We derive explicit, strain-coupled quantum hydrodynamics for electrons in graphene near a Dirac point, starting from the Wigner–BGK equation and using the quantum maximum entropy principle (QMEP) for closure. Working fully in differential operator form, we obtain the quantum Euler and quantum Navier–Stokes (QNS) equations via a Chapman–Enskog expansion to first order in the BGK relaxation-time \(\tau \) (small- \(\tau \) regime, denoted \(O(\tau )\) ) and a semiclassical expansion including terms up to order \(\hbar ^2\) , where \(\hbar \) is the reduced Planck constant. We treat two strain regimes: (I) linear (small) strain modeled by a valley-odd pseudogauge field \(\textbf{A}_s(\varvec{\varepsilon })\) with pseudomagnetic field \(B_s=\nabla \times \textbf{A}_s\) , leading to anisotropic viscosity and Hall-like (odd) viscosity contributions; and (II) nonlinear/geometric strain modeled by a curved-space Dirac Hamiltonian with metric \(g_{ij}=\delta _{ij}+2\varepsilon _{ij}+O(\varepsilon ^2)\) and spin-connection couplings, yielding covariant QNS with curvature/strain-gradient \(O(\hbar ^2)\) corrections to the quantum pressure (Bohm-like) and viscous stress. The \(O(\hbar ^2)\) Euler sector remains finite as the spectral gap \(\Delta \rightarrow 0\) , and we give explicit constitutive integrals for the strain-dependent viscosity tensor. We discuss relevance to viscous-electron experiments and Stokes analogies in graphene.
