= Solution
Use the nonrelativistic moving-conductor Ohm law in the form
$$
\mathbf E=\frac{\mathbf j}{\sigma}-\mathbf u\times\mathbf B.
$$
The <Maxwell equations> give <Faraday's law of induction>, $\partial_t\mathbf B=-\nabla\times\mathbf E$. For a nonmagnetic conductor with constant permeability $\mu_0$, the quasistatic <Ampere-Maxwell law> gives $\mathbf j\simeq(\nabla\times\mathbf B)/\mu_0$. This step neglects <displacement current>, requiring it to be small compared with conduction current; nonrelativistic velocity alone is not the complete justification.
For spatially uniform scalar <electrical conductivity> $\sigma$,
$$
\partial_t\mathbf B
=\nabla\times(\mathbf u\times\mathbf B)
-\frac{1}{\mu_0\sigma}\nabla\times(\nabla\times\mathbf B).
$$
Apply the identity $\nabla\times(\nabla\times\mathbf B)=\nabla(\nabla\cdot\mathbf B)-\nabla^2\mathbf B$ and <Gauss's law for magnetism>, $\nabla\cdot\mathbf B=0$. The <resistive induction equation> is
$$
\boxed{\frac{\partial\mathbf B}{\partial t}
\simeq\nabla\times(\mathbf u\times\mathbf B)
+\eta\nabla^2\mathbf B,\qquad
\eta=\frac1{\mu_0\sigma}.}
$$
The first term transports and deforms the <magnetic field> with the moving fluid; the second diffuses it. The <magnetic diffusivity> $\eta$ has units $\mathrm{m^2\,s^{-1}}$, so both terms have units tesla per second. No assumption of <incompressible flow> was needed. If $\sigma$ varies spatially, the <conductivity-gradient term in magnetic induction>, $-\nabla\eta\times(\nabla\times\mathbf B)$, must also be retained.
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