**Physical principle is Newton’s second law, which states that the rate of change of momentum of a body is equal to the net force acting on it**:

For a fluid element of unit volume moving within a flow, Applying material derivative ,

In the Cartesian coordinates,

We can distinguish between two kinds of forces acting on a fluid element:

**Body forces.***They act directly on the mass of the fluid and originate from a remote source. The examples are the gravity, electric*also belong to this list. The total body force acting on a fluid element is proportional to its mass. In the following, we will assume that the body forces are lumped together into a net force of strength f per unit mass, so that the force per unit volume is ρf.

(Coulomb), magnetic, and Lorentz forces. Fictitious centrifugal and Coriolis forces, which appear when the flow is described in a rotating reference frame,**Surface forces.***They are the pressure and friction forces acting between neighboring fluid elements and between a fluid element and an adjacent wall.*It is shown in the fluid dynamics books that the vector field of surface forces can be represented as divergence of a symmetric 3 × 3 tensor called the stress tensor τ . Its component τij can be seen as the i-component of the surface force acting on a unit area surface, which is normal to the j -axis of the Cartesian coordinate system. Here and in the rest of the book we assume that the values 1, 2, and 3 of indices i and j correspond to the Cartesian coordinates x, y, and z . The diagonal elements τii cause extension/ contraction of the fluid element, while the off-diagonal elements are responsible for its deformation by the shear

The Newton’s second law can be written for a fluid element of unit volume as

The stress tensor can be separated into the isotropic pressure part, which is always present, and the viscous (friction) part, which exists only in flowing fluid and must be zero if the fluid is at rest:

where

is the Kroneker delta-tensor.

For the equations to fully describe the flow, a model for the viscous stresses σij has to be introduced. ** Newton** was first to suggest that the shear stress must be proportional to the velocity gradient. This was later developed by Stokes into the linear model for the stress tensor:

where μ and λ are the first and second viscosity coefficients and we used ui with i = 1, 2, 3 for the velocity components u, v, w. Note that

in an incompressible fluid with ∇ · V = 0, the term with the second viscosity disappears. For compressible fluids, it is generally believed

that λ = −2/3μ is an accurate approximation except for interior of shock waves in hypersonic flows and for absorption and attenuation of acoustic waves.

The model does not have a fully satisfactory theoretical justification. It has, however, being validated in experiments and simply in everyday practice of applying the resulting equations.

The fluids whose behavior satisfies the model are called Newtonian. There are non-Newtonian fluids that behave quite differently (e.g., polymer melts and solutions, human blood at high shear stress, etc.).

Using the second viscosity assumption we obtain the final form of the momentum conservation

equations, the **Navier-Stokes equations**:

The equations can be written in a shorter form if we introduce the rate of strain tensor with components

and use the Einstein summation convention, according to which repeated indices in a term imply summation over all their possible values (1, 2, 3 in our case):

*For the special case of an incompressible fluid with constant viscosity coefficient μ, the Navier-Stokes equations become*

Another special case is that of an inviscid fluid with μ = λ = 0, for which the so-called Euler equations are valid:

Of course, both ABOVE EQUATIONS must be understood as idealizations, strictly speaking, achievable only as asymptotic limits of flows with very low compressibility and very small viscosity, respectively. This does not prevent them from being widely used as approximations