Cauchy momentum equation

In convective (or Lagrangian) form the Cauchy momentum equation is written as:

After an appropriate change of variables, it can also be written in conservation form:

The X component of the forces acting on walls of a cubic fluid element (green for top-bottom walls; red for left-right; black for front-back).

Adding forces (their X components) acting on each of the cube walls, we get:

In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations.

An example of convective acceleration. The flow is steady (time-independent), but the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow).

A significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.

In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:

that is, . This leads to a considerable simplification of the Euler momentum equation:

the mass conservation for a steady incompressible flow states that the density along a streamline is constant

The convenience of defining the total head for an inviscid liquid flow is now apparent:

That is, .

the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant

The effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure.

Substitution of these inverted relations in the Euler momentum equations yields:

and the coefficient of skin-friction or the one usually referred as 'drag' co-efficient in the field of aerodynamics:

Cauchy equations in the Froude limit Fr → ∞ (corresponding to negligible external field) are named free Cauchy equations: