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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 isIts 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 extensionThis 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: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 ${\eta }_m{\nabla }^{2}M$ alone.