Анизотропные жидкости. Биологические структуры. Петрова Г.П. - 49 стр.

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МГУ | Москва

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48
(
nn=−
1
k
2
0
k
=
.
2
0
k
≠
.
1
a
,
2
a
3
22 2
2
011 22 33
1
2
yy y
xxx
nn n
nnn
FF k k k
xy xy z z


∂∂ ∂

∂∂∂



=− + + + + +





∂∂ ∂∂ ∂ ∂




()
2
15
aa
+
()
()
2
24
aa
−+
22
36
aa
+
div
y
x
n
n
n
xy
∂
∂
+=
∂∂
,
() ()
001 0 rot
1
y
x
xy
ijk
n
n
nn
yx xy
nn
∂
∂
∂∂
−= =⋅
∂∂ ∂∂
,
()
2
2
div
y
x
n
n
n
zy
∂

∂
+=


∂∂

(splay)
()
2
2
rot
y
x
n
n
nn
yx
∂

∂
−+ =⋅


∂∂

(twist)
[]
2
2
2
rot
y
x
n
n
nn
zz
∂

∂

+=⋅



∂∂


(bend)
3
F
() ()
[]
222
11 22 33
11 1
div rot rot
22 2
Fk n knn knn=+⋅+⋅
. (27)
22
k
()
2
22 2 22
rot
kn nkk⋅+
      DhgklZglu NL b NLM gZau\Zxlky fh^meyfb mijm]hklb NjZgdZ
      J_amevlbjmxs__ mjZ\g_gb_ ^ey iehlghklb k\h[h^ghc wg_j]bb
^_nhjfbjh\Zggh]h h[jZapZ [m^_l aZ\bk_lv hl dhgdj_lgh]h \b^Z
kbff_ljbb `b^dh]h djbklZeeZ >ey g_fZlbq_kdbo djbklZeeh\
( n = − n  fhe_dmeu g_ objZevgu k1 b k2 = 0 .
      >ey ohe_kl_jbq_kdbo djbklZeeh\ k2 ≠ 0 .
      K mq_lhf agZq_gbc a1 , a2 b l^ iehlghklv k\h[h^ghc wg_j]bb
 \ wj]kf3 fh`_l [ulv \ujZ`_gZ
                        2                    2       ∂n 2  ∂n                      2 
        1   ∂nx ∂n y           ∂n y ∂nx 
                                                               y               
                                                                                       
F = F0 − k11    +        + k22      +       + k33  x  +                   
             
        2   ∂x   ∂y                                     
                                                       ∂z   ∂z                     
                                 ∂x    ∂y 
                                                                                     
               (a1 + a5 )          ( − ( a2 + a4 ) )
                        2                          2
                                                                     a32 + a62

     Fh`gh ij_^klZ\blv wlh mjZ\g_gb_ \ \_dlhjghc nhjf_ LZd dZd           

                             ∂nx ∂n y
                                +     = div n ,
                             ∂x   ∂y
                                  i          j     k
                ∂nx ∂n y         ∂          ∂
                   −     = (001)                   0 = ( n ⋅ rot n ) ,
                ∂y   ∂x          ∂x         ∂y
                                 nx         ny     1
lh \deZ^ ^_nhjfZpbc fh`gh aZibkZlv \ \_dlhjghc nhjf_
                               2
                    ∂n  ∂n y 
                               = (div n )
                                           2
                   x +                                 (sp la y)
                    ∂z   ∂y 
                                   2
                   ∂nx ∂n y 
                              = ( n ⋅ rot n )
                                               2
                   − +                                 (tw ist)
                   ∂y   ∂x 
                                       2
                   ∂nx   ∂n y 
                         2

                   ∂  +  ∂  = [n ⋅ rot n ] (b e n d )
                                                2

                   z   z 
>ey g_fZlbq_kdhc nZau wg_j]by ijboh^ysZyky gZ  kf3  iehl
ghklv wg_j]bb F \ujZ`Z_lky nhjfmehc
              1             2 1                 2 1
           F = k11 ( div n ) + k22 ( n ⋅ rot n ) + k33 [n ⋅ rot n ] .
                                                                   2
                                                                                      (2 7 )
              2               2                   2
    >ey ohe_kl_jbq_kdbo djbklZeeh\ bf__lky ihklhyggh_ djmq_
gb_ ihwlhfm \f_klh k22 bf__f k22 (n ⋅ rot n + k2 k22 )2  ]^_
48

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