Abstract

We initiate the systematic study of a particularly tractable and physically motivated hyperpressure constitutive relation for second-gradient incompressible viscous fluids. As the simplest relation within a broader family, it yields models that are physically and mathematically attractive. The resulting framework for second-gradient incompressible viscous fluids with constant viscosity is further extended in a novel way to incorporate pressure-dependent viscosities. We show that for the pressure-dependent viscosity model, the inclusion of second-gradient effects and the constitutive relation for the hyperpressure guarantee the ellipticity of the governing pressure equation, in contrast to previous models rooted in classical continuum mechanics. The constant viscosity model is applied to steady cylindrical flows, where explicit solutions are derived under both strong and weak adherence boundary conditions. In each case, we establish convergence of appropriately nondimensionalised velocity profiles to the classical Navier–Stokes solutions as the model’s characteristic nondimensionalised length scales tend to zero.

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