## Abstract

Although it is well known that Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity and that in the case of non-vanishing vorticity a first integral of them can be found using the Clebsch transformation for inviscid flow, generalization of the procedure for viscous flow has remained elusive. Accordingly, in this paper, a first integral of the Navier–Stokes equations for steady flow is constructed. In the case of a two-dimensional flow, this is made possible by formulating the governing equations in terms of complex variables and introducing a new scalar potential. Associated boundary conditions are considered, and an extension of the theory to three dimensions is proposed. The capabilities of the new approach are demonstrated by calculating a Reynolds number correction to the laminar shear flow generated in the narrow gap between a flat moving and a stationary wavy wall, as is often encountered in lubrication problems. It highlights the first integral as a suitable tool for the development of new analytical and numerical methods in fluid dynamics.

## 1. Motivation

By finding a general first integral of the equations governing the motion of a fluid, many problems in fluid dynamics can consequently be made more tractable; the most prominent example being Bernoulli’s equation, which emerges as the first integral of Euler’s equations on writing the velocity field ** v** as1.1where

*φ*is a scalar potential. However, despite its obvious advantages, the approach is restricted to inviscid and irrotational barotropic flows. In order to extend the methodology to encompass flows with non-vanishing vorticity, Clebsch (1859) proposed expressing

**in terms of three scalar potentials**

*v**φ*,

*α*and

*β*,1.2with the vorticity given by

**=∇×**

*ω***/2=∇**

*v**α*×∇

*β*/2. This so-called Clebsch transformation (1.2), as reported in standard textbooks (Lamb 1974; Panton 1996), facilitates the integration of Euler’s equations for barotropic flows leading to a set of three equations for the three potentials: a generalized Bernoulli equation for

*φ*and two transport equations for the vortex potentials

*α*and

*β*.

One benefit of the Clebsch transformation is that the resulting set of equations together with the continuity equation is self-adjoint, allowing for a variational formulation (e.g. Seliger & Whitham 1968; Scholle 2004*a*; Scholle & Anthony 2004). A generalization of the Clebsch transformation in the case of baroclinic flows has been formulated by Lin (1963) and Seliger & Whitham (1968), whereas that in the case of electrodynamics is reported by Wagner (2002). In contrast, no potential representation for velocity has yet been established which makes possible the integration of the Navier–Stokes equations.

A different strategy is proposed in this paper to address this deficit: as shown below, instead of employing a potential representation for velocity, a new scalar potential is introduced from which the pressure can be found. Using an alternative formulation of the two-dimensional Navier–Stokes equations in terms of complex variables, a first integral is constructed; a description of how the theory could be extended to the case of three-dimensional flow is provided in appendix B. Corresponding boundary conditions are considered in §3. Finally, in §4, the capability of the new approach is demonstrated by solving semi-analytically, as described in appendix A, a well-known shear flow problem. Conclusions are drawn in §5.

## 2. Field equations

In two dimensions, the Navier–Stokes and continuity equations governing a steady and incompressible flow, assuming that the external force on the fluid is conservative with a given potential energy density *U*(*x*,*y*), are2.1
2.2
and 2.3where *u*_{x} and *u*_{y} denote the velocity field and *p* the pressure field.

### (a) Complex variable formulation

The above field equations can be transformed and written in a more concise form by introducing the following complex variables:2.4The continuity equation (2.3) then becomes2.5where ℑ denotes the imaginary part of the subsequent complex expression and *u* is the complex velocity field given byBy taking the linear combination (2.1)+i(2.2) and using equation (2.5), followed by appropriate mathematical manipulation, the following complex Navier–Stokes equation2.6is obtained.

### (b) Potential representation and first integral

The complex equation (2.6) is transformed to an integrable form by introducing a new real-valued potential^{1} and writing the expression inside the square brackets of the first term as follows:2.7Note that in the case of a irrotational inviscid flow, the terms on the left-hand side of equation (2.7) vanish as they satisfy Bernoulli’s equation. In the general case of a viscous flow with non-vanishing vorticity, equation (2.7) is a potential representation of the deviation from Bernoulli’s equation and is very different from the classical Clebsch transformation.

The existence of the potential *Φ* is easily proved: for a given non-vanishing expression of the form (2.7), *Φ* is obtained in terms of *ξ* and as2.8As the above double integral is indefinite, the potential *Φ* is not unique: by adding any holomorphic function *f* and its complex conjugate according to2.9the expression on the left-hand side of equation (2.7) remains unchanged. By virtue of the gauge transformation (2.9), a degree of freedom exists which is used subsequently to simplify the resulting field equation.

After inserting equation (2.7) into the complex Navier–Stokes equation (2.6), the following integrable equation arises:which, after integration with respect to , becomes2.10with the unknown function *g* depending on the complex coordinate *ξ*. Next, applying the gauge transformation (2.9) to equation (2.10) gives2.11Obviously, by choosing the gauge function *f* as the second antiderivative of i*g*(*ξ*), i.e. *f*′′(*ξ*)=i*g*(*ξ*), the right-hand of equation (2.11) vanishes, and2.12forms a first integral of the Navier–Stokes equations.

The real-valued continuity equation (2.5) and the complex-valued first integral of the Navier–Stokes equation (2.12) give a complete set of field equations for the real-valued potential *Φ* and the complex velocity field *u*. Boundary conditions, necessary to close the problem, have to be formulated individually depending on the flow type, the two most common of which are discussed in §3.

Having solved the set of field equations (2.5) and (2.12) subject to attendant boundary conditions, the pressure as a dependent variable is obtained from equation (2.7).

### (c) The use of a streamfunction

In some cases, it is convenient to introduce a streamfunction. In terms of complex variables, the complex velocity field *u* is, according to,2.13represented by a real-valued streamfunction *ψ*, such that the continuity equation (2.5) is fulfilled identically. Having inserted equation (2.13) into equation (2.12), the first integral of the Navier–Stokes equations2.14remains the only field equation to be solved. The above complex-valued equation, for the two real-valued fields *ψ* and *Φ*, is of second order; whereas the original Navier–Stokes equations (2.1) and (2.2) become third-order equations if written in terms of the streamfunction.

### (d) Real-valued representation

In §2*b*, complex coordinates were used for convenience only. By re-transformation, the first integral of the Navier–Stokes equations can be represented alternatively in terms of the real-valued coordinates *x* and *y* and corresponding fields *u*_{x}, *u*_{y} and *Φ*. To this end, equation (2.12) is decomposed into its real and imaginary parts:2.15and2.16As is well known, the original Navier–Stokes equations (2.1) and (2.2) represent two components of a vector equation. In contrast to this, equations (2.15) and (2.16) are tensorial in character: when a rigid rotation is applied to the coordinates *x* and *y*, they can be interpreted as the two^{2} independent components of the tensor equation2.17where *i*,*j*,*k*∈{1,2}, Einstein’s summation convention is used for any index appearing twice in a product and *x*=*x*_{1}, *y*=*x*_{2}, *u*_{x}=*u*_{1} and *u*_{y}=*u*_{2}; *δ*_{ij} denotes the Kronecker-delta symbol.

The above tensor equation (2.17) is supplemented by the scalar continuity equation2.18which, if written in terms of the stream function, as defined in §2*c*, requires *u*_{1}=∂*ψ*/∂*x*_{2} and *u*_{2}=−∂*ψ*/∂*x*_{1}.

After the set of equations (2.17) and (2.18) has been solved, the scalar relation (2.7), which in tensor notation reads as2.19can be used to calculate the pressure *p* from the potential *Φ*.

### (e) Three-dimensional flow

Although the theory presented above is based upon a complex variable formulation of the field equations and at first glance appears restricted to two-dimensional problems, an extension of the theory to three-dimensional flow, based on tensor calculus, is proposed in appendix B.

## 3. Boundary conditions

Boundary conditions, as necessary, have to be formulated individually, depending on the type of boundary present for a particular problem. The two most prominent/common occurrences are discussed below.

### (a) Solid walls

Along solid walls, given as *x*_{i}=*b*_{i}(*s*) with arc length *s*, the no-slip condition3.1has to be fulfilled for the velocity field, where *U*_{Bi} is the velocity of the boundary. In contrast, no condition for the pressure field need be considered and consequently no condition for the potential *Φ* at such a boundary has to be formulated.

### (b) Free surfaces

Along a free surface, three boundary conditions have to be considered as its shape, denoted by *x*_{i}=*f*_{i}(*s*) with arc length *s*, is not known *a priori*. For the determination of the free surface shape, *f*_{i}, the kinematic boundary condition3.2must be fulfilled with the normal vector, *n*_{i}, given explicitly as and , where the prime denotes a first-order derivative with respect to *s*. Additionally, the dynamic boundary condition3.3takes into account the stress equilibrium at the fluid boundary, where *t*_{i}(*s*)=*f*_{i}^{′}(*s*) denotes the tangent vector along the surface and *p*_{0} the constant ambient pressure. By replacing the pressure *p* in the Navier–Stokes equations by the difference *p*−*p*_{0}, the ambient pressure is formally set to zero, where *p* is the pressure relative to the surroundings. Making use of equation (2.19), the pressure is eliminated from the dynamic boundary condition (3.3), giving3.4Consider next the first integral of the Navier–Stokes equations as the linear combination (3.4)–(2.17) · *n*_{j}3.5As ∂*u*_{k}/∂*x*_{k} vanishes according to the scalar continuity equation (2.18) and *u*_{j}*n*_{j} vanishes along the free surface according to the kinematic boundary condition (3.2), the velocity is eliminated; thus, equation (3.5) reads3.6Finally, the identitiesare used in order to rewrite equation (3.6) as the total differential3.7where *ε* denotes *ε*_{11}=*ε*_{22}=0, *ε*_{12}=1 and *ε*_{21}=−1. Hence, a first integral has been constructed for the dynamic boundary condition. Integration of the total differential (3.7) implieswith two constants of integration *C*_{i}. By applying the re-gauging for the potential *Φ*,which does not have any effect on the field equations (2.17) and (2.18), the integration constants vanish and the dynamic boundary condition takes the final form3.8of a vector equation for the first-order derivatives of the potential *Φ*.

## 4. Effects of Reynolds number on shear flow adjacent to a wavy wall

The capabilities of the above non-standard theoretical approach are demonstrated by example.

### (a) Formulation and solution

Consider the two-dimensional, steady, isothermal shear flow of an incompressible Newtonian liquid, confined between two horizontally aligned rigid walls, the upper one being flat and moving, the lower one stationary and having a sinusoidal profile, as in the problem investigated by Scholle *et al.* (2009). The solution domain and associated (*x*,*y*)-coordinate system are shown in figure 1 in terms of non-dimensional variables, where *λ*/2*π* is the length scaling employed with *λ* the wavelength. The line *y*=0 corresponds to the mean position of the wavy lower wall, the shape of which is given by . The upper flat wall, moving with speed *U*_{0}, is separated by a non-dimensional mean distance *h* from the stationary lower one.

Such flow has been investigated in the past by solving the full Navier–Stokes equations (a) using standard analytical asymptotic methods (e.g. Malevich *et al.* 2008), (b) numerically, via an accurate finite element formulation as in Scholle *et al.* (2009). In contrast, the problem is now solved in terms of the proposed first integral of the Navier–Stokes equations (2.15) and (2.16). After introducing a streamfunction *ψ* according to equation (2.13), using *U*_{0} for scaling of all velocities and *ηU*_{0}*λ*/2*π* for the scaling of the potential *Φ*, the field equations written in non-dimensional form read4.1and4.2with Reynolds number *Re*=*ϱλhU*_{0}/2*πη*. The corresponding boundary conditions are those of no-slip along the lower and upper walls,4.3
4.4
4.5
and 4.6supplemented by periodic boundary conditions to the left and right of the solution domain for the streamfunction.

The type of flow under investigation occurs in many problems associated with reducing friction between lubricated contacts having a non-flat profile. As for most such problems inertia effects are comparatively small, attention is focused on the calculation of small inertia corrections to the corresponding Stokes flow limit. The semi-analytic approach adopted for this purpose, involving a set of simplified equations, is described in detail in appendix A: a symmetry analysis results, to leading order, in a reduced set of self-adjoint equations facilitating a variational formulation. Using Ritz’s direct method, reasonably good approximate solutions can be obtained with considerably little effort.

### (b) Results and discussion

As the intention here is to establish the methodology rather than to reproduce the extensive parameter study reported in Scholle *et al.* (2009), only one representative case is considered. By choosing *h*=0.2, *a*=0.1 to define the flow geometry: (a) the validity of the lubrication solution (A7) for the corresponding Stokes flow is guaranteed due to a sufficiently small aspect ratio *a*; (b) as *a*/*h*=1/2, the appearance of an eddy within the flow is a certainty. As described in appendix A, the symmetry of the flow is broken by Reynolds number effects.

In figure 2*a*,*b*, the streamline pattern obtained via the semi-analytical approach developed in appendix A is shown for Reynolds number *Re*=100,^{3} and compared to the corresponding Stokes flow (*Re*=0) solution. Symmetry breaking due to inertia effects is clearly apparent: the eddy core is shifted in the flow direction, as reported in Scholle *et al.* (2009). A more quantitative validation of the semi-analytical result is provided with reference to figure 2*c*,*d*, which shows the streamline pattern resulting from a finite element solution of the full Navier–Stokes equations. In particular, the semi-analytic solution is observed to capture remarkably well both the shift of the eddy core and the migration of the two triple points due to inertia effects. For the semi-analytical calculations, only 20 coefficients (see appendix A), were used when applying Ritz’s direct method. By refining the procedure using more coefficients, the accuracy and hence agreement can be improved further.

Summarizing the benefits of the first integral approach with respect to Reynolds number corrections for shear flow adjacent to wavy walls, the following should be noted:

— A variational formulation is available that allows Ritz’s direct method to be applied. In the paper of Scholle

*et al.*(2009), Ritz’s direct method was also used, however, only for the case of Stokes flow.— Reasonably good semi-analytical solutions are obtained with considerably less effort compared to other (semi-)analytical methods, such as the laborious approach used by Malevich

*et al.*(2008) for the same flow problem.

## 5. Conclusions

It is shown that using a complex variable formulation together with the introduction of a new scalar potential *Φ*, for the case of steady, two-dimensional viscous flow, allows for a first integral of the Navier–Stokes equations: given conveniently as a complex-valued equation or, alternatively, as a tensor equation in terms of Cartesian coordinates. The convenience of this first integral is especially apparent if the velocity field is expressed in terms of a streamfunction: a set of second-order partial differential equations has to be solved for, instead of a third-order one as in the case of the original Navier–Stokes equations.

Attention is paid to the type of attendant boundary conditions that can arise; in particular, to the no-slip condition present along solid walls and to the kinematic and dynamic boundary conditions that apply at a free surface. The analysis presented also reveals a first integral of the dynamic boundary condition apropos the latter.

Laminar shear flow in the gap separating a moving flat wall from a stationary but sinusoidally varying one is investigated in terms of the constructed first integral of the Navier–Stokes equation. After establishing a variational formulation for the corresponding field equations, Ritz’s direct method is used, with a small number of coefficients, to obtain a semi-analytic solution to the problem for both Stokes flow and when the Reynolds number is non-zero. These are found to be in very good agreement with corresponding FEM solutions of the original full Navier–Stokes equations for the same flow conditions. The benefits from employing a variational formulation are not restricted to analytical approaches to flow problems: in principle, only a few modifications would be required to develop Ritz’s direct method further into a numerical FEM analogue that can be applied to various problems in fluid dynamics.

## Acknowledgements

P.H.G. is grateful to the Deutscher Akademischer Austauschdienst (DAAD) for their financial support and to the University of Bayreuth for its role as host.

## Footnotes

↵1 The term ‘potential’ is used here for an auxiliary field which is introduced in order to transform a given non-integrable field equation into an integrable one. It has to be distinguished from the classical meaning of a scalar field, the gradient of which represents a given vector field.

↵2 A two-dimensional, traceless and symmetric tensor contains only two independent components.

↵3 Note that for the approach to be valid

*Re**Ψ*has to be small compared to*ψ*_{0}not*Re*itself!

- Received March 19, 2010.
- Accepted May 26, 2010.

- This journal is © 2010 The Royal Society

## Appendix A. Semi-analytic solution for laminar shear flow adjacent to a wavy wall

#### (a) Symmetry analysis of the equations

It is known for the problem of interest (e.g. Scholle *et al.* 2009) that in the Stokes flow limit *Re*→0, the associated streamline pattern that exists is fully symmetric, corresponding to a streamfunction with even parity, i.e. *ψ*(−*x*,*y*)=*ψ*(*x*,*y*). Neglecting the Reynolds number terms in the field equations (4.1) and (4.2), it follows that the potential *Φ* must be of odd parity, i.e. *Φ*(−*x*,*y*)=−*Φ*(*x*,*y*). Reconsidering the full first integral of the Navier–Stokes equations, it becomes apparent that the Reynolds number terms induce symmetry breaking. Hence, it is convenient to make the decompositionA1 and A2for the fields into even and odd parts with capital letters indicating the fields with odd parity, i.e. *Φ*_{0}(−*x*,*y*)=−*Φ*_{0}(*x*,*y*) and *Ψ*(−*x*,*y*)=−*Ψ*(*x*,*y*), and small letters those with even parity, i.e. *ψ*_{0}(−*x*,*y*)=*ψ*_{0}(*x*,*y*) and *φ*(−*x*,*y*)=*φ*(*x*,*y*). By inserting equations (A1) and (A2) into each of the two field equations (4.1) and (4.2), as well as the four boundary conditions (4.3)–(4.6), they can be split into one equation with even parity and one with odd parity. The field equations then read
A3
A4
A5and
A6
with 𝒪(*Re*^{2} ∂*Ψ*^{2}) indicating terms containing squares of the Reynolds number and first-order derivatives of the inertia correction *Ψ* to the streamfunction. These terms are neglected due to the assumption of small Reynolds number corrections. As a consequence the two equations (A3) and (A4) are decoupled from the other equations. Together with the corresponding boundary conditions *ψ*=0 and ∂*ψ*/∂*y*=0, at and ∂*ψ*/∂*x*=0, ∂*ψ*/∂*y*=1 at *y*=*h*, these equations represent the Stokes flow limit *Re*→0 of the original equations (4.1) and (4.2). Their solution, *ψ*_{0}(*x*,*y*), has been studied carefully in the past both analytically and numerically (e.g. Pozrikidis 1987). A reasonable analytic approximation for *ψ*_{0}(*x*,*y*) in the case of moderate aspect ratio *a* up to 1/2 is given by Scholle (2004*b*) using a lubrication approximation. It can be written asA7using the abbreviations
A8and
A9
Hence, the problem remaining to be solved is given by the field equations (A5) and (A6) for the given Stokes flow solution (A7) with corresponding boundary conditionsA10
A11
A12
and A13

#### (b) Variational formulation and Ritz’s direct method

The two field equations (A5) and (A6) are self-adjoint, i.e. they can be obtained by variation with respect to *Ψ* and *φ* of a functional of the typeA14where *A* denotes the flow domain. The corresponding ‘Lagrangian density’ readsA15Variation with respect to *Ψ* leads to the field equation (A5) and variation with respect to *φ* to equation (A6). The Stokes flow solution *ψ*_{0} is given by equation (A7).

The existence of a variational principle allows for the use of Ritz’s direct method. In which case, reconsider the analytical form of the Stokes flow solution (A7) and assume a similar form for the Reynolds number corrections,A16
and A17with *Y* and *β* given by equations (A8) and (A9) and the polynomial functions *φ*_{k} and *Ψ*_{k} (*k*=0,1) given as
A18and
A19
() with unknown coefficients *Ψ*_{k,n} and *φ*_{k,n}, where *f*_{n}(*Y*) are given polynomials of order *n* (e.g. Chebyshev polynomials). The correction to the streamfunction, *Ψ*, vanishes by definition at the upper (*Y* =1) and at the lower (*Y* =0) walls. Hence, the two boundary conditions (A10) and (A12) are automatically fulfilled. As there are no boundary conditions to be fulfilled for the potential *φ*, the boundary values *φ*_{k}(0)=*φ*_{k,N+1} and *φ*_{k}(1)=*φ*_{k,N} are considered as independent variables, which according to classical variational calculus have to be fixed during variation.

The steps in the application of Ritz’s direct method are:

By inserting equations (A16)–(A19) for the fields into the Lagrangian (A15), the functional (A14) simplifies to a function of the 4(

*N*+1) coefficients*Ψ*_{k,n}(*k*=0,1;*n*=0,…,*N*−1) and*φ*_{k,n}(*k*=0,1;*n*=0,…,*N*+1). Setting its derivatives with respect to all coefficients except*φ*_{k,N}and*φ*_{k,N+1}to zero leads to 4*N*linear equations.The expansions (A17) and (A18) for the correction to the streamfunction are inserted in the two remaining boundary conditions (A11) and (A13); subsequent Fourier analysis produces four additional linear equations for the coefficients

*Ψ*_{k,n}.Combining the equations resulting from step (i) and step (ii), an algebraic set of 4(

*N*+1) equations for the 4(*N*+1) unknown coefficients is obtained.

Each of the above steps can be performed conveniently by making use of computer algebra (e.g. Maple). In producing the results provided in the main text, the order *N* of the polynomial functions was set to *N*=4. Hence, a set of only 4(*N*+1)=20 linear equations for the coefficients had to be solved for. The corresponding ‘stiffness matrix’ is of the size 20×20. Compared (i) to carrying out a full finite element method (FEM) calculation, the computational cost incurred is very low; (ii) with standard (semi-)analytical approaches, such as the one used by Malevich *et al.* (2008), the computational effort required is comparatively small.

## Appendix B. Extension of the theory to three-dimensional flow

The theory in its present state of development is restricted to incompressible steady two-dimensional flow. Attempts to relax these limitations are in progress. However, at the moment, these are at a very early stage. Accordingly, in what follows, the beginnings only of an extension of the theory towards steady incompressible three-dimensional flow is presented.

#### (a) A ‘naive’ three-dimensional adaptation and related problems

As complex numbers can only be identified with two-dimensional vectors, the case of a three-dimensional flow cannot be treated with the complex variable approach given in §2*a*. Furthermore, it is important to recognize that a streamfunction cannot be used in the three-dimensional case, unlike in §2*c*. Reconsidering the real-valued tensor representation (2.17) of the first integral, however, it is justifiable to modify this equation by replacing each two-dimensional traceless symmetric tensor by its analogous traceless symmetric three-dimensional one. This ‘naive’ replacement procedure leads to an equation of the formB1(*i*,*j*,*k*∈{1,2,3}) which is supplemented by the continuity equationB2By taking the divergence of the above tensor equation (B1) and considering the continuity equation (B2), one obtainsB3which are the Navier–Stokes equations if the potential *Φ* fulfills the identityB4Therefore, it is proved that each solution of equations (B1) and (B2) is also a solution of the three-dimensional Navier–Stokes equations together with the continuity equation. However, this does not mean that the converse is true: that each solution of the Navier–Stokes equations is also a solution of the equations (B1) and (B2). The crucial point here is that the number of independent equations, five from (B1) plus one from (B2), exceeds the number of fields, three velocities *u*_{i} plus one potential *Φ*. Thus, the 6−4=2 extra equations impose additional restrictions for the velocity field. Accordingly the above set of equations is ill-posed.

#### (b) Extended potential representation for three-dimensional flow and first integral

In order to replace the ill-posed set of equations (B1) and (B2) by a well-posed one, the number of potentials has to be increased by two additional ones, i.e. a minimum of three potentials, *φ*_{1},*φ*_{2} and *φ*_{3}, is required. Accordingly, the tensorin equation (B1) is replaced by a traceless symmetric tensor depending on the second-order derivatives of the three potentials *φ*_{1},*φ*_{2} and *φ*_{3}. At first glance, the potential representation, i.e. the detailed form of *A*_{ij}, is an open one. The resulting tensor equationB5and the continuity equation (B2) form a set of six independent equations for the six fields *u*_{1},*u*_{2},*u*_{3} and *φ*_{1},*φ*_{2},*φ*_{3}. Taking the divergence of the tensor equation (B5) and considering equation (B2), the vector equationB6results, which is equivalent to the Navier–Stokes equations if and only if the tensor *A*_{ij} fulfills the identityB7Next, a representation of the tensor *A*_{ij} in terms of the second-order derivatives of the potentials has to be chosen so as to fulfill condition (B7). Among various choices the following is preferred:B8
B9
B10
B11
B12
and B13Here the abbreviation ∂_{k}=∂/∂*x*_{k} has been used. By applying this potential representation, the condition (B7) results in the integrable equationB14which requires the expression in square brackets to be a constant, which by gauging of the potentials can be set to zero. Hence the equationB15results, which is the three-dimensional analogue of equation (2.19) for the two-dimensional case, giving the relationship of the three potentials to the pressure. The analogue to the first integral (2.15) and (2.16) is the set of five independent equations arising from tensor equation (B5). Considering the potential representation (B8)–(B13), these equations read explicitly asB16
B17
B18
B19and
B20
Equations (B16)–(B20) provide a first integral construct of the Navier–Stokes equations for three-dimensional steady incompressible flow, which together with the continuity equation (B2), completes the proposed extension of the theory.