1. Introduction

Three important discoveries are presented using aerodynamic theory for subsonic conditions in unbounded domains. The first discovery follows from proving that the expressions of special relativity are obtained exactly by solving the nonlinear convected wave equation for a steady, rotating source in compressible flow. The problem is solved with absolute 3D cylindrical-polar and absolute time coordinates in a non-inertial, fixed-to-body reference frame. This has previously [1] only been shown for a steady, translating source with absolute 3D Cartesian and absolute time coordinates in a fixed-to-body reference frame.

The second discovery involves adding a conservative body force, that should never have been ignored in the first place, from the Navier-Stokes equation – the induced velocity contribution. Its formulation is based on the localized induction approximation (LIA) theory for curved vortex filaments. This paper addresses the effect of vortices that remain attached to slender, solid bodies translating and rotating through a compressible fluid. Even though it is not discussed here, an induced velocity body force can also be generated by the presence of curved vortex filaments that terminate and remain attached to surface boundaries (e.g., solid-fluid or fluid1-fluid2) or to hydrodynamic surfaces such as Lamb surfaces that form, break free, and move with the fluid flow system [2]. This could be a potential source of seed-points for turbulence.

The third discovery comes about after laying bare what has always been present in plain sight – the embedding of the exact cubic nonlinear Schrödinger (NLS) equation within the convected wave equation for compressible flow. However, it requires one to resist the parsimonious logic of eliminating cross-derivatives from partial differential equations in the pursuit of mathematical beauty and over-emphasis on simplicity. The embedding of the cubic NLS equation means that frictionless solitons and other nonlinear vortex processes are predicted to be present within equations describing simple laminar flow systems. Such complex features can’t be simulated or even approximated by summing linear perturbations in either analytical or numerical simulations of incompressible flow equations.

2. Background & Review

2.1. Motivation

It was demonstrated in a 2017 paper [1] that the unsteady, nonlinear convected wave equation exactly represented the disturbance created by a steady translating source using an absolute 3D Cartesian and absolute time coordinate system. The source was assumed to translate at a constant speed along a straight line through an initially motionless, compressible fluid in an unbounded domain. Furthermore, it was demonstrated that the classical Lorentz transforms (i.e., special relativity) used for velocity, acceleration, momentum, energy, and mass are mathematical artefacts that arise from ignoring nonlinear cross-derivative terms in the convected wave equation and assuming the fluid was incompressible.

This paper will present two major findings for unbounded, compressible fluids using an absolute 3D cylindrical-polar and absolute time coordinate system. Both findings appear to have never been presented or suggested before:

1) To show that the convected wave equation analysis and findings of the 2017 paper [1] can be exactly extended to a source attached to a constant-speed rotating reference frame;

2) To derive the (0+2) cubic nonlinear Schrödinger equation for a harmonic oscillating source attached to a constant-speed rotating reference frame that simultaneously is also translating at a constant speed along the frame’s centerline axis.

The second finding is developed after including an additional body-force term in the Navier-Stokes equations. This conservative body-force term is the negative gradient of a potential. This term also represents a potential energy source that is proportional to the bending stiffness of an elastic, curved filament vortex subjected to a self-induced velocity using the theory of the localized induction approximation. In aerodynamics, this body force arises from the bounded horseshoe vortices that trail from each wingtip surface. These vortices are the ones that engine exhaust makes visible in high altitude contrails.

2.2. Navier-Stokes Equation

The 3D Navier-Stokes equation for an inviscid fluid can be written in the following form:

(1)

Term , with units , represents a body force per unit mass that acts on the fluid. Replace term with the equivalent vector expression . If we restrict ourselves to body forces that are conservative, we can then replace term with the negative gradient of a potential term , with units , (e.g., for gravity). The Navier-Stokes equation (1) reduces as:

(2)

The problem is further restricted if the term is replaced with a barotropic relationship when , such that:

(3)

Under the special condition of irrotational flow, the term (i.e., vorticity) vanishes. However, vorticity will vanish everywhere except along infinitesimally thin vortex lines. In addition, irrotational flow means the velocity vector in (3) can be replaced with the gradient of a scalar velocity potential term , with units , such that:

(4)

The following is a summary of assumptions for the fluid represented by the Navier-Stokes expression given in (4):

a) Inviscid (negligible viscosity)

b) Barotropic (density is a function of pressure)

c) Subjected to a conservative body force

d) Unsteady flow

e) Irrotational (except along thin vortex lines)

f) Potential based flow

g) Compressible (density varies in time and space)

h) Homentropic (uniform & constant entropy)

An isentropic flow means the entropy level of each infinitesimal fluid volume does not change with time but may vary from volume element to volume element. Thus, homentropic flow is isentropic but an isentropic flow is not necessarily homentropic.

2.3. Bernoulli Function

The expression within the brackets shown in (4) is called the Bernoulli function , with units , thus:

(5)

(6)

The condition means the Bernoulli function is independent of location along a streamline.

Compressibility of a fluid can be defined as the relative change in the local fluid density , with units , in response to a change in the local fluid pressure , with units . An adiabatic compression means the entropy content of the fluid remains approximately constant during a compression event. The freestream characteristic speed of the fluid, called the speed of sound , with units , can be defined [3], such that:

(7)

The subscript indicates fluid properties are to be evaluated outside the zone of disturbance.

2.4. Momentum and Continuity Equations

The momentum equation can be written for a fluid subjected to a conserved body force [4] in terms of a material derivative as:

(8)

(9)

The continuity equation represents the change in the quantity of fluid contained in a differential volume element in the time interval . This change is set equal to the amount of fluid flowing in the volume element minus the amount flowing out of the volume element [3], such that:

(10)

The gradient of a velocity potential can be expressed in terms of the fluid velocity for irrotational flow, such that:

(11)

2.5. Compressible, Nonlinear, Convected-Wave Equation

The momentum and continuity equations can be combined and flow velocity replaced with velocity potential for irrotational, inviscid, barotropic, isentropic flow [5] [6] subjected to a body force:

(12)

Consider for the moment a fixed-to-body Cartesian coordinate system. The above wave equation (12) is an exact expression for the unsteady, nonlinear flow of a compressible fluid. It is written in terms of velocity potential that is valid for both subsonic and supersonic flow conditions. However, (12) does not hold for transonic velocities since additional terms are needed to account for compression-shock and temperature loses. Replace the velocity potential term with a subscript symbol , to indicate compressible conditions. One can then write (12) as follows:

(13)

In the format given in (13), the reference frame is attached to the leading edge of the vehicle (or any slender, solid object). The positive X-axis is pointed towards the downstream end of the vehicle. Because of the fixed-to-body reference frame, the vehicle is stationary while the ambient fluid moves with freestream speed directed parallel to the positive X-axis. The wave equation (13) is classified as a hyperbolic partial differential equation in terms of spatial coordinate X and time coordinate for subsonic speeds.

2.6. Convected-Wave Equation in Cylindrical-Polar Coordinates

The unsteady convected wave equation (12) can also be written in a cylindrical-polar coordinate system, such that:

(14)

Term is the polar radius; is the azimuth angle; is the Cartesian X-coordinate; and is the Cartesian Y-coordinate. The cylindrical-polar velocity components are as follows:

(15)

Term is the longitudinal velocity component parallel to the longitudinal unit vector ; term is the circumferential velocity component parallel to the tangential unit vector ; and term is the radial velocity component parallel to the polar-radial unit vector . Hence, one can write the velocity vector for compressible flow conditions as the sum of the three vector components:

(16)

The quotient of a velocity component with that of the freestream speed of sound from (7) is defined as the Mach number for that velocity component, such that:

(17)

The total Mach number for a fluid velocity is defined as the sum of its squared components. It must be less than one for subsonic flow conditions:

(18)

(19)

Only the cylindrical-polar coordinate version of the wave equation (14) will be discussed in the remainder of the paper.

3. Case 1 - Rotating but Non-Translating Reference frame with a Vanishing Body Force for Subsonic Velocities

The first case will demonstrate that the 3D nonlinear convected wave equation in cylindrical-polar coordinates for compressible flow conditions with a non-inertial, fixed-to-body reference frame in an unbounded domain reduces to a 2D equation in Cartesian coordinates. This occurs when:

1) A source is assumed to consist of a small, slender, solid object;

2) Source is fixed to a non-inertial reference frame that rotates at a constant, subsonic, angular speed about the centerline axis;

3) Fluid in an unbounded domain is initially at rest;

4) Fluids are limited to those that are inviscid (i.e., viscosity ), irrotational (i.e., ), barotropic (i.e., ), isentropic (i.e., constant entropy), and compressible;

5) No body force acts on the surrounding fluid.

Using the above assumptions, we will set the following terms for a rotating but non-translating reference frame and a negligible body force:

(20)

3.1. Linearization of Wave Equation

The fluid density , fluid pressure , fluid velocity , and velocity potential will be linearized as follows with a perturbation component:

(21)

The subscript is used to indicate terms that are evaluated at freestream or undisturbed conditions.

It will be assumed that the magnitude of the perturbed velocity is much smaller than the magnitude of the freestream velocity and that the magnitude of the freestream velocity is much smaller than the characteristic speed of the fluid :

(22)

Substitute the linearized variables from (21) and the Mach number expression from (17) into the general wave equation (14) for compressible flow conditions. The unsteady, convected wave equation for the perturbed velocity potential in 3D cylindrical-polar coordinates will then reduce as follows for Case 1 conditions in a fixed-to-body reference frame:

(23)

Term , with units , is the polar-radius coordinate in the cylindrical-polar coordinate system for compressible flow. All third- and fourth-order perturbation terms have been dropped in (23).

3.2. Change in Variables

Define the dimensionless parameter , spatial variable , with units , and temporal variable , with units , for subsonic velocity conditions, where:

(24)

(25)

The variable represents an estimate of the circumferential arclength traversed when the azimuth angle changes by the angle .

Take the following partial derivatives in terms of the variable using the chain-rule of differentiation:

(26)

Take the following partial derivatives in terms of the variable using the chain-rule of differentiation:

(27)

The wave equation (23) reduces as follows for Case 1 after substitution of the derivatives from (26) and (27):

(28)

Consider a new perturbation velocity potential term and its derivatives, such that

(29)

Term in (29) is an arbitrary dimensionless coefficient.

Replace the potential and its derivatives in the wave equation (28) with the new potential and divide out the common term when finished, such that:

(30)

The first bracketed term in (30) can be reduced to a standard radial coordinate form by setting the coefficient equal to . The wave equation for a fixed-to-body reference frame (30) simplifies as follows upon back substitution with the new coefficient, such that:

(31)

The subscript has been added in (31) to remind us that this equation is based on a compressible fluid assumption.

3.3. Separation-of-Variables

Consider the following separation-of-variables solution , where only the component varies with time:

(32)

Substitute the transform (32) into the compressible wave (31) and divide the resultant expression by the term :

(33)

Term is an unknown constant coefficient. The and terms are independently set equal to giving the coupled differential equations:

(34)

(35)

3.4. Wave Equation for the Bessel Laplacian

The first equation (34), which is not a function of time, is called the wave equation for the Bessel Laplacian. It is of order which has non-integer values. The details of the solution will not be presented here but they will be shown in Case 2 to equal:

(36)

Term is an unknown, real valued constant. It should be obvious that the parameter must be positive valued for a physically meaningful solution when is based on the Bessel function of the first kind

3.5. 2D Wave Equation and Special Relativity

We will now examine the transient, 2D wave equation given in (35) that includes the term . It should be noted that (35) is in the proper form of a 2D Cartesian coordinate system representing a compressible fluid. The coordinate set for compressible flow conditions will be transformed to an equivalent coordinate set for incompressible flow conditions:

(37)

(38)

(39)

(40)

Term is defined as and are unknown coefficients of the compressible-to-incompressible coordinate transform.

The 2D wave equation with term can be written in Cartesian coordinates for an incompressible fluid medium with a fixed-to-body reference frame as:

(41)

The coordinate transform linking the partial differential equation (35) with (41) is written in matrix form as:

(42)

The matrix in (42) is in fact the Miles transform [6]. The classical Lorentz transform [7] from Special Relativity is related to (42) by pre-multiplying the Galilean transform matrix with that of the Miles transform matrix:

(43)

Note that the coefficient played no role in the derivation of the transform matrices shown in (42) and (43).

All details of the derivations for (42) and (43) can be found in the 2017 [1] paper. Furthermore, [1] shows how the resultant PDE can be converted between sixteen fixed-to-body, fixed-in-space, compressible, and incompressible reference frames.

3.6. Separation-of-Variables: Again

The wave equation in (41) can be further solved by using the separation-of-variables method a second time by replacing term with the following triple product:

(44)

Partial differential equations that can be solved using the method of separation-of-variables leads to solutions that are products of exponential functions with either real or imaginary arguments [8].

Substitute (44) into (41) and divide the resultant expression by the triple product :

(45)

Set the right-hand side expression equal to another constant since it is independent of the left-hand side expression:

(46)

Set the right-hand side term from the first line in (46) equal the right-hand side term from the second line in (46) and rearrange terms, such that:

(47)

There are four possible solutions to (47) that depend upon the sign of the terms and whether the Mach number vanishes or not. Three phase angles are defined in terms of the unknown constants:

(48)

The four possible analytical solutions for term are as follows:

i. If and

(49)

ii. If and

(50)

iii. If and

(51)

iv. If

(52)

The terms & are unknown constants of integration that are solved based on the boundary conditions in an unbounded domain.

Rearrange the left-hand side of equation (46) and solve for the component:

(53)

Set the right-hand side of the first line equal the right-hand side of the second line with unknown coefficient and solve for :

(54)

There are three possible solutions to (54), depending upon the sign of the expression :

i. If

(55)

ii. If

(56)

iii. If

(57)

The terms are unknown constants of integration that are solved based on the boundary conditions in an unbounded domain.

Rearrange the left-hand side of equation (53) and solve for the component:

(58)

There are three possible solutions to (58), depending upon the sign of the coefficient :

i. If

(59)

ii. If

(60)

iii. If

(61)

The coefficients are unknown constants of integration that are solved based on the boundary conditions in an unbounded domain.

3.7. Final Solution to Case 1

The final solution of Case 1 for the velocity potential of compressible flow in a fixed-to-body reference frame can be written in the following form:

(62)

The conversion of incompressible coordinates and to compressible coordinates and are listed in (39) and (42). Order of the Bessel function is defined as .

The purpose of presenting Case 1 is to show how the original 3D, nonlinear, convected wave equation (23) in cylindrical-polar coordinates for a compressible fluid in a rotating, but non-translating, fixed-to-body reference frame can be transformed to a system of two partial differential equations. One of the partial differential equation (PDE) expressions is solved as a steady wave equation for the Bessel Laplacian. The other PDE expression is solved as a 2D, transient, wave equation in Cartesian coordinates. This was made possible by converting the azimuth coordinate to a circumferential arclength coordinate . It was pointed out in a 2017 paper [1] that the resultant PDE could be converted between sixteen fixed-to-body, fixed-in-space, compressible, and incompressible reference frames. The transformations are based on the Miles, Galilean, and Lorentz matrices.

4. Case 2 - Rotating and Translating Reference Frame Subjected to a Conservative Body Force for Subsonic Velocities

4.1. Introduction to Case 2

The second case will demonstrate that the 3D nonlinear convected wave equation for compressible flow conditions with a non-inertial, fixed-to-body reference frame in an unbounded domain reduces exactly to the (0+2) focusing cubic NLS equation. This occurs when:

1) A source is assumed to consist of a small, slender, solid object;

2) Source is a harmonic oscillator of the form ;

3) Source is fixed to a non-inertial reference frame that rotates at a constant, subsonic, angular speed about the centerline axis;

4) Source simultaneously translates at a constant, subsonic speed in a direction parallel to the centerline axis;

5) Fluid in an unbounded domain is initially at rest;

6) Fluids are limited to those that are inviscid (i.e., viscosity ), irrotational (i.e., , except along infinitely thin vortex lines), barotropic (i.e., ), isentropic (i.e., constant entropy), and compressible;

7) One or more curved vortex filaments are instantly generated by a source. The filaments remain attached to the source, but they extend into the downwind direction;

8) Each curved vortex filament generates an induced velocity in a direction perpendicular to the centerline. The induced velocity produces a conservative body force that acts on the surrounding fluid.

4.2. Wave Equation to be Solved in Case 2

As previously stated, we shall assume in Case 2 a rotating and translating reference frame subjected to a conservative body force and a harmonic oscillator source:

(63)

The corresponding convected wave equation to be solved is given in (14) and after applying the conditions of (63):

(64)

4.3. Linearization of Wave Equation

The fluid density , fluid pressure , fluid velocity , and velocity potential will be linearized with a perturbation component in the same way as shown in (21) and (22) and substituted into (64). Upon dropping all third and fourth-order perturbation terms, the convected wave equation from (64) reduces to the following:

(65)

The Mach coefficients and are defined in (17).

Before proceeding with the solution of (65) for the nonlinear Schrödinger equation, it is worthwhile to review the Madelung transform that has been used since 1927.

4.4. Brief History of Madelung Transforms

The Madelung transform was first introduced by Madelung [9]. It defines the complex valued wavefunction in polar form as . The argument of the wavefunction equals term and the modulus is term . The fluid density is then identified with term and the fluid velocity is identified with the gradient operator as . The Madelung transform only satisfies the (1+1) linear Schrödinger equation. It also brings about an unusual term called the Bohm quantum potential that appears in the resultant momentum equation [10], where .

A more recent and relevant use of the Madelung transformation is given in [10]. He assumes an ideal gas law and temperature that is a function of time only. The resultant (1+1) nonlinear Schrödinger equation formulation contains the Bohm-quantum potential [11] and the Bialynicki-Birula logarithmic potential [12]. Several exact solutions for inviscid, irrotational, isentropic, and compressible flow are derived [10] but they are essentially restricted to finite domain problems. This is because the velocity and density functions increase indefinitely with distance from the origin.

Only the Hasimoto transform [13], in conjunction with LIA based flow theory of curved vortex filaments, is known to be consistent with the (1+1) and (0+2) cubic NLS equations. However, what has completely been absent in the literature is a rational theory demonstrating the derivation or origin of the cubic NLS equation itself using the classical Navier-Stokes equations for compressible flow. This paper will show, apparently for the first time, a derivation based on an aerodynamic application.

4.5. Development of Convected Wave Equation

4.5.1. Source as a Harmonic Oscillator

The source disturbance will be treated as a harmonic oscillator. The resultant 3D wave will vary harmonically in time. One can then write the perturbed velocity potential , with units , as a function of a steady-perturbation velocity potential , with units , and a harmonic component:

(66)

Term , with units , represents the spin angular speed of the harmonic oscillator source.

4.5.2. Including a New Conservative Body Force

A conservative body force is expressed as the negative gradient of a potential :

(67)

Potential , with units , is related to a potential energy source that is proportional , with units , to the bending stiffness of an elastic, curved filament vortex undergoing self-induction:

(68)

4.5.3. Localized Induction Approximation

Assuming the applicability of the localized induction approximation (LIA) for a curved filament or vortex, then the velocity induced, , with units , at position along the filament centerline is given by the time derivative of the position vector [13] [14] [15] [16] [17] [18]:

(69)

Term , with units of , is called the coefficient of local induction.

4.5.4. Frenet-Serret Formulas of Differential Geometry

Term in (69), with units of , is the curvature of the filament curve and the dimensionless unit vector is the binormal vector from the Frenet-Serret formulas of differential geometry [19]. If the dimensionless unit vector is the tangent vector parallel to the centerline of the filament curve and dimensionless unit vector is the normal vector, then the partial derivative of the tangent vector with respect to arc-length along the curve is:

(70)

Hence, the magnitude of the self-induced velocity is given as [15] [20] [21] [22] [23]:

(71)

Term in (69) and (71) is a function of the vortex strength, radius of the vortex core, and cut-off arclength [13].

4.5.5. Components of the Fluid Velocity Vector

At any point in the unbounded domain, the fluid velocity will be assumed to consist of freestream velocity ; the perturbation velocity due to the presence of the solid; slender body moving through the fluid; and the induced velocity component generated by the presence of the curved filament vortex that trails downwind of the slender body that behaves as a harmonic source:

(72)

Velocity with units is the steady-perturbation velocity component due to the presence of a slender solid body; and with units is the steady-perturbation velocity component due to the self-inducted velocity produced by the curved filament vortex (i.e., at point along the filament).

4.5.6. Gradient of the Steady-Perturbation Velocity Potential

The gradient of the steady-perturbation velocity potential consists of two contributions:

(73)

This means induced velocity vector vanishes if no curved filament vortex forms or if the vortex does not remain attached to the solid, slender translating body.

Take the dot product of fluid velocity and the term from (73):

(74)

Assume the following relative magnitudes between velocity components:

a) Steady-perturbation velocity is small compared to

freestream velocity:

(75)

b) Steady-perturbation velocity is small compared to

induced velocity:

(76)

c) Induced velocity is not parallel to freestream velocity:

(77)

The dot product of fluid velocity and given in (74) reduces as follows upon substitution of the assumptions given in (75), (76), and (77):

(78)

The gradient of the body force potential given in (68) can now be evaluate as follows:

(79)

Take the dot product of fluid velocity with the gradient expression given in (78) and (79):

(80)

4.5.7. Material Derivative of Body Force Potential

The time derivative of body force potential vanishes since the steady-perturbation velocity potential is not a function of time:

(81)

The material derivative of body force potential can now be completely evaluated using the results of (80) and (81)

(82)

4.5.8. Derivatives of the Perturbed Velocity Potential

Differentiate the perturbed velocity potential from (66) with respect to time for the case when the source is a simple harmonic oscillator with constant amplitude and a constant angular speed that does not depend on the amplitude:

(83)

Substitute the time derivates of the perturbed velocity potential from (83), the material derivative formula for the body force potential from (82) into the linearized wave equation (65), and divide out the common term when finished:

(84)

4.5.9. Introducing a New Perturbed Velocity Potential

Create a new perturbation velocity potential , with units , to replace the steady-perturbed velocity potential , with units , where the coefficient is unknown:

(85)

(86)

Substitute the formulas (85) and (86) into the wave equation (84); replace with term and with term ; and divide out the common term , such that:

(87)

The expression in (87) can be reduced to a standard radial coordinate form by setting coefficient equal to the following:

(88)

Substitute coefficient from (88) into (87):

(89)

4.5.10. Change in Spatial and Temporal Coordinates

Define the longitudinal spatial variable , with units , and the transverse spatial variable , with units , for subsonic velocity conditions, where:

(90)

(91)

The variable represents an estimate of the circumferential arclength travelled when the azimuth angle changes by the amount and then rescaled by The variable represents an estimate of the transvers arclength travelled when the Z coordinate changes by the distance and then rescaled by

Take the following partial derivatives in terms of the variable (90) using the chain-rule of differentiation:

(92)

(93)

Take the following partial derivatives in terms of the variable (91) using the chain-rule of differentiation:

(94)

(95)

Substitute the change of variables and derivatives from (90) to (95) into the wave equation of (89)

(96)

4.6. Separation-of-Variables

Consider the following separation-of-variables solution to the compressible, convected wave equation (96), where only the term varies with the polar radius :

(97)

The separation-of-variables term in (97) is called the wave function of the compressible, convected wave equation given in (96). Substitute the new transform (97) into the wave equation (96) and divide the resultant expression by the term :

(98)

Term , with units , is an unknown constant.

Introduce the following new dimensionless space variable to replace the polar radius in the bracketed expression when the coefficient is greater than zero:

(99)

(100)

Substitute the above relationships (99) to (100) into the bracketed expression in (98) after setting it equal to :

(101)

Term is defined by the following expression:

(102)

Multiply the bracketed expression in (101) by the term , such that:

(103)

The ordinary differential equation in (103), which is not a function of time, is called the wave equation for the Bessel Laplacian of order . It should be obvious from (102) that the order is a non-integer with a value always greater than zero for subsonic flow velocities. The general solution to (103) is:

(104)

Bessel functions of the first kind with positive valued orders behave [24] with as the argument with a fixed value for order when ; and where is the gamma function. With as the argument . Bessel functions of the first kind with negative valued orders behave with as the argument . Hence, the constant must vanish for physically meaningful solutions for finite valued induced velocities along the curved filament vortex:

(105)

Now return to the bracketed expression in (98) that was set equal to . Multiply all terms by the wave function and rearrange terms, such that:

(106)

The left side of the expression given in (106) is in the form of the (0+2) focusing cubic nonlinear Schrödinger (NLS) equation. It will be shortly shown how the remaining two terms on the right side of the expression in (106) are to be properly treated.

4.7. Solving for the Wave Function

In the previous section, we solved the separation-of-variable term from the convected wave equation as a function of the polar radius. The remaining separation-of-variable term , called the wave function, will now be evaluated.

4.7.1. Introducing the Hasimoto transform

Consider the Hasimoto transform [13], [23], [25] for the complex valued wave function when the torsion of the filament vortices is not constant with arclength :

(107)

Term with units of is the torsion of the filament vortex and is a reference torsion value. For the special case of constant torsion , the Hasimoto transform reduces to the following form:

(108)

It should be noted that both curvature and torsion will be considered general functions of arclength along the filament and the transverse distance from the filament’s centerline.

Let represent the complex conjugate of the wave function , such that from (107):

(109)

It then follows from (107) and (109) that:

(110)

4.7.2. Differential and Riemannian geometry

The (0+2) NLS equation is formulated in differential geometry using Riemannian geometry with and coordinate curves on the congruence surface with Riemannian metric [26]:

(111)

The metric coefficients and are given as and on the surface. Function is called [26] the abnormality of the vector N-field.

Two of the constraints for torsion and curvature functions on the manifold are called the Da Rios-Betchov equations [25]:

(112)

(113)

The coefficient is defined as . Terms and are integration constants.

4.7.3. Solution for Torsion Function

A solution for the torsion function that satisfies both the LIA assumption, the Hasimoto transform, and Da Rios-Betchov equations on the manifold surface is:

(114)

Coefficient is defined in terms of LIA based coefficients for the filament vortex:

(115)

Terms are integration and boundary constants.

4.7.4. Conservation Expressions for Torsion, Curvature, and Wave Function

Differentiate the torsion solution (114) with respect to the and then the coordinate curves:

(116)

Or, upon rearranging (116)

(117)

Differentiate the Hasimoto transform for wave function in (107) with respect to the and the coordinate curves:

(118)

Substitute the formulas for torsion from (114) and derivative from (116) into the first Da Rios-Betchov equation (112) to obtain the conservation expression:

(119)

Substitute the formulas for torsion from (114), derivatives and from (117) into the first Da Rios-Betchov equation (112) to obtain the conservation expression:

(120)

Finally, substitute derivatives from (119) and from (120) into the derivative of the Hasimoto transform given in the second line of (118); rearrange and replace the resultant terms using the derivative of the Hasimoto transform given in the first line of (118), such that:

(121)

The derivation for all three conservation form expressions given in (119), (120), & (121) for the curvature , torsion and wave function is apparently new for the (0+2) NLS equation.

4.7.5. Wave Function with Multiple Derivatives

Differentiate the wave function conservation form expression in (121) with respect to the and then the coordinate curves:

(122)

Substitute derivative from the second line of (122) back into the first line of (122), such that:

(123)

The derivative from (121), the derivative from the second line of (122), and the derivative from (123) can now be used to replace terms in the (0+2) NLS equation (106), such that:

(124)

4.7.6. A New Compressibility Constant

Define a compressibility coefficient , such that:

(125)

And define the proportionality coefficient from the body force term, such that:

(126)

Substitute the coefficient from (125) and the constant from (126) into the (0+2) NLS equation (124), such that:

(127)

4.7.7. Standard form of the (0+2) NLS

The standard form of the (0+2) NLS is as follows:

(128)

The dimensionless term is called the pseudo-speed coefficient. It is expressed in terms of the reference torsion and reference curvature values:

(129)

4.7.8. Standard form of the (0+1) NLS

After eliminating the derivative in (128) using the conservation form expression (121), the standard form of the (0+1) NLS is written as follows:

(130)

4.7.9. Association of Variables

Upon comparing the compressible (0+1) NLS in (124) with the standard form of the (0+1) NLS in (130), one can write the following three associations between variables:

(131)

(132)

(133)

Substitute the formula for pseudo-speed from (129) into the first association (131) and solve for the reference torsion , such that:

(134)

Solve for the reference curvature from the second association (132), such that:

(135)

The separation-of-variables coefficient is evaluated from the third association expression (133), such that:

(136)

Take the ratio of from expressions (134) and (135), such that:

(137)

The vortex curve reference coefficients and separation-of-variables coefficient have been expressed in terms of LIA based coefficients and aerodynamic source terms and

4.8. Solution to the NLS

4.8.1. Differential Equation for Curvature

Substitute for the derivative from the second conservation expression (120) into the integral term in the second Da Rios-Betchov equation (113) and then substitute in the LIA based torsion formula from (114), such that:

(138)

Rearrange terms in (138) and multiply the resultant expression by , such that:

(139)

4.8.2. Cubic Polynomial Equation for the Auxiliary Function

Define the dimensionless auxiliary function as . The following cubic polynomial can be obtained by substitution of the auxiliary into (139) and integration:

(140)

The integration constant in (140) is given by:

(141)

4.8.3. Elliptic function solution for the (0+2) cubic NLS

The solution to the nonlinear ordinary differential equation in (140) is the Jacobian elliptic sine function , such that:

(142)

Dimensionless term is the Jacobi modulus; dimensionless term is the Jacobian elliptic angle; and are the three cubic roots when solving the nonlinear ODE given in (140) in terms of the coefficients and . Term is Kida’s [27] sliding speed coefficient; term is a constant of integration for elevation of the resultant vortex filament; and is Kida’s [27] translation speed coefficient.

An extensive discussion is given in Ch. 6 of [28] for additional constraints imposed on the parameters for curve closure, periodicity, and knotting.

4.8.4. Final Solution to Case 2

The final solution of Case 2 for the velocity potential of compressible flow in a fixed-to-body reference frame can be written in the following form:

(143)

A subscript is listed with coordinates and torsion to remind us that they are being used under the assumption of a reference frame subjected to compressible flow conditions. Term in (143) is given by ; is the Bessel function of the first kind, order v; is the separation-of-variables integration constant evaluated in (136); arclength coordinate is given in (90); transverse arclength coordinate is given in (91); torsion function is given in (114); curvature function is given in (142); reference torsion is given in (134); and reference curvature is given in (135).

4.9. Summary of Case 2

Authors since 1927 have solved the logarithmic density version of the (0+1) NLS using the Madelung transform and quantum hydrodynamics. However, a derivation of the (0+1) & (0+2) focusing cubic NLS equation from the Navier-Stokes equation using the Hasimoto transform and an aerodynamic body force is apparently new. The derivation presented here gives the exact expression for the steady form of the cubic NLS that matches the curvature and torsion constraints derived from Riemann geometry for curved surfaces.

Case 2 involves the transient, 3D, nonlinear, convected wave equation (64) in a fixed-to-body reference frame for a compressible fluid with constant coefficients expressed in cylindrical-polar coordinates for a source that is both rotating and translating. There are three reasons for presenting this problem: the first is to show that the (0+2) cubic NLS is embedded within the PDE of the original wave equation (after converting the partial derivatives and ); the second is to show that the (0+1) cubic NLS is embedded within the PDE of the wave equation; and the third is to show that there is an analytically solution (143) to the wave equation (64). The conversion of the partial derivates , , and is based on the discovery of three new conservation form expressions (119), (120), & (121) for the curvature , torsion and wave function .

5. Discussion

This article examines the derivation and solution of unsteady convected waves for compressible fluids using an analogous problem from aerodynamics. The take-away conclusion is that one is able to scale from the smallest-to-largest and slowest-to-fastest processes in the universe using Newton’s classical laws for fluids based on a system of absolute space and time coordinates. This is made possible by recognizing and retaining the effects of fluid compressibility that are intrinsically associated with the cross-derivative terms between space and time. The retention of the cross-derivatives makes the resultant convected nonlinear wave equations more difficult to solve. In compensation, it eliminates the artificial contradictions and mysticisms imposed with using elastic space-time coordinates when one insists on assuming incompressible flow conditions for in vacuo problems.

This paper and the 2017 paper [1] don’t reject the scientific data obtained from 100 years of testing special relativity. What is presented here is a radically different physical interpretation of prior test results. Instead of interpreting the special relativity tests as proof-of-errors in ignoring the elasticity of relative space-time coordinates, it interprets prior tests as showing the error in dropping the nonlinear cross-derivatives from the convected wave equation that uses an absolute space and time coordinate system.

It might seem to be argumentative to reject the classical understanding on why speed affects the measurement of distance but it actual goes to the heart of science: progress is made by challenging theories that one takes for granted and replacing it with an improved version, with each iteration bringing humanity closer to the truth. The unmasking of the cubic NLS and special relativity relationships within the compressible, convected wave equation for laminar flow offers proof in the value of challenging what one thought was totally understood for more than a century.

6. Conclusions

For the first time, an exact derivation of the (0+2) cubic NLS equation is obtained after combining the 3D Navier-Stokes equation, the Hasimoto transform, and an aerodynamic body force for induced velocity. Authors have previously derived a special logarithmic density version of the NLS using the Madelung transform and quantum hydrodynamics. However, the derivation given here results in an exact expression for the steady form of the (0+2) cubic NLS that matches the curvature and torsion constraints derived from Riemann geometry for curved surfaces. In addition, it is shown how special relativity expressions are obtainable within the steady-rotating source problem of the convected wave equation when written in cylindrical-polar coordinates and a non-inertial fixed-to-body reference frame.

Nomenclature