A rephrased form of Navier-Stokes equations is performed for incompressible, three-dimensional, unsteady flows according to Eulerian formalism for the fluid motion. In particular, we propose a geometrical method for the elimination of the nonlinear terms of these fundamental equations, which are expressed in true vector form, and finally arrive at an equivalent system of three semilinear first order PDEs, which hold for a three-dimensional rectangular Cartesian coordinate system. Next, we present the related variational formulation of these modified equations as well as a general type of weak solutions which mainly concern Sobolev spaces.

The analytical solutions of the initial or boundary value problems of Mathematical Physics, although they cannot be directly inserted into the calculation procedure dealing with conceptual design by engineering viewpoint, still have the advantage compared with the numerical methods of domain discretization and boundary elements that help us to deduce qualitative information (e.g., asymptotic behaviour or stability) for the set of the solutions of the ODE or PDE which describe the evolution of a physical problem [

It is well known that Navier-Stokes equations are nonlinear in nature and therefore they are very difficult to be solved analytically. For a detailed study of the nonlinearity of these equations, one may refer to the concepts and results which are performed in excellent monographs by Ladyzhenskaya and Lions [

Unfortunately, only a few analytical works are currently present in literature regarding these equations. One of them is the transformation of Navier-Stokes equations to Schrödinger equation by an application of Riccati equation [

Besides, an analytical solution to three-dimensional incompressible Navier-Stokes equations in Cartesian form has been presented in [

Nevertheless, the existence of weak solutions to the incompressible Navier-Stokes equations in dimension 3 was proved long ago by Leray and Hopf [

On the other hand, for the application of Computational Fluid Dynamics (CFD) approaches to unbounded domains (e.g., a half space or the complementary set of a finite space), the contrivance of artificial boundaries is generally demanded. Therefore, one has to suppose primarily that either the shape of the real boundary or the rates of a field quantity along the artificial boundary are considered as known beforehand. Besides, the creation of a finite element grid constitutes a difficult geometrical problem, perhaps more difficult as it was proved than the circumstantial physical problem that a CFD method is called to solve.

The key point here is that a mesh must be predefined to provide a certain relationship between the nodes, which is the base of the formulation of these conventional numerical methods [

However, it is known from modern Algebra [

In fact, every numerical solution by means of domain discretization methods is the approximation of the functional values at the mesh points, which evidently are discrete points.

However, we have to elucidate that if one does not prove the independency of the numerical solution for any possible distribution of the mesh points, then the spatial coordinates will be functionally dependent on each other something opposite with Eulerian formalism.

In other words, the resulting numerical solution should not depend on the selection of the binary relation of equivalence which relates the discrete variables

To overcome this difficulty, new discretization methods have been introduced such as mesh-free methods and domain-free discretization method. The main concept for mesh-free methods is to establish a system of algebraic equations for the whole problem domain without the use of a predefined mesh. Mesh-free methods use a set of nodes scattered within the problem domain as well as sets of nodes scattered on the boundaries of the domain to represent (not discretize) the problem domain and its boundaries. These sets of scattered nodes do not form a mesh, which means that no information on the relationship between the nodes is required, at least for field variable interpolation [

Meanwhile, domain-free discretization method directly solves partial differential equations in the standard coordinate system. It can be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The primal idea of this method is inspired by the analytical method. Consequently, the differential equation and its discrete form are irrelevant to solution domain. Therefore, the discrete form of PDEs can involve some points outside the solution domain or may not be the mesh points. The functional values at these points can be calculated by finding the approximate form of the solution [

In a Cartesian frame of reference, the fundamental equations of mass and momentum conservation for the generic case of an unsteady, incompressible, and viscous flow field are represented in vector terms in a domain

In the sequel, let us operate at both members of (

Obviously, the vector

Letting the rate of the arc

Concurrently, it is also known [

Hence the following statement holds:

Moreover, let us also make the additional assumption that each time rate of the vector

It is known from Vector Calculus that provided two arbitrary position vectors

Subsequently, the following claim can be formulated:

Besides, the curl of velocity can be modified as follows:

Here, one may consider that these statements can be also applicable for each time rate of the vector function

Thus, (

Also, according to identity (

Then, we can perform (

Also, we can remark that streamlines indicate local flow direction, not speed, which usually varies along a streamline.

In continuing, let us focus on relationship (

Thus, the initial system of the governing equations (

We can also pinpoint that in (

To apply this equation to the distribution of velocity field of a particular flow, it is necessary to define boundary conditions on solid surfaces if they are present and at infinity if a uniform stream is involved. This uniform stream usually constitutes a rigid lid for the flow, through which only molecular mass transfer can be carried out [

One exception, when mass transfer across the free stream surface takes place, concerns two-dimensional Prandtl’s boundary layer flows [

For viscous flows, the no-slip and kinematic boundary conditions on a solid surface require that the fluid adjacent to the surface has the velocity of the surface which in a motionless Cartesian frame of reference is zero.

However, speaking for the no-slip condition we should elucidate that there are some exceptions concerning some types of laminar and turbulent flows [

Besides, since no flow occurs across a streamline, therefore a streamline meets the same stipulation as a solid boundary and also close to a boundary wall the flow direction must be parallel to the boundary; one can consider, without violating the nature of the investigated phenomenon, the solid boundary as a stream surface of the investigated flow.

This conclusion arises from the fact that in any three-dimensional unsteady flow field the instantaneous streamlines are the orbits of the vector

Hence, since according to (

On the other hand, it is known that for a given instant of time a stream surface of a flow field, which, roughly speaking, is defined as a bundle of neighbouring streamlines, is created by a streamline which moves along a plane curve remaining parallel to a given line. This curve is the directrix of the stream surface and the stream line is the generator. Obviously, the directrix curve is not unique because any plane intersects a given stream surface along a plane curve that can serve as directrix.

However, since there is only one streamline passing through any particular velocity point of a flow field, but infinite stream surfaces that can contain it, the notion of the principal stream surface is more appropriate to describe the solid boundary of a flow field [

This definition extends to the case of the solid boundary of a flow field for all rates of the variable of time on the interval

Furthermore, ((

Consequently, whenever the no-slip and kinematic conditions for a solid boundary hold, (

Meanwhile, we should mention that sometimes it is convenient to express a vector equation such as (

Nevertheless, in situations where an injective mapping is necessary between Cartesian and cylindrical or spherical polar coordinates, one should primarily exclude the

The equation of a curve in cylindrical or spherical polar coordinates may be satisfied by points whose ordered triples are not in accordance with the above restrictions. This may happen because such an equation is generally a binary relation of the variables

According to the above data and given that (

On the other hand, (

These flows are divided into the following two major categories.

When the inertial forces are not neglected: this implies that viscous forces have the same order of magnitude as inertial and pressure gradient forces [

When the inertial forces are a priori neglected (e.g., fully developed duct flows, flows through porous media, groundwater movement, etc.): then (

In addition, as is known from the literature [

A qualitative property which automatically emerges is that, along any streamline of such a flow field, the sum

Suggestively, let us examine an open channel creeping flow in a rectangular parallelepiped (length

Also, the pressure at the free surface coincides with atmospheric pressure which is conventionally taken to be zero.

Hence, the Dirichlet conditions which complete (

Consequently, by inserting (

In continuing, let us return to (

This basic assumption is actually in accordance with the validity of the general principle of separation of motion for each axis of a Cartesian coordinate system, which holds either for individual particles or for Continuum media.

Let us also remark that any multivalued function

In the case of bounded domains a set is referred to as compact if and only if it is closed and bounded. Hence, the conceptual supposition that

Therefore, to derive the variational forms of the semilinear differential equations ((

Hence, we deduce

Suggestively, let us focus on (

Thus, we place

However, a shortcoming of (

Nevertheless, the derivatives here are generalized derivatives and the choice of spaces is made to ensure the existence of all integrals. Therefore, every classical solution of (

As it was previously stated, (

On the other hand, a variational formulation of (

What classical differentiability properties does the weak solution fulfil?

In what sense does the weak solution satisfy the circumstantial boundary conditions?

Moreover, another item that has not been thoroughly investigated so far is whether a weak solution can develop singularities in finite time, even considering smooth initial data [

The objective of this paper was to propose a method to eliminate the nonlinear terms of Navier-Stokes equations for three-dimensional, incompressible viscous flows, resulting in a modified form which holds along the streamline network.

The main characteristic of this recasted form is that it does not contain the viscous terms, and therefore this equivalent representation can hold either for frictionless or for viscous flows as well as for any type of shear or free turbulence.

A variational formulation of this relationship was also presented; hence one may obtain weak solutions in Sobolev spaces.

In closing, we can also remark that a weak formulation of this original problem provides us with related methods (e.g., Galerkin method) for approximating its circumstantial weak solution in a properly chosen finite-dimensional subspace of functions with good approximation properties that are also suitable for numerical computations.

The author declares that there is no conflict of interests regarding the publication of this paper.