本文内容主要包括:
1. 物质导数与空间时间导数及二者的联系
考虑运动变形过程中代表性物质点的物理量 Φ\bold\PhiΦ(张量) 随时间的变化率。
- 在物质描述中,Φ\bold\PhiΦ 以 (X⃗,t)(\vec{X},t)(X,t) 为自变量;
- 在空间描述中,Φ\bold\PhiΦ 以 (x⃗,t)(\vec{x},t)(x,t) 为自变量。
物理量 Φ\bold \PhiΦ 随某一固定的物质点一起运动的时间变化率(称作:物质导数)可写作:
DΦDt=(∂Φ(X⃗,t)∂t)∣X⃗≜Φ∙\dfrac{D\bold \Phi}{Dt} =\left. \left(\frac{\partial \bold\Phi(\vec{X},t)}{\partial t}\right)\right|_{\vec X} \triangleq \overset{\bullet}{\bold\Phi}DtDΦ=(∂t∂Φ(X,t))X≜Φ∙
物理量 Φ\bold \PhiΦ 在某一固定的空间坐标上的时间变化率(称作:空间时间导数/局部导数)可写作:
(∂Φ(x⃗,t)∂t)∣x⃗≜Φ′\left. \left(\frac{\partial \bold\Phi(\vec{x},t)}{\partial t}\right)\right|_{\vec x} \triangleq \bold\Phi'(∂t∂Φ(x,t))x≜Φ′
根据复合函数的求导法则可推出:
Φ∙={∂Φ[x⃗(X⃗,t),t]∂t}∣X⃗=(∂Φ∂t)∣x⃗+(∂Φ∂xr)(∂xr∂t)∣X⃗=Φ′+(∂Φ∂xr⊗g⃗r)⋅(g⃗s∂xs∂t)∣X⃗=Φ′+(Φ▽)⋅(∂x⃗∂t)∣X⃗=Φ′+(Φ▽)⋅(∂u⃗∂t)=Φ′+(Φ▽)⋅v⃗=Φ′+v⃗⋅(▽Φ)\begin{aligned} & \overset{\bullet}{\bold\Phi} =\left. \left\{\frac{\partial \bold\Phi[\vec{x}(\vec{X},t),t]}{\partial t}\right\}\right|_{\vec X} \\\\ &\quad=\left.\left(\dfrac{\partial\bold\Phi}{\partial t}\right)\right|_{\vec{x}}+\left(\dfrac{\partial\bold\Phi}{\partial {x}^r}\right)\left.\left(\dfrac{\partial x^r}{\partial t}\right)\right|_{\vec{X}}\\\\ &\quad=\bold\Phi'+\left(\dfrac{\partial\bold\Phi}{\partial {x}^r}\otimes\vec{g}\ ^r\right)\cdot\left.\left(\vec{g}_s\dfrac{\partial x^s}{\partial t}\right)\right|_{\vec{X}}\\\\ &\quad=\bold\Phi'+\left(\bold\Phi\triangledown\right)\cdot\left.\left(\dfrac{\partial\vec x}{\partial t}\right)\right|_{\vec{X}}\\\\ &\quad=\bold\Phi'+\left(\bold\Phi\triangledown\right)\cdot\left(\dfrac{\partial\vec u}{\partial t}\right)\\\\ &\quad=\bold\Phi'+\left(\bold\Phi\triangledown\right)\cdot\vec{v}\\\\ &\quad=\bold\Phi'+\vec{v}\cdot\left(\triangledown\bold\Phi\right) \end{aligned}Φ∙={∂t∂Φ[x(X,t),t]}X=(∂t∂Φ)x+(∂xr∂Φ)(∂t∂xr)X=Φ′+(∂xr∂Φ⊗g r)⋅(gs∂t∂xs)X=Φ′+(Φ▽)⋅(∂t∂x)X=Φ′+(Φ▽)⋅(∂t∂u)=Φ′+(Φ▽)⋅v=Φ′+v⋅(▽Φ)
2. 空间坐标系相关量的物质导数
2.1. 空间坐标系基矢的物质导数
随时间变化,某一固定物质点将映射至空间坐标系中的不同位置。因此,“空间坐标系基矢的物质导数”是指:某一物质点所在处的基矢变化率。故
g⃗i∙=(g⃗i)′+v⃗⋅▽g⃗i=v⃗⋅▽g⃗i=vj∂g⃗i∂xj=vjΓijkg⃗k=vjΓij,kg⃗k\overset{\bullet}{\vec{g}_i} =(\vec{g}_i)'+\vec{v}\cdot\triangledown\vec{g}_i =\vec{v}\cdot\triangledown\vec{g}_i =v^j\dfrac{\partial \vec{g}_i}{\partial x^j} =v^j\Gamma_{ij}^k\vec{g}_k =v^j\Gamma_{ij,k}\vec{g}^kgi∙=(gi)′+v⋅▽gi=v⋅▽gi=vj∂xj∂gi=vjΓijkgk=vjΓij,kgk
式中,Γijk、Γij,k\Gamma_{ij}^k、\Gamma_{ij,k}Γijk、Γij,k 分别为空间坐标系的第二类、第一类 Christoffel 符号。又由于
DDt(g⃗i⋅g⃗j)=g⃗i∙⋅g⃗j+g⃗i⋅g⃗j∙=0⟹g⃗i⋅g⃗j∙=−g⃗i∙⋅g⃗j\dfrac{D}{Dt}(\vec{g}_i\cdot\vec{g}^j) =\overset{\bullet}{\vec{g}_i}\cdot\vec{g}^j+\vec{g}_i\cdot\overset{\bullet}{\vec{g}^j} =0 \Longrightarrow \vec{g}_i\cdot\overset{\bullet}{\vec{g}^j}=-\overset{\bullet}{\vec{g}_i}\cdot\vec{g}^jDtD(gi⋅gj)=gi∙⋅gj+gi⋅gj∙=0⟹gi⋅gj∙=−gi∙⋅gj
令 g⃗j∙=βijg⃗i\overset{\bullet}{\vec{g}^j}=\beta^j_i\vec{g}^igj∙=βijgi ,那么:
g⃗i⋅βkjg⃗k=βij=−g⃗i∙⋅g⃗j=−vkΓikj\vec{g}_i\cdot\beta^j_k\vec{g}^k =\beta^j_i =-\overset{\bullet}{\vec{g}_i}\cdot\vec{g}^j =-v^k\Gamma^j_{ik}gi⋅βkjgk=βij=−gi∙⋅gj=−vkΓikj
故,
g⃗j∙=−vkΓikjg⃗i\overset{\bullet}{\vec{g}^j}=-v^k\Gamma^j_{ik}\vec{g}^igj∙=−vkΓikjgi
2.2. 空间坐标系协变基矢混合积的 g\sqrt{g}g 的物质导数
由空间坐标系基矢的物质导数可知:
gij∙=g⃗i∙⋅g⃗j+g⃗i⋅g⃗j∙=vr(Γirkgkj+Γjrkgki)=vr(Γir,j+Γjr,i)\overset{\bullet}{g_{ij}} =\overset{\bullet}{\vec{g}_i}\cdot\vec{g}_j+\vec{g}_i\cdot\overset{\bullet}{\vec{g}_j} =v^r(\Gamma^k_{ir}g_{kj}+\Gamma^k_{jr}g_{ki}) =v^r(\Gamma_{ir,j}+\Gamma_{jr,i})gij∙=gi∙⋅gj+gi⋅gj∙=vr(Γirkgkj+Γjrkgki)=vr(Γir,j+Γjr,i)
由于,
1det([A])[A∗]=[A]−1\dfrac{1}{det([A])}[A^*]=[A]^{-1}det([A])1[A∗]=[A]−1
其中,[A∗][A^*][A∗] 为 [A][A][A] 的伴随矩阵。则
1g∂g∂gji=gij,g=det(gij)\dfrac{1}{g}\dfrac{\partial g}{\partial g_{ji}}=g^{ij},g=det(g_{ij})g1∂gji∂g=gij,g=det(gij)
故,det(gij)det(g_{ij})det(gij) 的物质导数为:
g∙=∂g∂gjigji∙=ggijgji∙=gvr(Γiri+Γjrj)=2gvrΓiri\overset{\bullet}{g} =\dfrac{\partial g}{\partial g_{ji}}\overset{\bullet}{g_{ji}} =gg^{ij}\overset{\bullet}{g_{ji}} =gv^r(\Gamma_{ir}^i+\Gamma_{jr}^j) =2gv^r\Gamma_{ir}^ig∙=∂gji∂ggji∙=ggijgji∙=gvr(Γiri+Γjrj)=2gvrΓiri
式中,Γiri\Gamma_{ir}^iΓiri 为空间坐标系的第二类Christoffel 符号。进一步:
g∙=gvrΓiri\overset{\bullet}{\sqrt{g}}=\sqrt gv^r\Gamma_{ir}^ig∙=gvrΓiri
上式也可利用第二类Christoffel符号与协变基矢的混合积 g\sqrt{g}g 的关系与物质导数和局部导数的关系得到:
g∙=v⃗⋅(▽g)=vi∂g∂xi=gvrΓiri\overset{\bullet}{\sqrt{g}} =\vec{v}\cdot(\triangledown\sqrt{g}) =v^i\dfrac{\partial \sqrt{g}}{\partial x^i} =\sqrt gv^r\Gamma_{ir}^ig∙=v⋅(▽g)=vi∂xi∂g=gvrΓiri
3. 随体坐标系 {XA,t}\{X^A,t\}{XA,t} 相关量的物质导数
3.1. 随体坐标系 {XA,t}\{X^A,t\}{XA,t} 基矢的物质导数
随时间的变化,特定的物质点在随体坐标系 {XA,t}\{X^A,t\}{XA,t} 中的基矢不断改变。其协变基矢的变化率可写作:
C⃗A∙=[∂∂t(∂x⃗∂XA)]∣X⃗=[∂∂XA(∂x⃗∂t)]∣X⃗=∂∂XA(∂u⃗∂t)=∂v⃗∂XA=vB∣∣AC⃗B=∂v⃗∂xi∂xi∂XA=x,Ai∂v⃗∂xi=x,Aivj∣ig⃗j\begin{aligned} &\overset{\bullet}{\vec{C}_A} =\left.\left[\dfrac{\partial}{\partial t}\left(\dfrac{\partial \vec{x}}{\partial X^A}\right)\right]\right|_{\vec{X}} =\left.\left[\dfrac{\partial}{\partial X^A}\left(\dfrac{\partial \vec{x}}{\partial t}\right)\right]\right|_{\vec{X}} \\\ \\ &\quad\ =\dfrac{\partial}{\partial X^A}\left(\dfrac{\partial \vec{u}}{\partial t}\right) =\dfrac{\partial\vec{v}}{\partial X^A} =v^B||_A\vec{C}_B \\\ \\ &\quad\ =\dfrac{\partial\vec{v}}{\partial x^i}\dfrac{\partial x^i}{\partial X^A} =x^i_{,A}\dfrac{\partial\vec{v}}{\partial x^i} =x^i_{,A}v^j|_i\vec{g}_j \end{aligned} CA∙=[∂t∂(∂XA∂x)]X=[∂XA∂(∂t∂x)]X =∂XA∂(∂t∂u)=∂XA∂v=vB∣∣ACB =∂xi∂v∂XA∂xi=x,Ai∂xi∂v=x,Aivj∣igj
同理可知
C⃗A∙∙=∂a⃗∂XA=aB∣∣AC⃗B\overset{\bullet\bullet}{\vec{C}_A} =\dfrac{\partial\vec{a}}{\partial X^A} =a^B||_A\vec{C}_BCA∙∙=∂XA∂a=aB∣∣ACB
又
DC⃗A⋅C⃗BDt=C⃗A∙⋅C⃗B+C⃗A⋅C⃗B∙=0\dfrac{D{\vec{C}_A\cdot\vec{C}^B}}{Dt} =\overset{\bullet}{\vec{C}_A}\cdot\vec{C}^B+{\vec{C}_A}\cdot\overset{\bullet}{\vec{C}^B} =0DtDCA⋅CB=CA∙⋅CB+CA⋅CB∙=0
故
C⃗A∙=−(C⃗B∙⋅C⃗A)C⃗B=−vA∣∣BC⃗B=−X,jAvj∣ig⃗i\overset{\bullet}{\vec{C}^A} =-(\overset{\bullet}{\vec{C}_B}\cdot\vec{C}^A)\vec{C}^B =-v^A||_B\vec{C}^B =-X^A_{,\ j}v^j|_i\vec{g}^iCA∙=−(CB∙⋅CA)CB=−vA∣∣BCB=−X, jAvj∣igi
3.2. 随体坐标系 {XA,t}\{X^A,t\}{XA,t} 协变基矢混合积的 C\sqrt{C}C 的物质导数
C∙AB=C⃗∙A⋅C⃗B+C⃗A⋅C⃗∙B=vB∣∣A+vA∣∣B\overset{\bullet}{C}_{AB} =\overset{\bullet}{\vec C}_{A}\cdot{\vec C}_{B}+\vec{C}_A\cdot\overset{\bullet}{\vec C}_{B} =v_B||_A+v_A||_BC∙AB=C∙A⋅CB+CA⋅C∙B=vB∣∣A+vA∣∣B
又
1C∂C∂CBA=C−1AB,C=det(CAB)\dfrac{1}{C}\dfrac{\partial C}{\partial C_{BA}}=\overset{-1}{C}\ ^{AB},C=det(C_{AB})C1∂CBA∂C=C−1 AB,C=det(CAB)
则
C∙=∂C∂CBAC∙AB=CC−1ABC∙AB=C(vA∣∣A+vB∣∣B)=2CvA∣∣A\overset{\bullet}{C} =\dfrac{\partial C}{\partial C_{BA}}\overset{\bullet}{C}_{AB} =C\overset{-1}{C}\ ^{AB}\overset{\bullet}{C}_{AB} =C(v^A||_A+v^B||_B) =2Cv^A||_AC∙=∂CBA∂CC∙AB=CC−1 ABC∙AB=C(vA∣∣A+vB∣∣B)=2CvA∣∣A
进一步知:
C∙=12CC∙=CvA∣∣A\overset{\bullet}{\sqrt C} =\dfrac{1}{2\sqrt C}\overset{\bullet}{C} =\sqrt Cv^A||_AC∙=2C1C∙=CvA∣∣A
3.3. J\mathscr{J}J 的物质导数
由于,
J=det(F)=CG\mathscr{J}=det(\bold F)=\sqrt{\dfrac{C}{G}}J=det(F)=GC
故,
J∙=C∙G=CGvA∣∣A=JvA∣∣A=J▽⋅v⃗\overset{\bullet}{\mathscr{J}} ={\dfrac{\overset{\bullet}{\sqrt C}}{\sqrt G}} =\dfrac{\sqrt C}{\sqrt G}v^A||_A =\mathscr{J}v^A||_A =\mathscr{J}\triangledown\cdot\vec{v}J∙=GC∙=GCvA∣∣A=JvA∣∣A=J▽⋅v
4. 任意张量在空间坐标系与随体坐标系 {XA,t}\{X^A,t\}{XA,t} 中的物质导数
以三阶张量为例:
Φ=Φ∙∙kijg⃗i⊗g⃗j⊗g⃗k=Φ∙∙MABC⃗A⊗C⃗B⊗C⃗M\bold\Phi =\varPhi^{ij}_{\bullet\bullet k}\ \vec{g}_i\otimes\vec{g}_j\otimes\vec{g}^k =\varPhi^{AB}_{\bullet\bullet M}\ \vec{C}_A\otimes\vec{C}_B\otimes\vec{C}^MΦ=Φ∙∙kij gi⊗gj⊗gk=Φ∙∙MAB CA⊗CB⊗CM
则
Φ∙=Φ∙∙∙MABC⃗A⊗C⃗B⊗C⃗M+Φ∙∙MABC⃗A∙⊗C⃗B⊗C⃗M+Φ∙∙MABC⃗A⊗C⃗B∙⊗C⃗M+Φ∙∙MABC⃗A⊗C⃗B⊗C⃗∙M=(Φ∙∙∙MAB+Φ∙∙MNBvA∣∣N+Φ∙∙MANvB∣∣N−Φ∙∙NABvN∣∣M)C⃗A⊗C⃗B⊗C⃗M\begin{aligned} &\overset{\bullet}{\bold\Phi} =\overset{\bullet}{\varPhi}\ ^{AB}_{\bullet\bullet M}\ \vec{C}_A\otimes\vec{C}_B\otimes\vec{C}^M +\varPhi^{AB}_{\bullet\bullet M}\ \overset{\bullet}{\vec{C}_A}\otimes\vec{C}_B\otimes\vec{C}^M +\varPhi^{AB}_{\bullet\bullet M}\ \vec{C}_A\otimes\overset{\bullet}{\vec{C}_B}\otimes\vec{C}^M +\varPhi^{AB}_{\bullet\bullet M}\ \vec{C}_A\otimes\vec{C}_B\otimes\overset{\bullet}{\vec{C}}\ ^M \\\\ &\ \ \ =(\overset{\bullet}{\varPhi}\ ^{AB}_{\bullet\bullet M}+\varPhi^{NB}_{\bullet\bullet M}\ v^A||_N+\varPhi^{AN}_{\bullet\bullet M}\ v^B||_N-\varPhi^{AB}_{\bullet\bullet N}\ v^N||_M)\ \vec{C}_A\otimes\vec{C}_B\otimes\vec{C}^M \end{aligned}Φ∙=Φ∙ ∙∙MAB CA⊗CB⊗CM+Φ∙∙MAB CA∙⊗CB⊗CM+Φ∙∙MAB CA⊗CB∙⊗CM+Φ∙∙MAB CA⊗CB⊗C∙ M =(Φ∙ ∙∙MAB+Φ∙∙MNB vA∣∣N+Φ∙∙MAN vB∣∣N−Φ∙∙NAB vN∣∣M) CA⊗CB⊗CM
或
Φ∙=Φ∙∙∙kijg⃗i⊗g⃗j⊗g⃗k+Φ∙∙kijg⃗∙i⊗g⃗j⊗g⃗k+Φ∙∙kijg⃗i⊗g⃗∙j⊗g⃗k+Φ∙∙kijg⃗i⊗g⃗j⊗g⃗∙k=(Φ∙∙∙kij+Φ∙∙ksjvrΓrsi+Φ∙∙kisvrΓrsj−Φ∙∙sijvrΓrks)g⃗i⊗g⃗j⊗g⃗k=[(Φ∙∙kij)′+(vrΦ∙∙k,rij+Φ∙∙ksjvrΓrsi+Φ∙∙kisvrΓrsj−Φ∙∙sijvrΓrks)]g⃗i⊗g⃗j⊗g⃗k=[(Φ∙∙kij)′+vrΦ∙∙kij∣r)]g⃗i⊗g⃗j⊗g⃗k=Φ′+v⃗⋅▽Φ\begin{aligned} &\overset{\bullet}{\bold\Phi} =\overset{\bullet}{\varPhi}\ ^{ij}_{\bullet\bullet k}\ \vec{g}_i\otimes\vec{g}_j\otimes\vec{g}^k +\varPhi^{ij}_{\bullet\bullet k}\ \overset{\bullet}{\vec{g}}_i\otimes\vec{g}_j\otimes\vec{g}^k +\varPhi^{ij}_{\bullet\bullet k}\ \vec{g}_i\otimes\overset{\bullet}{\vec{g}}_j\otimes\vec{g}^k +\varPhi^{ij}_{\bullet\bullet k}\ \vec{g}_i\otimes\vec{g}_j\otimes\overset{\bullet}{\vec{g}}\ ^k \\\\ &\ \ \ =(\overset{\bullet}{\varPhi}\ ^{ij}_{\bullet\bullet k}+{\varPhi}\ ^{sj}_{\bullet\bullet k}v^r\Gamma^i_{rs}+{\varPhi}\ ^{is}_{\bullet\bullet k}v^r\Gamma^j_{rs}-{\varPhi}\ ^{ij}_{\bullet\bullet s}v^r\Gamma^s_{rk})\ \vec{g}_i\otimes\vec{g}_j\otimes\vec{g}^k \\\\ &\ \ \ =[({\varPhi}\ ^{ij}_{\bullet\bullet k})'+(v^r{\varPhi}\ ^{ij}_{\bullet\bullet k,r}+{\varPhi}\ ^{sj}_{\bullet\bullet k}v^r\Gamma^i_{rs}+{\varPhi}\ ^{is}_{\bullet\bullet k}v^r\Gamma^j_{rs}-{\varPhi}\ ^{ij}_{\bullet\bullet s}v^r\Gamma^s_{rk})]\ \vec{g}_i\otimes\vec{g}_j\otimes\vec{g}^k \\\\ &\ \ \ =[({\varPhi}\ ^{ij}_{\bullet\bullet k})'+v^r{\varPhi}\ ^{ij}_{\bullet\bullet k}|_r)]\ \vec{g}_i\otimes\vec{g}_j\otimes\vec{g}^k \\\\ &\ \ \ =\bold\Phi'+\vec{v}\cdot\triangledown\bold\Phi \end{aligned}Φ∙=Φ∙ ∙∙kij gi⊗gj⊗gk+Φ∙∙kij g∙i⊗gj⊗gk+Φ∙∙kij gi⊗g∙j⊗gk+Φ∙∙kij gi⊗gj⊗g∙ k =(Φ∙ ∙∙kij+Φ ∙∙ksjvrΓrsi+Φ ∙∙kisvrΓrsj−Φ ∙∙sijvrΓrks) gi⊗gj⊗gk =[(Φ ∙∙kij)′+(vrΦ ∙∙k,rij+Φ ∙∙ksjvrΓrsi+Φ ∙∙kisvrΓrsj−Φ ∙∙sijvrΓrks)] gi⊗gj⊗gk =[(Φ ∙∙kij)′+vrΦ ∙∙kij∣r)] gi⊗gj⊗gk =Φ′+v⋅▽Φ