9.2.1 The deformation or rate of strain tensor

Consider two infinitesimally close fluid parcels with material coordinates $\zeta^{s}$ and $\zeta^{s} + d\zeta^{s}$, where s=1,2,3. The components (u¹,u²,u³) of the velocity for these two parcels differ by the increment

$$\frac{\partial u^{m}}{\partial \zeta^{s}} \, d\zeta^{s}$$ d u^m =

$$\frac{\partial u^{m}}{\partial \zeta^{s}} \, \frac{\partial \zeta^{s}}{\partial x^{n}} \, dx^{n}$$ =

$$\frac{\partial u^{m}}{\partial x^{n}} \, dx^{n}.$$ = (9.1)

The velocity derivatives $\partial u^{m}/\partial x^{n}$ form the components to a second order tensor. In order to attach physical significance to this tensor, it is useful to separately consider its symmetric and anti-symmetric components, which are written

$$\Omega_{mn} + e_{mn},$$ u_m,n = (9.2)

where

$$2 \, \Omega_{mn}$$ = u_m,n - u_n,m (9.3)

$$2 \, e_{mn}$$ = u_m,n + u_n,m, (9.4)

and $u_{m} = g_{mn} \, u^{n}$ are the covariant components to the velocity vector. Note that in curvilinear coordinates, the partial derivatives appearing in $\Omega_{mn}$ and e_mn generalize to covariant derivatives. The anti-symmetric piece of the velocity derivative tensor is related to the vorticity through

$$2 \, \omega^{i} - \epsilon^{ijk} \, \Omega_{jk}$$ =

$$\epsilon^{ijk} \, u_{k,j},$$ = (9.5)

where $\epsilon^{ijk}$ is the Levi-Civita symbol defined in Section 4.6.3. In conventional Cartesian vector notation, this result takes the form

$${\bf\omega} \frac{1}{2} \, \nabla \wedge {\bf u}.$$ = (9.6)

Standard results from fluid mechanics establish the connection between vorticity and rigid body rotation of a fluid parcel. If the motion is completely rigid, which means that it consists of a translation plus a rotation, then the symmetric part of the velocity derivative tensor vanishes. Consequently, the symmetric tensor e_mn is called the deformation or rate of strain tensor since it represents deviations from rigid body motion. To provide a further interpretation of the strain tensor, consider the squared distance between two infinitesimally close material parcels of fluid

$$g_{mn} \, dx^{m} \, dx^{n}$$ ds² =

$$g_{mn} \, \frac{\partial x^{m}}{\partial \zeta^{p}} \frac{\partial x^{n}}{\partial \zeta^{q}} \, d\zeta^{p} \, d\zeta^{q},$$ = (9.7)

where $g_{mn} = \delta_{mn}$ since we are working with Cartesian coordinates. The material time derivative of this distance is given by

$$\frac{D (ds^{2})}{Dt} g_{mn} \, \left( \frac{\partial u^{m}}{\partial \zeta^{p}} \frac{... ...} \frac{\partial u^{n}}{\partial \zeta^{q}} \right) \, d\zeta^{p} \, d\zeta^{q}$$ = (9.8)

where $D\zeta^{p}/Dt = 0$ since $\zeta^{p}$ are material coordinates. Use of the chain rule in the forms

$$\frac{\partial u^{m}}{\partial \zeta^{p}} \, d\zeta^{p} u^{m}_{\; ,n} \, dx^{n}$$ = (9.9)

$$\frac{\partial x^{m}}{\partial \zeta^{p}} \, d\zeta^{p}$$ = dx^m (9.10)

renders

$$\frac{D (ds^{2})}{Dt} 2 \, e_{mn} \, dx^{m} \, dx^{n},$$ = (9.11)

or equivalently

 

$$\frac{1}{ds} \, \frac{D (ds)}{Dt} e_{mn} \, \frac{dx^{m}}{ds} \, \frac{dx^{n}}{ds}.$$ = (9.12)

Now dx^m/ds is a component of the unit vector which points from one fluid parcel to the other. Hence, equation (9.12) says that the rate of change of the infinitesimal distance separating the two parcels, as a fraction of the distance, is related to the relative position of the parcels through the strain tensor. Section 4.42 of Aris (1962), amongst other places, provides further elaboration.