Analysis of Navier-Stokes equations
The mathematics is not possible to find a solution of the Navier-Stokes equations
up to now. Could be found any reasons?
To show a possible reason, in the following the shear stress of the Navier-Stokes
equations will be analysed. This occurs applying the Navier-Stokes model in an area
of a simple flow field.
Needed basics and formula.
For this proof the following condition will be used:
The 3. Newtonian law „actio = reactio“ describes mechanical
This means, that the forces and momenta
at the section plane are equal in absolut value and directly opposed.
The method of sections confirms to this law. The inner forces and momenta
come at the section plane become outer ones. If a System is separated into subsystems
and the physical model is applied to them, the addition of the forces and momenta
must be equal to the forces and momenta applying to the origin system.
First the occurring shear stresses will be described. This based on the derivation
of the book “Prandtl's Essentials of Fluid Mechanics“  from the chapter
5 „Fundamental Equations of Fluid Mechanics“ in the subchapter 5.2 „Navier-Stokes
The occurring stresses at a volume element in a flow are shown in figure 1.
Figure 1: Shear vectors at the system boundaries 
The normal stresses can be converted with the term 
Newtonian fluids will be described by following equations :
Shear stresses :
with the symmetry condition :
Specification of the flow field
To simplify subjects and without loss of generality a laminar and steady flow will
be considered with
The flow field will have only a velocity component in the x-direction. Outer forces
won't be taken into account in this example.
The equations 1.0.0 to 1.0.2 will be simplified as:
The normal stresses yield to -p. They have no significance in this treatment and
won't be considered in the following part.
The equations 1.1.0 to 1.1.2 can be expressed as:
The shear stresses exist only in the xz-pane in this example.
Shear stresses at the origin system
In figure 2 the shear stresses at the system which described in equations 2.2.1,
2.3.1 and 1.2.0 are plotted. The shear stresses only depend on the velocity gradient
Figure 2: Stresses in a simple flow field
Separation into subsystems
For the examination the considered physical model will be orthographically separated
to the flow direction into two subsystems “1” and “2”. The
generated subsystems have a common section plane.
Figure 3: System separation
Shear stresses at the subsystems
At both subsystems the shear stresses will be determined. The equations and the
physical model above are also valid for the subsystems. Due to the configuration
of the flow field in 2.0.0 and with the equations 2.2.1 and 1.2.0 the shear tresses
can be described as:
Figure 4: Shear stresses of the subsystems
Superposing the subsystems
After determination the shear stresses at the subsystems, they will be superposed.
This is shown in figure 5. At the section plane of both subsystems the following
stresses must be considered:
Figure 5: Superposition of subsystem "1" and "2"
The resulting stress follows to:
This is shown in figure 6.
Figure 6: Resulting inner stress
At the section plane of the subsystems yields a resulting difference stress dτxz. This phenomenon doesn't
occur at a separation parallel to the x-direction.
The resulting shear stress dτxz
doesn't exist in the origin system.
That means, that the flow model of “Navier-Stokes” can't meet the requirement
of the 3. Newtonian law “Actio = Reactio“. The sums of the subsystems
stresses are not equal to the particular stresses of the complete system.
From mathematically sight it can be interpreted as a discontinuous of the shear
stresses at the systems section planes. The shear stresses always contain a constant
component being independent from the direction of their normal. This will be caused
by the symmetry condition.
As shown above the physical model of “Navier-Stokes” violates the 3.
Newtonian law. The symmetry condition of the shear stresses can be detected as the
reason. Due to this, the components of the shear stresses behaviour orthographically
to the flow direction is discontinuous. This caused that no complete solutions can
be found for this system of equations.
μ : dynamic viscosity [Ns/m2]
ρ : density [kg/m3]
σ : normal stress [N/m2]
τ : shear stress [N/m2]
p : pressure [N/m2]
u : component of velocity x-direction [m/s]
v : component of velocity y-direction [m/s]
w : component of velocity z-direction [m/s]
: velocity vector
denotes direction of the surface normal
denotes direction of vector
In a system from the stresses yield:
the normal stresses can be written as
- Prandtl's Essentials of Fluid Mechanics SE /Springer