Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 36E (Fluid Dynamics II)

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.

Put $Q=F\cdot x$. Differentiate the given Stokeslet velocity:Symmetrizing and subtracting the pressure term cancels the diagonal terms, leaving the Cauchy stress tensorOn the radius-$R$ surface with outward sphere normal $n=x/R$, the force exerted by the exterior fluid on the sphere is the tractionUse $dS=R^{2}d\Omega$ and ${\int }_{S^2}{n}_i{n}_{j}d\Omega =(4\pi /3){\delta }_{ij}$ to findThe $R^2$ factors cancel, proving independence of radius. This calculation uses the printed far-field Stokeslet; higher multipoles of the exact sphere flow have zero net force through enclosing surfaces.

For constant viscosity, differentiating $\sigma =-pI+\mu (\nabla u+\nabla u^T)$ gives $\nabla \cdot \sigma =-\nabla p+\mu {\nabla }^{2}u$, since $\nabla \cdot u=0$. This is zero by the body-force-free Stokes equations.

Apply the divergence theorem to the annular fluid. Its outer normal is $+n_R$, but its inner normal is minus the sphere's outward normal. Therefore $0={\int }_{S_R}\sigma n_{R}dS-{\int }_{S_a}\sigma n_{a}dS$. The second integral is the hydrodynamic force on the inner sphere, $-F$, because that sphere exerts $F$ on the fluid. Thus again $G=-F$, without differentiating the far field. Local momentum conservation carries the same total force through every enclosing surface.