
R(P p Ss) +12
S(Rr P p) =12
Sr 12
Rs.
dxdydz= (Qr Rq)+ (SqQs)+ (Rs Sr),
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1
6(Qr Rq)+1
6(SqQs) +1
6(Rs Sr).
Q= dx + Ldxdt q = Mdxdt = Ndxdt
R= ldydt r= dt + mdydt = ndydt
S=dzdt s= dzdt = dz+ dzdt.
Qr Rq = dxdy(1 + Ldt + mdt + LMdt2 MLdt2)
SqQs = dxdz(dt Ldt2 + Mdt2)
Rs Sr = dydz(dtmdt2 + ldt2),
1
6dxdydz
1 +
+Ldt +Lmdt2 +Lmdt3
+mdt Mldt2 Mldt3
+dt +Ldt2 Lndt3
+mdt2 +Mndt3
ndt2 Nmdt3
Ndt2 +Nldt3
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o= 16
dxdydz
dt
0 = L+ m + +dt(Lm + L+ mMl N n)
+dt2(Lm+ Mn+ Nl LnMlNl).
L + m +
= 0
u, v, w L= du/dx m= dv/dy =
dw/dz,
x, y, z u, v, w
du
dx+
dv
dy+
dw
dz = 0.
du/dx + dv/dy+dw/dz= 0
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AL = x, Ll = y
l =
du=Ldx+ ldy+ dz+ Ldt,
dv=Mdx+ mdy+ dz+ Mdt,
dw=Ndx+ ndy+ dz+Ndt.
L+ m + = 0,
dt = 0.
A, L, l,
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du
dx+
dv
dy+
dw
dz = 0.
dt
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AL = x, Ll = y;
AL AB
du=Ldx+ ldy+ Ldt,
dv=Mdx+ mdy+ Mdt;
L + m= 0.
= udt AL
= vdt AB;
dt, dx udt dy vdt,
du=Ludt + lvdt +
Ldt,
dv=Mudt + mvdt+Mdt,
L = ut, M =v/t.
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alongAL: 2(Lu+ lv+ L)
alongAB : 2(Mu+ mv+ M)
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dp= Rdx + rdy+Rdt, lm = no = dx
ln= mo= dy ; =dxdy.
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= p, = p+Rdx,
=p + rdy =p + Rdx + rdy.
=dx(p + 12
Rdx).
=dx(p + 12
Rdx + rdy).
=rdxdy. dy(p + 1
2rdy) dy(p + Rdx + 1
2rdy)
= Rdxdy.
=R, =r.
R.
R= 2Lu+ 2lv+ 2L
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R= 2Lu 2lv 2L,
r= 2Mu+ 2mv+ 2M
r= 2Mu 2mv 2M
dp= dx 2(Lu + lv+ L)dx 2(Mu+ mv+ M)dy+Rdt.
dx
Rdt,
d(Lu+ lv+ L)dy
=d(Mu+ mv+ M)
dx
du/dx= L, du/dy= l, dv/dx= M, dv/dy= m,
Ll+ udLdy
+ lm+ vdldy
+dLdy
=M L+ udMdx
+ mM+vdmdx
+dMdx
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(L+ m)(l M) + u
dL
dy dM
dx
+ v
dl
dydm
dx
+
dLdy dM
dx = 0.
Ldx+ldy+Ldt Mdx+mdy+Mdt
dL
dy
= dl
dx
; dm
dx
=dM
dy
; dL
dy
=dl
dt
and dM
dx
=dM
dt
,
(L + m)(l M) + u
dl dMdx
+ v
dl dM
dy
+
dl dMdt
= 0
l = M : du/dy = dv/dx.
udx+vdy
udx + vdy
=du dv
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udx + vdy
p,
M=l,
dp= dx
2u(Ldx + ldy)
2v(ldx + mdy)
2
Ldx
2
Mdy+
Rdt.
Ldx + ldy = du Ldt ldx+ mdy= dv Mdt,
dp= dx 2udu 2vdv + 2Ludt + 2Mvdt 2Ldx 2Mdy+ Rdt.
dp= dx 2udu 2vdv 2Ldx 2Mdy
L =du/dt M =dv/dt,
dp= dx
2udu
2vdv
2
du
dt
dx
2
dv
dt
dy.
M =l.
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udx + vdy
t S
dS=udx+vdy dt= 0.
dS= udx + vdy + Udt.
du/dt= dU/dx dv/dt= dU/dy. U=dS/dt.
du
dt dx +dv
dt dy=dU
dx dx +dU
dydy
= U.
dU=K dx + kdy, dU/dx= K dU/dy= k.
dU
dxdx+
dU
dydy = K dx + kdy = dU.
U=dS/dt,
dp= dx 2udu 2vdv 2dU
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p= Const.+ x uu vv 2dSdt
,
S dt= 0, udx + vdy.
=V = (uu + vv) p = Const.+ x uu vv 2dS/dt. S dS S=
(udx + vdy),
ds. =p/f.
dS/dt= U,
dp= dx pf
ds 2V dV 2dU.
p= e
s/f e
s/f
(dx 2V dV 2dU)
dV
dU.
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p= x V V 2U1f
es/f
es/f(x V V 2U)ds.
udx+ vdy
du/dx+ dv/dy = 0, udy +vdx
udx+ vdy
udy+ vdx
,
AL, AB, AC,
du = Ldx+ ldy+ dz+ Ldtdv = Mdx+ mdy+ dz+ Mdtdw = Ndx+ ndy+ dz+Ndt;
L+ m+ = 0
du
dx+
dv
dy+
dw
dz = 0.
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dt, .
=udt AL, =vdt
AB, = wdt AC.
alongAL = u + Ludt + lvdt + wdt +
Ldt,
alongAB = v+ Mudt+ mvdt + wdt +Mdt,alongAC = w+ Nudt + nvdt + wdt +Ndt.
alongAL = 2(Lu+ lv+ w+ L),alongAB = 2(Mu+ mv+ w+
M),
alongAC = 2(Nu+ nv+ w +N).
AC
AL AB AC,
= 1 p
dp= Rdx+ rdy+ dz,
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AL: R, AB: r, AC: .
R= 2(Lu+ lv+ w+ L),r= 2(Mu+ mv+ w+ M),= 1 2(Nu+ nv+ w +N).
dp = Rdx+ rdy +dz
dR/dy= dr/dx; dR/dz= d/dx; dr/dz= d/dy.
I
uLy+ vly+ wy+ Ly +Ll+ lm+ n =uMx+ vmx+ wx+ Mx +ML+ mM+ N,
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II
uLz+ vlz+ wz+ Lz +L + l+ =uNx+ vnx+ wx+Nx +NL+ nM+ N ,
III
uMz+ vmz+ wz+ Mz +M+ m+ =uNy+ vny+ wy+Ny +Nl+ nm+ n.
Ly =lx; mx=My; y =lz; x= Mz ; Ly =lt; Mx= Mt,Lz =x; lz =y; nx = Ny; x = Nz; Lz =t; Nx= Nt,Mz =x; Ny =nx; mz =y; y =nz; Mz =t; Ny =nt.
dl dM
dt
+udl
dM
dx
+vdl
dM
dy
+wdl
dM
dz
+(l M) (L+ m)+nN=
d dN
dt
+u
d dN
dx
+v
d dN
dy
+w
d dN
dz
+(N) (L+ )+lnM=
d dn
dt
+u
d dn
dx
+v
d dn
dy
+w
d dn
dz
+( n) (m + )+MNl= 0.
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l= M; = N; = N;
dudy =
dvdx;
dudz =
dwdx ;
dvdz =
dwdy .
udx+vdy+
wdz
u, v, w x, y, z
udx + vdy + wdz
dt= 0
du=Ldx + Mdy+ Ndz
dv=Mdx+ mdy+ ndz
dw=Ndx+ ndy+ dz
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R, r,
R= 2(Lu+ Mv+ Nw+ L),r= 2(Mu+ mv+ nw+ M),= 1 2(Nu+ nv+ w +N).
dp= dz 2u(Ldx+ Mdy+ Ndz)2v(Mdx+ mdy+ ndz)2w(Ndx + ndy+ dz)2Ldx 2Mdy 2Ndz
=dz 2udu 2vdv 2wdw 2Ldx 2Mdy 2Ndz.
L =du/dt; M =dv/dt; N =dw/dt,
p= C z uu vv ww 2
du
dtdx +
dv
dtdy+
dw
dtdz
.
udx+vdy + wdz
=S.
t
dS= udx + vdy + wdz+ Udt,
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du/dt= dU/dx; dv/dt= dU/dy; dw/dt= dU/dz.
dU
dxdx+
dU
dydy+
dU
dzdz= dU
p= C
z
uu
vv
ww
2U,
p= C z uu vv ww 2dSdt
.
uu+ vv + ww V
,
p= C z V V 2dSdt
.
S udx+vdy +wdz.
t dt,
dS/dt,
U =S/t.
dS/ds.
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udx + vdy + wdz
du/dx+dv/dy+dw/dz= 0.
udx+ vdy + wdz+ Udt
p= C z uu vv ww 2U.
udx+vdy +wdz
du/dx + dv/dy + dw/dz= 0.
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p = Cz uu vv ww 2U.
udx +
vdy + wdz
du/dx+dv/dy+ dw/dz = 0. u = dS/dx, v = dS/dy,
w= dS/dz,
ddS
dx2 +
ddS
dy2 +
ddS
dz2 = 0.
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S= (Ax+ By + Cz)n.
dS/dx= nA(Ax+ By + Cz)n1 ddS/dx2 =n(n1)AA(Ax+ By +Cz)n2, ddS/dy2 ddS/dz2.
n(n 1)(Ax+ By + Cz)n2(AA + BB + CC) = 0.
n= 0 n= 1.
S=Ax + By +
Cz,
AA + BB +
CC= 0.
S= (Ax+ By + Cz)n
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S=+ x + y + z+ (Ax+ By + Cz)n
+ (Ax+ By+ Cz)n
+ (Ax+ By+ Cz)n
+ (Ax+ By+ Cz)n
etc.
AA + BB + CC= 0;
AA + BB + CC = 0;
AA + BB + CC = 0 etc.
I. S= A,
II. S= Ax+ By + Cx,
III. S= Axx+ Byy+ Czz+ 2Dxy+ 2Exz+ 2F yz
(A + B+ C= 0),
IV. S= Ax3 + By3 +Cz3 + 3Dxxy+ 3Fxxz+ 3Hyyz+ 6Kxyz
A+ E+ G= 0; B+ D+ I= 0; C+ F+ H= 0;
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+Ax4 + 6Dxxyy+ 4Gx3y+ 4Hxy3 + 12Nxxyz
V. S= +By4
+ 6Exxzz+ 4Ix3
z+ 4Kxz3
+ 12Oxyyz+By4 + 6Exxzz+ 4Ix3z+ 4Kxz3 + 12Oxyyz
where A + D+ E= 0 G + H+ P = 0,
B+ D+ F = 0 I+ K+ O= 0,
C+ E+ F = 0 L+ M+ N= 0.
x+y+z
LZxx + MZyy +
NZzz,
L + M+ N
+Ax5 + 5Dx4y+ 5Dx4z+ 10Gx3yy + 10Gx3zz+ 20Kx3yz+ 30NxyyzzS= +By5 + 5Exy4 + 5Ey4z+ 10Hx2y3 + 10Hy3zz+ 20Lxy3z+ 30Oxxyzz
+Cz5 + 5F xz4 + 5Fyz4 + 10Ixxz3 + 10Iyyz3 + 20Mxyz3 + 30Pxxyyz,
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A + G + G = 0; D+ H+ O= 0; D + I+ P = 0B+ H+ H = 0; E+ G + N= 0; E+I+ P = 0; K+ L + M= 0C+ I+I= 0; F+ G + N= 0; F+ H+ O= 0.
S=A,
p= C2dAdtz.
S=Ax + By + Cz,
u= A; v= B &w = C.
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,
p= CzAABBCC2xdAdt 2y dBdt 2zdCdt.
S=Axx+Byy +Czz+2Dxy +2Exz+2F yz A+ B + C = 0 :
u= 3Ax + 2Dy+ 2Ez;v = 2By + 2Dx + 2F z;w = 2Cz+ 2Ex + 2F y,
w= 2Ex + 2F y 2(A+ B)z.
p = C z AACC 2xdAdt 2y dB
dt z dC
dt.
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o;
== dx =
QQ + qq+
= = dy =
RR+ rr+
= o= dz =
SS+ ss +
= = dx2 + dy2 = (QR)2 + (q
r)2 + (
)2
= o =
dx2 + dz2 =
(Q S)2 + (q s)2 + ( )2
= ro =
dy2 + dz2 =
(R S)2 + (r s)2 + ( )2.
QR+ qr+ = 0;QS+ qs + = 0;
RS+ rs + = 0.
, ,
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1 = 1 + 2Ldt; l+ M= 0
1 = 1 + 2mdt; + N= 0
1 = 1 + 2dt; + n= 0
L = 0, m = 0, = 0, M =l,N= n= .
du= + ldy+ dz
dv= ldx + dz
dw= dx dy.
l=M, = N
n = .
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=dxdydz,
AL: = 2dxdydz(Lu + lv+ w+L
) = 2dxdydz(uux+ vuy+ wuz+ ut)
AB: = 2dxdydz(Mu+ mv+ w+M) = 2dxdydz(uvx+ vvy+ wvz+ vt)AC: = 2dxdydz(Nu+ nv+ w +N) = 2dxdydz(uwx+ vwy+ wwz+ wt)
udx+ vdy+ wdz
= S.
dS = udx+ vdy+ wdz+ Udt. du/dy = dv/dx;
du/dz= dw/dx;du/dt= dU/dx
AL:= 2dxdydzudu + vdv+ wdw+ dU
dx ,
ux du
dx .
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AB:= 2dxdydzudu + vdv + wdw+ dUdy
,
AC := 2dxdydzudu + vdv+ wdw+ dU
dz .
uu+vv +ww +2U=T ,
dT =K dx+ kdy + dz
dxdydz
AL: =Kdxdydz
AB: =kdxdydz
AC: =dxdydz.
,
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T =uu + vv + ww + 2U
p= CzT,
,
p= C+ (Qdx + qdy+ dz) T.
Qdx + qdy+ dz
.
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u= 0, v= 0
w= 0. T = 2U,
p= C+
(Qdx+ qdy+ dz) 2U.
U t,
U C,
p= C+
(Qdx+ qdy+ dz) ,
,
(Qdx+ qdy+ dz)
Qdx+qdy + dz =P,
p= C+ P.
p= C
z.
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p= C+ P
P =
P =E, C=
E;
p= PE. p= Cz, z= C,
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=rr,
=f f, =, dt +d. t, d/dt. V = f f/rr. dy = dx dz = dx;
u=f frr
11 + +
;
v=f frr
1 + +
;
w=f frr
1 + +
;
. f f
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uu+vv + ww = V V = f4/r4. rr
udx+vdy +wdz
=S,
dS=f f
rr
dx (1 + + )1 + +
=f f
rr dx
1 + + .
dx
1 + +
= ds, dS = ffdsrr . t, .
S= ffdsrr .
U=dS/dt. ffdsrr
dS/dt = U = ddt ffdsrr ;
T =f4
r4 +
2ddt
ffds
rr .
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p= C+
(Qdx + qdy+ dz)f
4r4 2d
dt
ffds
rr .
Qdx+qdy + dz
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12
0 0 1
dx + Ldxdt Mdxdt 1
ldydt dy+ mdydt 1
= 12
dxdy+ 12
(L+ m)dxdydt + 12
(Lm lM)dxdydtdt 0 =L + m= u/x + v/y.
udx+vdy
=d, ,
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L+ m = 0.
l M
d
dt
u
y v
x
=
d
dt= 0,
= 0
l M, N, n = U,
DDt U + [ U]= 0,
D/Dt
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U, S=
1
2
[U] + [U]T
,
U= S.
D
Dt = S,
lM= N= n= 0
A+B+C = 0,
AA, BB, C C
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p= C+
(Qdx + qdy+ dz) T
Qdx+ qdy+ dz