Incompressible isotropic case

For an incompressible and isotropic Newtonian fluid, the viscous stress is related to the strain rate by the simpler equation

where

is the shear stress ("drag") in the fluid,

is a scalar constant of proportionality, the shear viscosity of the fluid

is the derivative of the velocity component that is parallel to the direction of shear, relative to displacement in the perpendicular direction.

If the fluid is incompressible and viscosity is constant across the fluid, this equation can be written in terms of an arbitrary coordinate system as

where

is the th spatial coordinate

is the fluid's velocity in the direction of axis

is the th component of the stress acting on the faces of the fluid element perpendicular to axis .

One also defines a total stress tensor ) that combines the shear stress with conventional (thermodynamic) pressure . The stress-shear equation then becomes

For anisotropic fluids

More generally, in a non-isotropic Newtonian fluid, the coefficient that relates internal friction stresses to the spatial derivatives of the velocity field is replaced by a nine-element viscosity tensor .

There is general formula for friction force in a liquid: The vector differential of friction force is equal the viscosity tensor increased on vector product differential of the area vector of adjoining a liquid layers and rotor of velocity:

where - viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components - turbulence eddy viscosity

42. Pressure coefficient at stagnation point for incompressible continuous flow and for hypersonic flow according Newtonian theory.

The pressure coefficient is a parameter for studying the flow of incompressible fluids such as water, and also the low-speed flow of compressible fluids such as air. The relationship between the dimensionless coefficient and the dimensional numbers is ^{[1] [2]}

where:

is the pressure at the point at which pressure coefficient is being evaluated

is the pressure in the freestream (i.e. remote from any disturbance)

is the freestream fluid density (Air at sea level and 15 °C is 1.225 )

is the freestream velocity of the fluid, or the velocity of the body through the fluid

Using Bernoulli's Equation, the pressure coefficient can be further simplified for incompressible, lossless, and steady flow:^[3]

where V is the velocity of the fluid at the point at which pressure coefficient is being evaluated.

This relationship is also valid for the flow of compressible fluids where variations in speed and pressure are sufficiently small that variations in fluid density can be ignored. This is a reasonable assumption when the Mach Number is less than about 0.3.

• of zero indicates the pressure is the same as the free stream pressure.

• of one indicates the pressure is stagnation pressure and the point is a stagnation point.

• of minus one is significant in the design of gliders because this indicates a perfect location for a "Total energy" port for supply of signal pressure to the Variometer, a special Vertical Speed Indicator which reacts to vertical movements of the atmosphere but does not react to vertical maneuvering of the glider.

In the fluid flow field around a body there will be points having positive pressure coefficients up to one, and negative pressure coefficients including coefficients less than minus one, but nowhere will the coefficient exceed plus one because the highest pressure that can be achieved is the stagnation pressure. The only time the coefficient will exceed plus one is when advanced boundary layer control techniques, such as blowing, is used.