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The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum – a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure, flow velocity, density, and temperature are at least weakly differentiable.
This simple model is the basis for the "law of the wall", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small pressure gradients. More general turbulence models have evolved over time, with most modern turbulence models given by field equations similar to the Navier–Stokes equations.
The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law.It deals with the case of linear elastic materials.Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used.
This is constitutive equation is also called the Newtonian law of viscosity. Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state .
This means that the capillary pressure for a drainage process is different from the capillary pressure of an imbibition process with the same fluid phases. Hysteresis does not change the shape of the governing flow equation, but it increases (usually doubles) the number of constitutive equations for properties involved in the hysteresis.
This constitutive equation is also called the Newton law of viscosity. The total stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} can always be decomposed as the sum of the isotropic stress tensor and the deviatoric stress tensor ( σ ′ {\displaystyle {\boldsymbol {\sigma }}'} ):
For small-scale flows (such as those seen in porous media mechanics), operating with the true speed of sound can lead to unacceptably short time steps. It is therefore common to raise the lattice Mach number to something much larger than the real Mach number, and compensating for this by raising the viscosity as well in order to preserve the ...
In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a non-linear convection term and a non-linear and non-local pressure gradient term. These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.