Covariant basis

Description

Velocity-gradient tensor

The gradient of a velocity vector is

(jEjXj)(iEiUi)=ij(EjEiXj)Ui+ij(EjEi)UiXj.

By using the changes of the basis vectors:

EiXj=kδijHj2HkHkHiXkEkkδijHi2HkHkHjXkEk+1HiHiXjEi+1HjHjXiEj,

I find that the vector gradient is given by the sum of the following elements.

ij(EjEi)1HjHj1HiHiXjUi.
j(EjEj)1HjHj1HjkHjXkUk.
ijk(EjEk)δijHj2HkHkHiXkUiijk(EjEk)δijHi2HkHkHjXkUi=jk(EjEk)Hj2HkHkHjXkUjik(EiEk)Hi2HkHkHiXkUi=ij(EjEi)1HjHjHjHiHiHjXiUj=ij(EjEi)1HiHi1HjHjXiUj.
ij(EjEi)1HjHjUiXj.

Note that the negative of the third element is the transpose of the first one.

In summary, the velocity gradient is given by the sum of

(E1E2E3)(1H1H1U1X11H1H1U2X11H1H1U3X11H2H2U1X21H2H2U2X21H2H2U3X21H3H3U1X31H3H3U2X31H3H3U3X3)(E1E2E3),
(E1E2E3)(1H1H1k1H1H1XkUk0001H2H2k1H2H2XkUk0001H3H3k1H3H3XkUk)(E1E2E3),
(E1E2E3)(01H1H11H2H2X1U21H2H21H1H1X2U11H1H11H3H3X1U31H3H31H1H1X3U11H2H21H1H1X2U11H1H11H2H2X1U201H2H21H3H3X2U31H3H31H2H2X3U21H3H31H1H1X3U11H1H11H3H3X1U31H3H31H2H2X3U21H2H21H3H3X2U30)(E1E2E3).

Strain-rate tensor

The strain-rate tensor is defined as the symmetric part of it; namely the sum of

(E1E2E3)(1H1H1U1X1121H1H1U2X1+121H2H2U1X2121H1H1U3X1+121H3H3U1X3sym.1H2H2U2X2121H2H2U3X2+121H3H3U2X3sym.sym.1H3H3U3X3)(E1E2E3),
(E1E2E3)(1H1H1k1H1H1XkUk00sym.1H2H2k1H2H2XkUk0sym.sym.1H3H3k1H3H3XkUk)(E1E2E3).

Example

Cylindrical coordinates

The velocity-gradient tensor is the sum of

(E1E2E3)(U1X1U2X1U3X11X1X1U1X21X1X1U2X21X1X1U3X2U1X3U2X3U3X3)(E1E2E3),
(E1E2E3)(00001X1X1U1X10000)(E1E2E3),
(E1E2E3)(0U2X10U2X100000)(E1E2E3).

The strain-rate tensor is

(E1E2E3)(U1X112U2X1+121X1X1U1X212U3X1+12U1X3sym.1X1X1U2X2+1X1X1U1X1121X1X1U3X2+12U2X3sym.sym.U3X3)(E1E2E3).
Rectilinear coordinates

The velocity-gradient tensor is the sum of

(E1E2E3)(1H1H1U1X11H1H1U2X11H1H1U3X11H2H2U1X21H2H2U2X21H2H2U3X21H3H3U1X31H3H3U2X31H3H3U3X3)(E1E2E3),
(E1E2E3)(1H1H1k1H1H1XkUk0001H2H2k1H2H2XkUk0001H3H3k1H3H3XkUk)(E1E2E3).

The strain-rate tensor is

(E1E2E3)(1H1H1U1X1+1H1H1k1H1H1XkUk121H1H1U2X1+121H1H1U2X1121H1H1U3X1+121H3H3U1X3sym.1H2H2U2X2+1H2H2k1H2H2XkUk121H2H2U3X2+121H3H3U2X3sym.sym.1H3H3U3X3+1H3H3k1H3H3XkUk)(E1E2E3).
Application

The velocity-gradient tensor is the sum of

(E1E2E3)(1H1H1U1X11H1H1U2X11H1H1U3X11H2H2U1X21H2H2U2X21H2H2U3X21H3H3U1X31H3H3U2X31H3H3U3X3)(E1E2E3),
(E1E2E3)(00001H2H21H2H2X1U10000)(E1E2E3),
(E1E2E3)(01H1H11H2H2X1U201H1H11H2H2X1U200000)(E1E2E3).

The strain-rate tensor is

(E1E2E3)(1H1H1U1X1+1H1H11H1H1X1U1121H1H1U2X1+121H1H1U2X1121H1H1U3X1+121H3H3U1X3sym.1H2H2U2X2+1H2H21H2H2X1U1121H2H2U3X2+121H3H3U2X3sym.sym.1H3H3U3X3)(E1E2E3).