Cauchy momentum equation

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

Surface forces act on walls of the cubic fluid element.

For each wall, the X component of these forces was marked in the figure with a cubic element (in the form of a product of stress and surface area e.g.

The minus sign is due to the fact that a vector normal to this wall

(we use similar reasoning for stresses acting on the bottom and back walls, i.e.:

Applying Newton's second law (ith component) to a control volume in the continuum being modeled gives:

Then, based on the Reynolds transport theorem and using material derivative notation, one can write

Here j and s have same number of dimensions N as the flow speed and the body acceleration, while F, being a tensor, has N2.

Regardless of what kind of continuum is being dealt with, convective acceleration is a nonlinear effect.

Convective acceleration is represented by the nonlinear quantity u ⋅ ∇u, which may be interpreted either as (u ⋅ ∇)u or as u ⋅ (∇u), with ∇u the tensor derivative of the velocity vector u.

[7] The convective acceleration (u ⋅ ∇)u can be thought of as the advection operator u ⋅ ∇ acting on the velocity field u.

The vector calculus identity of the cross product of a curl holds:

where the Feynman subscript notation ∇a is used, which means the subscripted gradient operates only on the factor a. Lamb in his famous classical book Hydrodynamics (1895),[8] used this identity to change the convective term of the flow velocity in rotational form, i.e. without a tensor derivative:[9][10]

In fact, in case of an external conservative field, by defining its potential φ:

(where I is the identity tensor), and the Euler momentum equation in the steady incompressible case becomes:

that is, 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 Lamb form is also useful in irrotational flow, where the curl of the velocity (called vorticity) ω = ∇ × u is equal to zero.

Here ∇p is the pressure gradient and arises from the isotropic part of the Cauchy stress tensor.

The anisotropic part of the stress tensor gives rise to ∇ ⋅ τ, which usually describes viscous forces; for incompressible flow, this is only a shear effect.

where I is the identity matrix in the space considered and τ the shear tensor.

Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the flow motion.

[12] For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density.

The vector field f represents body forces per unit mass.

Typically, these consist of only gravity acceleration, but may include others, such as electromagnetic forces.

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

The limit of high Froude numbers (low external field) is thus notable for such equations and is studied with perturbation theory.

For asymmetric stress tensors, equations in general take the following forms:[2][3][4][14]

Below, we write the main equation in pressure-tau form assuming that the stress tensor is symmetrical (

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).
In the top graph we see approximation of function (blue line) using a finite difference (yellow line). In the bottom graph we see "infinitely many times enlarged neighborhood of point " (purple square from the upper graph). In the bottom graph, the yellow line is completely covered by the blue one, thus not visible. In the bottom figure, two equivalent derivative forms have been used: ], and the designation was used.
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).