  The Navier-Stokes (NS) partial differential equations, as the governing
equation of fluid dynamics, are based on particles with zero volume. In such a
condition, conservation law of moment of momentum is automatically satisfied,
and thus NS equations only contain conservation of mass, momentum and energy.
Because of the difficulty to get an analytical solution of NS equations,
scientists develop finite element method (FEM) and finite volume method (FVM)
to obtain approximate solutions. However, these methods are dependent on the
size of finite volumes, which is not zero. Considering the size of the finite
volume, conservation law of moment of momentum is no longer automatically
satisfied and it is necessary to develop a new control equation to take effect
of momentum into account. In this paper, new relations of reciprocal shear
stresses are derived from conservation law of moment of momentum, and so are
new constitutive relations and modified NS equations. The Liutex based NS
equation may be the universal governing equations for both laminar and
turbulent flow which is dominated by vortices and conservation of moment of
momentum must be satisfied.
