  The motion of water is governed by the Navier-Stokes equations, which are
complemented by the continuity equation to ensure local mass conservation. In
this work, we construct the relativistic generalization of these equations
through a gradient expansion for a fluid with conserved charge in a curved
$d$-dimensional spacetime. We adopt a general hydrodynamic frame approach and
introduce the Irreducible-Structure (IS) algorithm, which is based on
derivatives of both the expansion scalar and the shear and vorticity tensors.
By this method, we systematically generate all permissible gradients up to a
specified order and derive the most comprehensive constitutive relations for a
charged fluid, accurate to third-order gradients. These constitutive relations
are formulated to apply to ordinary, non-conformal, and conformally invariant
charged fluids. Furthermore, we examine the hydrodynamic frame dependence of
the transport coefficients for a non-conformal charged fluid up to the third
order in the gradient expansion. The frame dependence of the scalar, vector,
and tensor parts of the constitutive relations is obtained in terms of the
field redefinitions of the fundamental hydrodynamic variables. Managing these
frame dependencies is challenging due to their non-linear character. However,
in the linear regime, these higher-order transformations become tractable,
enabling the identification of a set of frame-invariant coefficients. An
advantage of employing these coefficients is the possibility of studying the
linear equations of motion in any chosen frame and, hence, we apply this
approach to the Landau frame. Subsequently, these linear equations are solved
in momentum space, yielding dispersion relations for shear, sound, and
diffusive modes for a non-conformal charged fluid, expressed in terms of the
frame-invariant transport coefficients.
