Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 36E / i / Solution

An isotropic fourth-order tensor has the form $A_{ijkl}=a{\delta }_{ij}{\delta }_{kl}+b{\delta }_{ik}{\delta }_{jl}+c{\delta }_{il}{\delta }_{jk}$. Contracting with the velocity gradient gives $a{\delta }_{ij}\nabla \cdot u+b{\partial }_j{u}_i+c{\partial }_i{u}_j$. Incompressibility removes the first term. Symmetry of the stress for arbitrary incompressible velocity gradients, including antisymmetric ones, forces $b=c$. Writing their common value as the dynamic viscosity $\mu$ yieldsThe asserted constant is the homogeneous material coefficient of this Newtonian constitutive law. Isotropy alone fixes its tensor form, not spatial constancy if the material's viscosity were allowed to vary.