Past exam of the advanced fluid dynamics course of the University of Oxford / 2016 / 2 / a / Solution

Label a trajectory by $\mathbf{a}$ and write its flow map as $\mathbf{X}(\mathbf{a},t)$, with ${\partial }_t\mathbf{X}=\mathbf{u}(\mathbf{X},t)$. An infinitesimal material line joining neighboring labels has tangent ${\ell }_i=F_{ij}d{a}_j$, where $F_{ij}=\partial X_i/\partial a_j$ is the deformation gradient. Differentiating at fixed labels givesThis identity holds for an unnormalized material line tangent even in a compressible flow. Incompressibility also makes an advected magnetic field obey the same stretching law. A unit tangent would instead require a projection removing its change of length.

Use the source's velocity gradient convention $G_{ij}={\partial }_i{u}_j$. For a outer product $A_{ij}={\ell }_i{\ell }_j$, the product rule yieldsThe same law holds for a sum or locally transported average of such outer products. Thus the dyadic transport of material line elements motivatesThis upper-convected derivative vanishes for a symmetric tensor transported solely by stretching of material lines. It subtracts both advection and geometric deformation, allowing a constitutive equation to isolate intrinsic relaxation or production. More generally $\mathsf{A}=F{\mathsf{A}}_{\mathrm{ref}}{F}^T$ gives $\stackrel{▽}{\mathsf{A}}=F{\dot{\mathsf{A}}}_{\mathrm{ref}}{F}^T$. Under a time-dependent rigid rotation $Q$, its value transforms as $Q\stackrel{▽}{\mathsf{A}}{Q}^T$, making it an objective time derivative. This is a suitable derivative for this transport type; symmetry alone does not make it the unique possible objective time derivative.