  The moving discontinuous Galerkin finite element method with interface
condition enforcement (MDG-ICE) is applied to the case of viscous flows. This
method uses a weak formulation that separately enforces the conservation law,
constitutive law, and the corresponding interface conditions in order to
provide the means to detect interfaces or under-resolved flow features. To
satisfy the resulting overdetermined weak formulation, the discrete domain
geometry is introduced as a variable, so that the method implicitly fits a
priori unknown interfaces and moves the grid to resolve sharp, but smooth,
gradients, achieving a form of anisotropic curvilinear $r$-adaptivity. This
approach avoids introducing low-order errors that arise using shock capturing,
artificial dissipation, or limiting. The utility of this approach is
demonstrated with its application to a series of test problems culminating with
the compressible Navier-Stokes solution to a Mach 5 viscous bow shock for a
Reynolds number of $10^{5}$ in two-dimensional space. Time accurate solutions
of unsteady problems are obtained via a space-time formulation, in which the
unsteady problem is formulated as a higher dimensional steady space-time
problem. The method is shown to accurately resolve and transport viscous
structures without relying on numerical dissipation for stabilization.
