= Solution
The common part of the <magnetohydrodynamic-viscoelastic analogy> is stretching and advection of a line element. With the component-first <velocity gradient> $L_{ij}=\partial_j u_i$, the ideal tensor equation is
$$
D_tM-LM-ML^T=0.
$$
Its left side is an <upper-convected derivative>, the same deformation law that appears for a <polymer> <conformation tensor> when its relaxation is neglected.
There are two important physical differences. First, the <solenoidal magnetic-field constraint> prevents a <magnetic field line> from having an isolated source or sink inside the fluid. A <polymer> strand can have molecular endpoints, so there is no corresponding no-endpoint constraint on the strands. Second, a stretched <polymer> has a preferred relaxed distribution of configurations and a finite <polymer relaxation time>; it can recoil relative to the solvent through molecular drag and thermal motion. An ideal <magnetic field> has no analogous local tendency to return to a prescribed coil size. Real <polymers> also have a finite contour length, a limitation absent from the ideal field-stretching law.
For example, a normalized <Hookean dumbbell conformation equation> has the dissipative extension
$$
\boxed{D_tC-LC-CL^T=-\frac1\tau(C-I).}
$$
This is local relaxation toward an isotropic thermal configuration. It remains nonzero for a spatially uniform stretched $C$. By contrast, constant <magnetic diffusivity> $\eta_m$ adds $\eta_m\nabla^2\mathbf B$ to the <resistive induction equation>. The product rule gives the <resistive evolution of the magnetic second-moment tensor>:
$$
\boxed{D_tM_{ij}-L_{in}M_{nj}-M_{in}L_{jn}
=\eta_m\nabla^2M_{ij}
-2\eta_m(\partial_nB_i)(\partial_nB_j).}
$$
The magnetic correction involves spatial gradients, vanishes for a uniform <magnetic field>, and does not create field-line endpoints: $\partial_t(\nabla\cdot\mathbf B)=\eta_m\nabla^2(\nabla\cdot\mathbf B)$. It permits slippage and <magnetic reconnection> while preserving the no-monopole constraint. After averaging, its gradient-product term generally needs additional information beyond the averaged $M$. One cannot replace it indiscriminately by the local <polymer> relaxation term, or even by $\eta_m\nabla^2M$ alone.
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