Periodic Navier Stokes Equations for a 3D Incompressible Fluid with Liutex Vortex Identification Method

Abstract

The Incompressible Navier-Stokes Equations (NSEs) are on the list of Millennium Problems, to prove their existence and uniqueness of solutions. The NSEs can be formulated in a periodic 3D domain, where they are termed the Periodic Navier Stokes (PNS) Equations, and can be treated on a subspace spanning a 3-dimensional torus, or T3. Treating the PNS Equations in T3-space, this article demonstrates that a decaying of turbulence occurs in the 3D case for the z component of velocity when non-smooth initial conditions are considered for x, y components of velocity and that ‘vorticity’ sheets in the small scales of 3D turbulence dominate the flow to the extent that non-smooth temporal solutions exist for the z velocity for smooth initial data for the x, y components of velocity. Unlike the Navier-Stokes equations, which have no anti-symmetric vorticity tensor, there are new governing equations which have vorticity tensor and can be decomposed into a rotational part(Liutex), antisymmetric shear and compression and stretching. It is shown that under these recent findings, that there is a strong correlation between vorticity and vorticies for (PNS).

1. Introduction

This chapter gives a general model using specific periodic special functions, that is elliptic Weierstrass P functions. The definition of vorticity in [1], is that vorticity is a rotational part added to the sum of antisymmetric shear and compression and stretching. Satisfying a divergence free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term $f\left(\overrightarrow{x},t\right)$ which is smooth and periodic, then the existence of solutions which blowup in finite time can occur. On the other hand if $u_0$ is not smooth, then there exist globally in time solutions on $t\in \left[0,\infty \right)$ with a possible blowup at $t=\infty$. The control of turbulence is possible to maintain when the initial conditions and boundary conditions are posed properly for (PNS) [2, 3, 4]. This leads to the following two questions for (PNS),

1. Is there a decaying of turbulence in the 3D case for the z component of velocity when non-smooth initial conditions are considered for x, y components of velocity?

   and

2. Are the vorticity sheets in the small scales of 3D turbulence dominating the flow to the extent that non-smooth temporal solutions exist for the z velocity for smooth initial data for the x, y components of velocity?

A positive answer exists for both of the above questions [4, 5]. In this chapter it is shown explicitly that for smooth forcing that is both spatial and temporal and Weierstrass P product functions in space for velocity $u_x$ and $u_y$ that the equivalent form of the Navier-Stokes equations derived in [6, 7, 8], has as one of the possible solutions for $u_z$ a separable product of spatial functions in the three space variables together multiplied by a general function of t which is a blowup at infinity. On the other hand if $\mathbf{f}\left(\overrightarrow{x},t\right)$ is a smooth reciprocal function of a general Weierstrass P function defined on the 3-Torus, then when $u_x$ and $u_y$ in 3D Navier Stokes equations are both in the smooth reciprocal form of the Weierstrass P function then this implies that $u_z$ is not smooth in time. In [6, 7, 8] the z component of vorticity was chosen to be constant. Extending the vorticity definition, in particular in this chapter, $u_x,u_y$ satisfy a non-constant spatial or time dependent vorticity for 3D vorticity $\overrightarrow{\omega }$. Finally new eqs. [1] are conjectured to possess smooth solutions appearing to not have finite time singularities using the correct definition of vorticity in this study. For (PNS) it is shown that there exists a vortex in each cell of the lattice associated with ${\mathbb{T}}^3$ using the decomposition of pure rotation(Liutex), antisymmetric shear and compression and stretching. Furthermore it is observed that a singular cusp bifurcation occurs along a principle main axis for the case of smooth and non-smooth initial inputs of velocity.

2. Mathematics preliminaries

Let $s\in \mathbb{R}$, the homogenous Sobolev space is,

$${\dot{H}}^s\left({\mathbb{T}}^3\right)≔\left\{f=\sum _{\mathbf{k}\in {\mathbb{Z}}^{\mathbf{3}}}{a}_k{e}^{i\mathbf{k}\cdot \mathbf{x}};a_0=0\text{and}\sum _{k\ne 0}{\left|k\right|}^{2s}{\left|a_k\right|}^2<\infty \right\}$$E1

with associated norm,

$${‖f‖}_{{\dot{H}}^s}≔{\left(\sum _{k\ne 0}{\left|k\right|}^{2s}{\left|a_k\right|}^2\right)}^{1/2}$$

The inhomogeneous Sobolev Space is,

$$H^s\left({\mathbb{T}}^3\right)≔\left\{f=\sum _{\mathbf{k}\in {\mathbb{Z}}^3}{a}_k{e}^{i\mathbf{k}\cdot \mathbf{x}};a_0\ne 0\text{and}\sum _k{\left|k\right|}^{2s}{\left|a_k\right|}^2<\infty \right\}$$E2

with associated norm,

$${‖f‖}_{H^s}≔{\left(\sum _{k\in \mathbb{Z}}{\left|k\right|}^{2s}{\left|a_k\right|}^2\right)}^{1/2}$$E3

The particular inhomogeneous Sobolev space $H^{\frac{1}{2}}\left({\mathbb{T}}^3\right)$ is a scale invariant space for (PNS).

Theorem 1 (Preköpa-Leindler) Let $0<\lambda <1$ and let f, g, and h be nonnegative integrable functions on ${\mathbb{R}}^n$ satisfying,

$$h\left(\left(1-\lambda \right)x+\mathit{\lambda y}\right)\ge f{\left(x\right)}^{1-\lambda }g{\left(y\right)}^{\lambda },$$

for all $x,y\in {\mathbb{R}}^n$. Then

$${\int }_{{\mathbb{R}}^n}h\left(x\right)\mathit{dx}\ge {\left({\int }_{{\mathbb{R}}^n}f\left(x\right)\mathit{dx}\right)}^{1-\lambda }{\left({\int }_{{\mathbb{R}}^n}g\left(x\right)\mathit{dx}\right)}^{\lambda }$$

Theorem 2 (Gagliardo-Nirenberg) Let $1\le q\le \infty$ and $j,k\in \mathrm{\mathbb{N}},j<k,$ and either,

$$\left\{\begin{array}{l}r=1\\ \frac{j}{k}\le \theta \le 1\end{array}\right.\left\{\begin{array}{l}1<r<\infty \\ k-j-\frac{n}{r}=\mathrm{0,1,2},\dots \\ \frac{j}{k}\le \theta <1\end{array}\right.$$

If we set $\frac{1}{p}=\frac{j}{n}+\theta \left(\frac{1}{r}-\frac{k}{n}\right)+\frac{1-\theta }{q}$, then there exists constant C independent of u such that

$${‖{\nabla }^{j}u‖}_p\le C{‖{\nabla }^{k}u‖}_r^{\theta }{‖u‖}_q^{1-\theta },\text{for}\mathrm{all}u\in L^q\left({\mathbb{R}}^n\right)\cap W^{k,r}\left({\mathbb{R}}^n\right)$$

3. Equivalent form of 3D incompressible Navier stokes equations

The 3D incompressible unsteady Navier-Stokes Equations (NSEs) in Cartesian coordinates may be expressed [6, 7, 8] as the coupled system Eqs. (4)–(9) below, for the velocity field ${\mathbf{u}}^{\ast }=u^{\ast i}{\overrightarrow{\mathbf{e}}}_i,u^{\ast i}=\left\{u_x^{\ast },u_y^{\ast },u_z^{\ast }\right\}$ from the original NSE’s,

$$\mathcal{G}={\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+3{\mathcal{G}}_{\delta 3}+{\mathcal{G}}_{\delta 4}=0$$E4

where,

$$\begin{array}{c}{u}_x^{\ast }\to \frac{u_x}{\delta },u_y^{\ast }\to \frac{u_y}{\delta },u_z^{\ast }\to \frac{u_z}{\delta }\\ x^{\ast }\to \mathit{\delta x},y^{\ast }\to \mathit{\delta y},z^{\ast }\to \mathit{\delta z},t^{\ast }\to {\delta }^{2}t\\ \frac{\partial }{\partial x^{\ast }}\to {\delta }^{-1}\frac{\partial }{\partial x},\frac{\partial }{\partial y^{\ast }}\to {\delta }^{-1}\frac{\partial }{\partial y},\frac{\partial }{\partial z^{\ast }}\to {\delta }^{-1}\frac{\partial }{\partial z},\frac{\partial }{\partial t^{\ast }}\to {\delta }^{-2}\frac{\partial }{\partial t}\end{array}$$

and where $\delta =A+1$ with A being arbitrarily small when $\delta \approx 1$. In [6] $\mathcal{G}$ there is defined without the tensor product term Q. Calculating the tensor product term Q in Eq. (14) see [6] and using Eq. (22) in [6] shows it’s volume integral to approach zero due to ${‖\nabla u_z‖}_2^2$ approaching zero. See Theorem 1 and 2, where the Preköpa-Leindler and Gagliardo-Nirenberg inequality is used to show this when $r=1$ and $\lambda =\theta =\frac{1}{2}$. In this paper it is shown that the problem in [6] can be extended to all three velocity components $u_i$. The G.A. decomposition used there shows that there is a missing term (intended to be present) when multiplying Eq. 5 (there) by $u_z$ and adding to the product of $\overrightarrow{b}$ and the z momentum equation. This sum is precisely ${‖\nabla u_z‖}^2$ which is bounded by ${‖\nabla u_z‖}_2^2$ which is shown below to approach zero as the volume $\mathrm{\Omega }$ approaches infinity. Also the pressure due to conservation of forces theorem is a regular function in t. As a result it is assumed that it can be written as $P=\tilde{P}\left(x,y,z\right)P_{t_0}\left(t-t_0\right)$ where $P_{t_0}\left(t-t_0\right)$ approaches zero as $t\to t_0$ and $\nabla \cdot \left(\frac{1}{\mathit{\delta \rho }}{u}_z^2{\nabla }_{\mathit{xy}}P+\overrightarrow{b}\frac{1}{\rho }{u}_z\frac{\partial P}{\partial z}\right)=3\mathrm{\Phi }\left(t\right)$ where $\overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$. In Eq. (7) of [6], for the vector $\overrightarrow{B}=u_z\nabla \cdot \left(u_z\overrightarrow{b}\right)\overrightarrow{b},L\overrightarrow{u}=\overrightarrow{B}.$ Furthermore in Eq. (11) [6] there, the term ${\mathrm{\Omega }}_4$ is the divergence of the vector $\overrightarrow{B}$. Using Ostrogradsky’s formula in terms of the vorticity $\overrightarrow{\omega }$ and velocity $\overrightarrow{b},{\int }_{\mathrm{\Omega }}{u}_z\overrightarrow{b}\nabla \cdot \left({\overrightarrow{u}}_z\overrightarrow{\omega }|\overrightarrow{r}|\right)d\overrightarrow{x}=-{\int }_{\mathrm{\Omega }}\mid \overrightarrow{r}\mid u_z\overrightarrow{\omega }\cdot \nabla \left(\overrightarrow{b}{u}_z\right)d\overrightarrow{x}.$ Now for a specific pressure P on an R-sphere, ${\int }_{\mathrm{\Omega }}{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}d\overrightarrow{x}=3\mathrm{\Phi }\left(t\right)$, where $\mathrm{\Phi }\left(t\right)$ assumed to be bounded and contain the pressure terms in Eq. (7). The sphere is $\mid \overrightarrow{r}\mid =R$. Since the 3-Torus is compact there are m closed sets covering it. The outer measure is used where the infimum is taken over all finite subcollections $\mathrm{ℳ}$ of closed spheres ${\left\{E_j\right\}}_{j=1}^n$ covering a specific space associated with ${\mathbb{T}}^3$. This space is $S_{{\mathbb{T}}^3}$ and is within $ϵ$ measure of the 3-Torus and is obtained by minimally smoothening the vertices of ${\left[-L,L\right]}^3$ and slightly puffing out it’s facets. Also inner measure is used where the supremum is taken over all finite subcollections $\mathcal{N}$ of closed spheres ${\left\{F_j\right\}}_{j=1}^p$ inside $S_{{\mathbb{T}}^3}$. Generally by Hölder’s inequality, since ${\mathcal{G}}_{\delta 3}$ is positive for sufficiently sharp increases in radial pressure where $u_z$ does not blowup in finite time (but will be shown to be non-smooth), and for sufficiently small $c>0$,

$$\begin{array}{l}\underset{\mathcal{N}}{sup}\left|c\left[{\int }_{F=\underset{j=1}{\overset{p}{\cup }}{F}_j}\overrightarrow{r}\times {\nabla }_D^{-1}\left[{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}\right]d\overrightarrow{x}\right]\right|\\ \le \underset{\mathrm{ℳ}}{inf}\left|c{\int }_{{\mathrm{\Omega }}_R=\underset{j=1}{\overset{m}{\cup }}{E}_j}\overrightarrow{r}\times {\nabla }_D^{-1}\left[{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}\right]d\overrightarrow{x}\right|\\ =\left|c{\int }_{\mathrm{\Omega }}|\overrightarrow{r}\Vert {\nabla }_D^{-1}\left[{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}\right]|sin\left(\theta \right)\overrightarrow{n}d\overrightarrow{x}\right|\\ \le \left|c{\int }_{\mathrm{\Omega }}|\overrightarrow{r}\Vert {\nabla }_D^{-1}\left[{\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 3}+{\mathcal{G}}_{\delta 4}\right]|d\overrightarrow{x}\right|\\ =\left|-c{\int }_{\mathrm{\Omega }}{\left|\overrightarrow{r}\right|}^2{\overrightarrow{\omega }}_1\cdot \nabla \left(u_z\overrightarrow{b}\right)d\overrightarrow{x}\right|\end{array}$$

$$\begin{array}{l}=c\left|{\int }_{\mathrm{\Omega }}{\left|\overrightarrow{r}\right|}^2{\overrightarrow{\omega }}_1\cdot \nabla \left(u_z\overrightarrow{b}\right)d\overrightarrow{x}\right|\\ \le c{\int }_{\mathrm{\Omega }}\left|{\left|\overrightarrow{r}\right|}^2{\overrightarrow{\omega }}_1\cdot \nabla \left(u_z\overrightarrow{b}\right)\right|d\overrightarrow{x}\\ \le c{\left({\int }_{\mathrm{\Omega }}{\left|\overrightarrow{r}\right|}^{4}d\overrightarrow{x}\right)}^{\frac{1}{4}}\left[{\left({\int }_{\mathrm{\Omega }}{\left|{\overrightarrow{w}}_1\cdot \nabla \left(u_z\overrightarrow{b}\right)\right|}^2\right)}^{\frac{3}{4}}\right]\\ \le \begin{array}{l}c{\left({\int }_{\mathrm{\Omega }}{\left|\overrightarrow{r}\right|}^{4}d\overrightarrow{x}\right)}^{\frac{1}{4}}[{\left[{\left({\int }_{\mathrm{\Omega }}{\left|{\overrightarrow{\omega }}_1\cdot \left(\nabla u_z\otimes \overrightarrow{b}\right)\right|}^{2}d\overrightarrow{x}\right)}^{1/2}\right]}^{\frac{3}{2}}\\ +{\left[{\left({\int }_{\mathrm{\Omega }}{\left|u_z\nabla \left(\overrightarrow{b}\right)\cdot {\overrightarrow{\omega }}_1\right|}^{2}d\overrightarrow{x}\right)}^{1/2}\right]}^{\frac{3}{2}}]\end{array}\\ \le \begin{array}{l}c{\left({\int }_{\mathrm{\Omega }}{\left|\overrightarrow{r}\right|}^{4}d\overrightarrow{x}\right)}^{\frac{1}{4}}[{\left[{\int }_{\mathrm{\Omega }}{\left|{\overrightarrow{\omega }}_1\right|}^2{\left|\overrightarrow{u}\right|}^2{\left|\nabla u_z\right|}^{2}d\overrightarrow{x}\right]}^{\frac{3}{4}}\\ +{\left[{\int }_{\mathrm{\Omega }}{\left|u_z\right|}^2{\left|\nabla \overrightarrow{b}\right|}^2{\left|{\overrightarrow{\omega }}_1\right|}^{2}d\overrightarrow{x}\right]}^{\frac{3}{4}}]\end{array}\\ \le \begin{array}{l}c{\left({\int }_{\mathrm{\Omega }}{\left|\overrightarrow{r}\right|}^{4}d\overrightarrow{x}\right)}^{\frac{1}{4}}[{\left[{‖{\overrightarrow{\omega }}_1‖}_4^2{‖\overrightarrow{u}‖}_8^2{‖\nabla u_z‖}_{16}^2\right]}^{\frac{3}{4}}\\ +{\left[{\int }_{\mathrm{\Omega }}{\left|u_z\right|}^2{\left|\nabla \overrightarrow{b}\right|}^2{\left|{\overrightarrow{\omega }}_1\right|}^{2}d\overrightarrow{x}\right]}^{\frac{3}{4}}]\end{array}\\ \le \frac{152}{15}{\mathit{cL}}^{7/4}\left[\frac{1}{c}{‖{\overrightarrow{\omega }}_1‖}_4^{\frac{3}{2}}{‖\overrightarrow{u}‖}_8^{\frac{3}{2}}{‖\nabla u_z‖}_2^3+{\left[\frac{1}{c}{\int }_{\mathrm{\Omega }}{‖u_z‖}_8^{\frac{3}{2}}{‖\nabla \overrightarrow{b}‖}_2^3{‖{\overrightarrow{\omega }}_1‖}_4^{\frac{3}{2}}d\overrightarrow{x}\right]}^{\frac{3}{4}}\right]\\ \boxed{\le 0(\mathrm{due}\text{to}2-\text{norms\hspace{0.17em}\hspace{0.17em}approaching\hspace{0.17em}\hspace{0.17em}zero}}\end{array}$$

Considering inscribed spheres in each puffed cell of the perturbed lattice $S_{{\mathbb{T}}^3}$ and integrating over the union of all such spheres where each cell is of arbitrary small measure containing spheres results in a finite well defined integral bounded by a now finite supremum. The supremum on the left most side of above chain of inequalities may exist for certain Elliptic functions defined on the 3-Torus with limiting parameters as the present study aims to show. Then integration over ${\mathbb{T}}^3$ is well defined and we can replace $S_{{\mathbb{T}}^3}$ by ${\mathbb{T}}^3$. Now ${\overrightarrow{\omega }}_1$ is vorticity $\overrightarrow{\omega }$ multiplied with $u_z$ and to begin with ${\nabla }_D^{-1}$ is the inverse of the divergence operator integrated over an arbitrary cell of the 3-Torus lattice. Here the identity $A\times B=\mid A\Vert B\mid sin\left(\theta \right)\overrightarrow{n}$ was used and in the previous two integrals over a periodic lattice, the first one approaches zero as R becomes arbitrarily large, where the 3-Torus being a compact manifold is bounded by an R-Ball with the norm of the gradient of $u_z$ approaching zero and the second integral consisting of $\mid \nabla \overrightarrow{b}\mid$ approaches zero for the finite but large volume 3-Torus. The Preköpa-Leindler and Gagliardo-Nirenberg Inequalities have been used for each

$${\left|\mathrm{\Omega }\right|}^{\frac{1}{2}}{‖\nabla u_j‖}_2\le C{‖u_j‖}_q^{1-\theta }\left[{\left|\mathrm{\Omega }\right|}^{\frac{1}{2}}{‖{\nabla }^2{u}_j‖}_r^{\theta }\right]\le C{‖u_j‖}_q^{1-\theta }{\int }_{\mathrm{\Omega }}{\nabla }^2{u}_{j}d\overrightarrow{x}$$

Here ${\sum }_{j=1}^3{u}_j=\mathrm{\Psi }\ast w$, where * is convolution and ${\int }_{{\mathbb{T}}^3}wd\mathbf{x}=\mathbf{s}\in \mathbb{R}$. If $\mathrm{\Psi }$ is the fundamental solution of the scalar Laplacian on the 3-Torus ${\mathbb{T}}^3=S^1\times S^1\times S^1$ noting that $\Delta \left(\mathrm{\Psi }\ast w\right)=w$ then the integral of the Laplacian is in general non zero [9]. We must rely on the dimension of the Lattice to ensure the limit value is zero upon dividing by large enough $\mid \mathrm{\Omega }\mid$. Off of the associated compact set the velocity is zero or the velocity has compact support. The chain of inequalities at top of this page imply that, in general ${\nabla }_D^{-1}\left({\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 4}\right)=\overrightarrow{r}\mathrm{\Phi }\left(t\right)$ since the two vectors $\overrightarrow{r}$ and ${\nabla }_D^{-1}$ can be in the same direction. A term $\mathrm{\Phi }\left(t\right)$ is also multiplied by $\overrightarrow{r}$. A group G of transformations of $\overrightarrow{u}\left(\overrightarrow{r},t\right)$ is a symmetry group of NS if for all $g\in G,\overrightarrow{u}$ a NS solution implies $g\overrightarrow{u}$ is a NS solution. The group is $\mathbb{Z}$. The Navier Stokes equations are invariant under the dilation group as shown after Eq. (4). Next there will be an application of the group transformations seen in Eqs. (5)–(8):

$${\mathcal{G}}_{\delta 1}=-\left(1-\frac{1}{\delta }\right)\left(\frac{\partial u_z}{\partial t}\right)\left(\frac{\partial u_z}{\partial t}-\frac{\mu }{\rho }{\nabla }^2{u}_z+\frac{1}{\rho }\frac{\partial P}{\partial z}\right)$$E5

$${\mathcal{G}}_{\delta 2}=u_z\frac{\partial u_z}{\partial z}\frac{\partial u_z}{\partial t}+u_z^2\frac{{\partial }^2{u}_z}{\partial z\partial t}+\frac{2u_z}{\delta }\left(\frac{\partial \mathbf{u}}{\partial t}\cdot \nabla u_z\right)$$E6

$${\mathcal{G}}_{\delta 3}={\iint }_{\mathrm{\partial \Omega }}\frac{u_z}{\rho }\left(\frac{1}{\delta }{u}_z{\nabla }_{\mathit{xy}}P+\frac{\partial P}{\partial z}\overrightarrow{b}\right)\cdot \overrightarrow{n}\mathit{dS}-{\int }_{\mathrm{\Omega }}\left|\left|\left(\frac{\overrightarrow{b}}{\Vert \overrightarrow{b}\Vert }\cdot \nabla u_z\right)\frac{\partial u_z}{\partial t}\overrightarrow{b}\right|\right|\mathit{dV}$$E7

$${\mathcal{G}}_{\delta 4}={\delta }^2{\overrightarrow{F}}_T\cdot \nabla u_z^2-{\delta }^3{u}_z\frac{\partial u_z}{\partial z}{F}_z+{\delta }^3\overrightarrow{b}\cdot \nabla \left(u_z{F}_z\right)+{\delta }^3\left(1-\frac{1}{\delta }\right)\frac{F_z}{u_z}\frac{\partial u_z}{\partial t}$$E8

where $\overrightarrow{b}=\frac{1}{\delta }\left(u_x\overrightarrow{i}+u_y\overrightarrow{j}+u_z\overrightarrow{k}\right)$, and $\overrightarrow{i},\overrightarrow{j}$ and $\overrightarrow{k}$ are the standard unit vectors. For Poisson’s Equation seen in Eq. (9) (see [6, 8]), the second derivative $P_{\mathit{zz}}$ is set equal to the second derivative obtained in the ${\mathcal{G}}_{\delta 1}$ expression as part of $\mathcal{G}$,

$$\begin{array}{c}{P}_{\mathit{zz}}=-2u_z{\nabla }^2{u}_z-{\left(\frac{\partial u_z}{\partial z}\right)}^2+\frac{1}{\eta }\frac{\partial }{\partial z}\left(\frac{\partial u_z}{\partial x}+\frac{\partial u_z}{\partial y}\right)-\delta u_x\frac{{\partial }^2{u}_z}{\partial z\partial x}-\\ \delta u_y\frac{{\partial }^2{u}_z}{\partial z\partial y}+{\left(\frac{\partial u_x}{\partial x}\right)}^2+2\frac{\partial u_x}{\partial y}\frac{\partial u_y}{\partial x}+{\left(\frac{\partial u_y}{\partial y}\right)}^2\end{array}$$E9

where the last three terms on rhs of Eq. (9) can be shown to be equal to $-\left(P_{\mathit{xx}}+P_{\mathit{yy}}\right)$. Along with Eqs. (4)–(9), the continuity equation in Cartesian co-ordinates, is ${\nabla }^i{u}_i=0$. Furthermore the right hand side of the one parameter group of transformations are next mapped to $\eta$ variable terms,

$$u_i=\frac{1}{\eta }{v}_i,P=\frac{1}{{\eta }^2}Q,x_i=\eta y_i,t={\eta }^{2}s,i=\mathrm{1,2,3}.$$E10

and Eq. (4) becomes,

$$\mathcal{G}\left(\eta \right)=\mathcal{G}{\left(\eta \right)}_{\delta 1}+\mathcal{G}{\left(\eta \right)}_{\delta 2}+\mathcal{G}{\left(\eta \right)}_{\delta 3}+\mathcal{G}{\left(\eta \right)}_{\delta 4}=0$$E11

where,

$$\mathcal{G}{\left(\eta \right)}_{\delta 1}=\frac{1}{{\eta }^6}\left[\left({\delta }^{-1}-1\right){\left(\frac{\partial v_3}{\partial s}\right)}^2+\frac{\mu \left(\frac{\partial v_3}{\partial s}\right)\left(\frac{{\partial }^2{v}_3}{\partial {y_1}^2}+\frac{{\partial }^2{v}_3}{\partial {y_2}^2}+\frac{{\partial }^2{v}_3}{\partial {y_3}^2}\right)}{\rho }\left(1-{\delta }^{-1}\right)+\frac{\left({\delta }^{-1}-1\right)\left(\frac{\partial v_3}{\partial s}\right)\frac{\partial Q}{\partial y_3}}{\rho }\right]$$E12

$$\mathcal{G}{\left(\eta \right)}_{\delta 2}=\frac{v_3}{{\eta }^6}\left(\frac{\partial v_3}{\partial y_3}\right)\frac{\partial v_3}{\partial s}+\frac{{\left(v_3\right)}^2}{{\eta }^6}\frac{{\partial }^2{v}_3}{\partial y_3\partial s}+\frac{2\left(\frac{\partial v_1}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_1}+2\left(\frac{\partial v_2}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_2}+2\left(\frac{\partial v_3}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_3}}{{\mathit{\delta \eta }}^6}$$E13

$$\mathcal{G}{\left(\eta \right)}_{\delta 3}=\frac{1}{{\eta }^3}\times \left[{\iint }_S\left(\frac{1}{\mathit{\delta \rho }}{v}_3^2{\nabla }_{y_1{y}_2}Q+\frac{1}{\delta }\overrightarrow{v}\frac{1}{\rho }{v}_3\frac{\partial Q}{\partial y_3}\right)\cdot \overrightarrow{n}\mathit{dS}-{\int }_{\mathrm{\Omega }}\frac{\Vert \frac{\partial v_3}{\partial s}\overrightarrow{b}\cdot \left(\overrightarrow{b}\otimes \nabla v_3\right)\Vert }{\Vert \overrightarrow{b}\Vert }\mathit{dV}\right]$$E14

$$\mathcal{G}{\left(\eta \right)}_{\delta 4}=\frac{1}{{\eta }^3}\left[{\delta }^2{\overrightarrow{F}}_T\cdot {\nabla }_{y_1{y}_2}{v}_3^2-{\delta }^3{v}_3\frac{\partial v_3}{\partial y_3}{F}_z+{\delta }^2\overrightarrow{v}\cdot \nabla \left(v_3{F}_z\right)\right]$$E15

where $\overrightarrow{v}=\left(v_1,v_2,v_3\right)$ and ${\overrightarrow{F}}_T=F_{T_1}\overrightarrow{i}+F_{T_2}\overrightarrow{j}$. The body force ${\mathbf{F}}^{\ast }={\mathrm{F}}^{\ast \mathrm{i}}{\mathbf{e}}_{\mathrm{i}}$, with ${\overrightarrow{F}}_T=\left(F_{T_1}\left(y_1,y_2,y_3,s\right),F_{T_2}\left(y_1,y_2,y_3,s\right)\right)$ and $F_z$ is the z–component of the force vector. P depends on $\eta$ as $P=\frac{1}{{\eta }^2}Q$. Thus $\frac{\partial P}{\partial z}=\frac{1}{{\eta }^3}\frac{\partial Q}{\partial y_3}$. We solve for $\frac{\partial P}{\partial z}$ and using Poisson’s equation Eq. (9), set second derivatives of P w.r.t. z equal to each other, and then set $\delta \approx 1$ after multiplying a factor of $\delta -1$ out of the equation. This makes $A=\delta -1$, a (small) perturbation parameter. Considering the Kinematic Viscosity $\nu =\mu /\rho$, since it was shown that there exists a C such that ${\left(\frac{\partial u_z}{\partial t}\right)}^2=C^2{\nu }^2{\left[\frac{{‖\nabla u_z‖}_2^2}{u_z}\right]}^2$ in earlier work ([6], page 392) then since it was also demonstrated ${‖\nabla u_z‖}_2^2=O\left(\mathit{ϵ\eta }\right)=\frac{\eta }{\parallel \mathrm{\Omega }\parallel }$, due to increasing measure of $\mathrm{\Omega }$, it can be seen that ${\left(\frac{\partial u_z}{\partial t}\right)}^2=\frac{{\nu }^2{ϵ}^2{\eta }^2}{{\eta }^2{v}_3^2}=\frac{{\nu }^2{ϵ}^2}{v_3^2}=\frac{{\zeta }^2}{v_3^2}$. This implies that the constant C is given as in [10]:

$$C=inf\frac{{\int }_{\mathrm{\Omega }}\parallel \nabla u{\parallel }^2\mathit{dv}{\left({\int }_{\mathrm{\Omega }}{u}^2\mathit{dv}\right)}^{2/d}}{{\int }_{\mathrm{\Omega }}\parallel u-u_{\mathrm{\Omega }}{\parallel }^{2+4/d}\mathit{dv}}$$

where $d=3$ and the infimum is taken over functions $u\in W^{1,1}\left(\mathrm{\Omega }\right)$ and $u_{\mathrm{\Omega }}$ is its average $\parallel \mathrm{\Omega }{\parallel }^{-1}{\int }_{\mathrm{\Omega }}\mathit{udv}$. Calculation of above integration terms leads to the identification that $C\approx O\left(1\right)$, giving ${\eta }^6$ order when transforming with Eq. (10) in $C^2{\nu }^2{\left[\frac{{‖\nabla u_z‖}_2^2}{u_z}\right]}^2$. The final operators become independent of $\eta$ and the equation is in the form,

$$\begin{array}{c}\mathcal{L}={\mathcal{L}}_1+{\mathcal{L}}_2+\frac{{\eta }^3}{ϵ}{\mathcal{L}}_3+\frac{{\eta }^3}{ϵ}{\mathcal{L}}_4=0\end{array}$$E16

where the four components are defined as:

$${\mathcal{L}}_1={\nu }^2{ϵ}^2\left({\delta }^{-1}-1\right)v_3^{-2}+\left(1-{\delta }^{-1}\right)\left(\frac{\partial v_3}{\partial s}\right)\left(\frac{{\partial }^2{v}_3}{\partial {y_1}^2}+\frac{{\partial }^2{v}_3}{\partial {y_2}^2}+\frac{{\partial }^2{v}_3}{\partial {y_3}^2}\right)+\left({\delta }^{-1}-1\right)\frac{\left(\frac{\partial v_3}{\partial s}\right)\frac{\partial Q}{\partial y_3}}{\rho }$$E17

$${\mathcal{L}}_2=v_3\left(\frac{\partial v_3}{\partial y_3}\right)\frac{\partial v_3}{\partial s}+{\left(v_3\right)}^2\frac{{\partial }^2{v}_3}{\partial y_3\partial s}+\frac{1}{\delta }\left[2\left(\frac{\partial v_1}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_1}+2\left(\frac{\partial v_2}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_2}+2\left(\frac{\partial v_3}{\partial s}\right)v_3\frac{\partial v_3}{\partial y_3}\right]$$E18

$${\mathcal{L}}_3={\iint }_S\left(\frac{1}{\mathit{\delta \rho }}{v}_3^2{\nabla }_{y_1{y}_2}Q+\frac{1}{\delta }\overrightarrow{v}\frac{1}{\rho }{v}_3\frac{\partial Q}{\partial y_3}\right)\cdot \overrightarrow{n}\mathit{dS}-{\int }_{\mathrm{\Omega }}\frac{\Vert \frac{\partial v_3}{\partial s}\overrightarrow{b}\cdot \left(\overrightarrow{b}\otimes \nabla v_3\right)\Vert }{\Vert \overrightarrow{b}\Vert }\mathit{dV}$$E19

$${\mathcal{L}}_4={\delta }^2{\overrightarrow{F}}_T\cdot {\nabla }_{y_1{y}_2}{v}_3^2-{\delta }^3{v}_3\frac{\partial v_3}{\partial y_3}{F}_z+{\delta }^2\overrightarrow{v}\cdot \nabla \left(v_3{F}_z\right)$$E20

It can be seen that the expressions in square brackets in Eqs. (14) and (15) are ${\mathcal{L}}_3$ & ${\mathcal{L}}_4$ in Eqs. (19) and (20), respectively. Finally, in subsequent sections the Weierstrass degenerate elliptic P function will be used. Letting the Weierstrass P function be denoted by $\wp \left(z,g2,g3\right)$, the degenerate case can be denoted as $P_m\left(z\right)=\wp \left(z,3m^2,m^3\right)$.

3.1 Decomposition of NSEs

For Eqs. (4)–(9) the Dirichlet condition ${\overrightarrow{u}}^{\ast }\left({\overrightarrow{x}}^{\ast },0\right)=\overrightarrow{\xi }\left({\overrightarrow{x}}^{\ast }\right)$ such that $\nabla \cdot \overrightarrow{\xi }=0$ describes the NSEs together with an incompressible initial condition. Considering periodic boundary conditions defined on 3-torus with associated Lattice is a periodic BVP for the NSEs. Solutions were found to be in the form,

$$\mathbf{u}=\left(u_x,u_y,u_z\right):{\mathbb{R}}^{+}\times {\mathbb{R}}^3/{\mathbb{Z}}^3\to {\mathbb{R}}^3$$E21

where $u_x,u_y$ and $u_z$ satisfy Eqs. (4)–(9).

3.2 Liutex vector and respective governing equations

Theorem 1 in [6] is used in the above decomposition of (PNS) and is the basis theorem of this paper. By using the generalized divergence theorem for scalar products, the term $\nabla u_z^2\cdot \frac{\partial \overrightarrow{b}}{\partial t}$ in Eq. (6) is extended to 3-components of velocity. (see Eq. (13) in [8] which only incorporated a 2-component velocity field). No finite time blowup was obtained for the simplified case there. Solutions are obtained symbolically with Maple 2021 software, and with the use of Poisson’s equation, Eqs. (4)–(9), lead to,

$$L=L_1+2\left(L_2+L_3\right)\frac{\partial v_3}{\partial s}=0$$

with $L_1,L_2$ and $L_3$ expressions given in Appendix 1. Solving symbolically for $L=0$ individually for the mixed partial derivatives in the expression $\left(\frac{{\partial }^2{v}_2}{\partial y_3\partial s}-\frac{{\partial }^2{v}_1}{\partial y_3\partial s}\right)$, using the following definition of $\kappa$ in terms of $v_1,v_2$ and $v_3$, [see Eq. (33) in [1]. The new equation there has a viscous term reduced to half and one only needs to calculate 3 elements.]

$$\begin{array}{c}-\left(\frac{{\partial }^2{v}_2}{\partial y_3\partial s}-\frac{{\partial }^2{v}_1}{\partial y_3\partial s}\right)=\frac{\partial }{\partial s}\kappa \left(y_1,y_2,y_3,s\right)-\frac{{\partial }^2{v}_3}{\partial y_2\partial s}+\frac{{\partial }^2{v}_3}{\partial y_1\partial s}\end{array}$$E22

Both,

$$M_1=\frac{{\partial }^2{v}_1}{\partial y_3\partial s}$$E23

and

$$M_2=\frac{{\partial }^2{v}_2}{\partial y_3\partial s}$$E24

are nonlinear partial differential equations. The $v_1$ and $v_2$ velocities are chosen respectively as the following general spatial–temporal functions, which are assumed to fulfill compatibility conditions in [11, 12],

$$v_1\left(y_1,y_2,y_3,s\right)=\left(U_1\left(y_1,y_2,y_3\right)+A\right)\times f_1\left(s\right)$$E25

and

$$v_2\left(y_1,y_2,y_3,s\right)=\left(U_2\left(y_1,y_2,y_3\right)+A\right)\times f_2\left(s\right)$$E26

where $A\ll 1$ and positive. Note that the magnitude of Liutex (scalar form) is obtained in the plane perpendicular to the local axis, which is twice the angular speed of local fluid rotation,

$${\overrightarrow{\omega }}_L=\frac{2}{{\left|\overrightarrow{r}\right|}^2}\left(\overrightarrow{r}\times \overrightarrow{v}\right)$$E27

where ${\omega }_L$ is associated with the Liutex vector part of vorticity, $r=y_1\overrightarrow{i}+y_2\overrightarrow{j}+y_3\overrightarrow{k}$ and where the Liutex magnitude difference is calculated as follows,

$$\begin{array}{c}\kappa \left(y_1,y_2,y_3,s\right)=2\frac{y_2{v}_3-y_3{v}_2-y_1{v}_3+y_3{v}_1}{{y_1}^2+{y_2}^2+{y_3}^2}\end{array}$$E28

Substituting $M_1-M_2$ expressions in Eqs. (23) and (24) and $\kappa$ from Eq. (28) into Eq. (22) gives a new PDE,

$$\mathcal{F}=\frac{{\mathcal{F}}_1}{H}+2\frac{{\mathcal{F}}_2}{H}+{\mathcal{F}}_3=0,$$E29

$$H=\left(\frac{\partial v_3}{\partial s}\right)v3\left(\frac{\partial v_3}{\partial y1}\right)\frac{\partial v_3}{\partial y2}$$

$$\begin{array}{c}{\mathcal{F}}_1=\left[\frac{1}{2}\left(\frac{\partial v_3}{\partial s}\right)v_3\frac{{\partial }^3{v}_3}{\partial {y_3}^2\partial s}+\left(\frac{{\partial }^2{v}_3}{\partial y_3\partial s}{v}_3\right)(-\frac{1}{2}{v}_3\frac{{\partial }^2{v}_3}{\partial y_3\partial s}+\left(-F_{T_1}-U_1\right)\frac{\partial v_3}{\partial y_1}{v}_3+\right.\\ \left.\left(-U_2-F_{T_2}\right)\frac{\partial v_3}{\partial y_2}-\frac{\mathrm{\Phi }\left(s\right)}{2}+\left(\frac{\partial v_3}{\partial s}\right)\frac{\partial v_3}{\partial y_3})\right]\times \left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\end{array}$$E30

$$\begin{array}{c}{\mathcal{F}}_2=\left[v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\left(F_{T_1}+U_1\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial y_1}+v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\left(F_{T_2}+U_2\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial y_2}+\right.\\ \frac{3}{2}\left(\frac{\partial v_3}{\partial s}\right)v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^2{v}_3}{\partial {y_3}^2}+\frac{3}{2}{\left(\frac{\partial v_3}{\partial y_3}\right)}^2\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{\partial v_3}{\partial s}+\\ \left(\left(F_{T_1}+U_1\right)\frac{\partial v_3}{\partial y_3}+v_3\left(\frac{\partial U_1}{\partial y_3}{U}_1+\frac{\partial F_{T_1}}{\partial y_3}\right)\right){\left(\frac{\partial v_3}{\partial y_1}\right)}^2-\left(\frac{\partial v_3}{\partial y_2}\right)\left(\left(U_1-U_2+F_{T_1}-F_{T_2}\right)\frac{\partial v_3}{\partial y_3}+\right.\\ \left.v_3\left(\frac{\partial F_{T_1}}{\partial y_3}-\frac{\partial F_{T_2}}{\partial y_3}\right))\frac{\partial v_3}{\partial y_1}-{\left(\frac{\partial v_3}{\partial y_2}\right)}^2\left(\left(F_{T_2}+U_2\right)\frac{\partial v_3}{\partial y_3}+v_3\left(\frac{\partial U_2}{\partial y_3}+\frac{\partial F_{T_2}}{\partial y_3}\right)\right)\right]\frac{\partial v_3}{\partial s}\end{array}$$E31

$$\begin{array}{c}{\mathcal{F}}_3=-\left[\frac{\left(-2y_1+2y_2\right)\frac{\partial v_3}{\partial s}+2y_3\left(U_1\left(y_1,y_2,y_3\right)-U_2\left(y_1,y_2,y_3\right)\right)}{{y_1}^2+{y_2}^2+{y_3}^2}-\frac{{\partial }^2{v}_3}{\partial y_2\partial s}+\frac{{\partial }^2{v}_3}{\partial y_1\partial s}\right]\end{array}$$E32

$$\begin{array}{c}{F}_{T_1}\left(y_1,y_2,y_3,s\right)=f_0\left(s\right)\left(F\left(y_1,y_2,y_3\right)+A\right)\\ F_{T_2}\left(y_1,y_2,y_3,s\right)=f_0\left(s\right)\left(G\left(y_1,y_2,y_3\right)+A\right)\end{array}$$E33

3.3 Case 1

An example of a smooth force $f_0\left(s\right)$ at $s=s_0+\alpha$ is considered with it’s accompanying solution of Eq. (29),

$$\begin{array}{c}{f}_0\left(s\right)=sech{\left(s-s_0\right)}^2\\ \frac{{\mathit{dF}}_4\left(s\right)}{\mathit{ds}}=\sqrt[3]{3c_4\int \mathrm{\Phi }\left(s\right)\mathrm{d}s+C_1},\text{smooth}{f}_0\left(s\right)\end{array}$$E34

For small $m\ll 1$,

$$\begin{array}{c}F\left(y_1,y_2,y_3\right)=\mathcal{P}{\left(y_1,3m^2,m^3\right)}^{-1}\times \mathcal{P}{\left(y_2,3m^2,m^3\right)}^{-1}\times \mathcal{P}{\left(y_3,3m^2,m^3\right)}^{-1}\\ \\ G\left(y_1,y_2,y_3\right)=\mathcal{P}{\left(y_1,3m^2,m^3\right)}^{-1}\times \mathcal{P}{\left(y_2,3m^2,m^3\right)}^{-1}\times \mathcal{P}{\left(y_3,3m^2,m^3\right)}^{-1}\end{array}$$E35

Setting $v_1$ and $v_2$, as

$$\begin{array}{c}{v}_1=f_{a_1}\left(s\right)\left(\mathcal{P}{\left(y_1,3m^2,m^3\right)}^{-1}\times \mathcal{P}(y_2,3m^2,m^3){}^{-1}\times \mathcal{P}(y_3,3m^2,m^3){}^{-1}+A\right)\\ \\ v_2=f_{a_2}\left(s\right)\left(\mathcal{P}{\left(y_1,3m^2,m^3\right)}^{-1}\times \mathcal{P}(y_2,3m^2,m^3){}^{-1}\times \mathcal{P}(y_3,3m^2,m^3){}^{-1}+A\right)\end{array}$$E36

where as an example $f_{a_1}\left(s\right)=f_{a_2}\left(s\right)=tanh\left(s-s_0\right)$, a tan hyperbolic linearization in s at $s_0$. Here the relationship between $f_0$ and each of $v_i$ is $f_0=\frac{d}{\mathit{ds}}{f}_{a_i}.{\mathcal{P}}^{-1}$ is the reciprocal of the degenerate Weierstrass P function with parameter m plotted for some m values listed in captions in Figure 1. The definition of degenerate function is,

Figure 1.

Plots of the reciprocal of the degenerate Weierstrass P functions in two dimensions $\left(y_1,y_2\right)$ given relative to the canonical Weierstrass P functions $\wp \left(y_i,g2,g3\right)$ as ${\mathcal{P}}_{m=n}\left(y_i\right)=\wp \left(y_i,3n^2,n^3\right)$. (a) Reciprocal of Weierstrass degenerate P function for g2 = 3m², g3 = m³, m = 1. (b) Reciprocal of Weierstrass degenerate P function for g2 = 3m², g3 = m³, m = $\frac{1}{2}$. (c) Reciprocal of Weierstrass degenerate P function for g2 = 3m², g3 = m³, m = $\frac{7}{20}$. (d) Reciprocal of Weierstrass degenerate P function for g2 = 3m², g3 = m³, m = $\frac{3}{20}$.

$$\mathcal{P}\left(z,3m^2,m^3\right)=-\frac{m}{2}+\frac{3}{2}mcsc{\left(\frac{z\sqrt{6}\sqrt{m}}{2}\right)}^2$$E37

3.4 Case 2

An example of a smooth force $f_0\left(s\right)$ at $s=s_0$ is considered with it’s accompanying solution of Eq. (29),

$$\begin{array}{c}{f}_0\left(s\right)=sech{\left(s-s_0\right)}^2\\ \\ \frac{{\mathit{dF}}_4\left(s\right)}{\mathit{ds}}=c_4\frac{{\mathit{df}}_0\left(s\right)}{\mathit{ds}}\text{smooth},f_0\left(s\right)\end{array}$$E38

Setting $v_1$ and $v_2$, as

$$\begin{array}{c}{v}_1=f_{a_3}\left(s\right)\mathcal{P}\left(y_1,3m^2,m^3\right)\times \mathcal{P}\left(y_2,3m^2,m^3\right)\times \mathcal{P}\left(y_3,3m^2,m^3\right)\\ \\ v_2=f_{a_4}\left(s\right)\mathcal{P}\left(y_1,3m^2,m^3\right)\times \mathcal{P}\left(y_2,3m^2,m^3\right)\times \mathcal{P}\left(y_3,3m^2,m^3\right)\end{array}$$E39

$$f_{a_3}\left(s\right)=-f_{a_4}\left(s\right)$$E40

where $\mathcal{P}$ is the degenerate Weierstrass P function with parameter m. Also for case 2, $\mathrm{\Phi }\left(s\right)=0$ and the relationship between $f_0$ and each of $v_i$ is $f_0=\frac{d}{\mathit{ds}}{f}_{a_i}$. The general reducing solution in Section 4 will also be considered for the two cases $\mathrm{\Phi }\left(s\right)=0$ and $\mathrm{\Phi }\left(s\right)\ne 0$. It follows that $\mathrm{\Phi }\left(s\right)=0$ iff $v_1=-v_2$, from which it follows that ${\iint }_S{v}_3^2\left(n_3\frac{\partial P}{\partial y_3}+\left({\nabla }_{y_1{y}_2}P\cdot \overrightarrow{n}\right)\right)\mathit{dS}=0$ and the slope of the time dependent linear solution of $v_3$ is arbitrarily small.

For case 2, for spatially non-smooth $v_1$ and $v_2$, the solution of Eq. (29) is in the form $v_3=F_4\left(s\right)F_5\left(y_1,y_2,y_3\right)$ which satisfies,

$$\frac{\mathrm{d}}{\mathrm{d}t}{F}_4\left(s\right)=c_4\frac{{\mathit{df}}_0}{\mathit{ds}}$$E41

$$\begin{array}{l}{\left(P\left(y_2,3m^2,m^3\right)\right)}^2\left(\left(\frac{{\partial }^2{F}_5}{\partial y_3\partial y_1}\right)F_5+\left(\frac{{\partial }^2{F}_5}{\partial y_3\partial y_2}{F}_5\right)F_5+1/2\left(\frac{\partial F_5}{\partial y_3}\right)\left(\frac{\partial F_5}{\partial y_1}+\frac{\partial F_5}{\partial y_2}\right)\right)\times \\ {\left(P\left(y_1,3m^2,m^3\right)\right)}^2\left({y_1}^2+{y_2}^2+{y_3}^2\right)\left(\frac{\partial F_5}{\partial y_1}-\frac{\partial F_5}{\partial y_2}\right){\left(P\left(y_3,3m^2,m^3\right)\right)}^3+P\left(y_2,3m^2,m^3\right)\\ \times \left(P\left(y_2,3m^2,m^3\right)P^{\prime }\left(y_3,3m^2,m^3\right)\left(\frac{\partial F_5}{\partial y_1}-\frac{\partial F_5}{\partial y_2}\right)\left(\frac{\partial F_5}{\partial y_1}+\frac{\partial F_5}{\partial y_2}\right)\right.\\ \times \left({y_1}^2+{y_2}^2+{y_3}^2\right)P\left(y_1,3m^2,m^3\right)+1/4\left(7F_5\left({y_1}^2+{y_2}^2+{y_3}^2\right)\left(\frac{\partial F_5}{\partial y_1}-\frac{\partial F_5}{\partial y_2}\right)\frac{{\partial }^2{F}_5}{\partial {y_3}^2}\right.\\ -2\left(\frac{\partial F_5}{\partial y_2}\right)\left({y_1}^2+{y_2}^2+{y_3}^2\right){\left(\frac{\partial F_5}{\partial y_1}\right)}^2+7\left(\left(2/7{y_1}^2+2/7{y_2}^2+2/7{y_3}^2\right){\left(\frac{\partial F_5}{\partial y_2}\right)}^2\right.\\ \left.+4/7F_5\left(y_1-y_2\right)\frac{\partial F_5}{\partial y_2}+{\left(\frac{\partial F_5}{\partial y_3}\right)}^2\left({y_1}^2+{y_2}^2+{y_3}^2\right)\right)\frac{\partial F_5}{\partial y_1}\\ \left.\left.-7\left(\frac{\partial F_5}{\partial y_2}\right){\left(\frac{\partial F_5}{\partial y_3}\right)}^2\left({y_1}^2+{y_2}^2+{y_3}^2\right)\right)c_4\right)F_5\times P\left(y_1,3m^2,m^3\right){\left(P\left(y_3,3m^2,m^3\right)\right)}^2\\ -\left(\left(\frac{{\partial }^2{F}_5}{\partial y_3\partial y_1}\right)F_5+\left(\frac{{\partial }^2{F}_5}{\partial y_3\partial y_2}\right)F_5+1/2\left(\frac{\partial F_5}{\partial y_3}\right)\left(\frac{\partial F_5}{\partial y_1}+\frac{\partial F_5}{\partial y_2}\right)\right)\left({y_1}^2+{y_2}^2+{y_3}^2\right)\\ \times \left(\frac{\partial F_5}{\partial y_1}-\frac{\partial F_5}{\partial y_2}\right)P\left(y_3,3m^2,m^3\right)+P^{\prime }\left(y_3,3m^2,m^3\right)F_5\left(\frac{\partial F_5}{\partial y_1}-\frac{\partial F_5}{\partial y_2}\right)\left(\frac{\partial F_5}{\partial y_1}+\frac{\partial F_5}{\partial y_2}\right)\\ \times \left({y_1}^2+{y_2}^2+{y_3}^2\right)=0\end{array}$$E42

The solution of Eq. (41) for $F_4\left(s\right)$ is,

$$F_4=c_4{f}_0\left(s\right)+C_1$$E43

where it is observed that it is smooth at $s=s_0$ and the limit of $F_4$ as $s\to \infty$ should be bounded. (In general $f_0\left(s\right)$ is assumed to be bounded and the same for both case 1 and 2.) The solution of Eq. (42) is given in Appendix 2.

For Case 1, associated with smooth $v_1$ and $v_2$ with $f_0$ and $f_1$ given by Eq. (34) and $f_{a_i}=tanh\left(s-s_0\right)$, the following solutions exist,

$$\frac{\mathrm{d}}{\mathrm{d}s}{F}_4\left(s\right)=\frac{c_4\mathrm{\Phi }\left(s\right)}{F_4^2\left(s\right)}$$E44

with solution,

$$F_4\left(s\right)=\sqrt[3]{3c_4\int \mathrm{\Phi }\left(s\right)\mathrm{d}s+C_1}$$E45

If $\mathrm{\Phi }\left(s\right)=-\lambda$ where $\lambda >0$, then,

$$F_4=\sqrt[3]{-c_4\lambda s+C_1}$$

and a non-smooth solution in time s is observed, that is a finite time blowup at $s=s_0$ for this $\mathrm{\Phi }\left(s\right)$ starting with the first derivative of $F_4$ and higher. Define $\mathrm{\Phi }\left(s\right)$ such that $F_4=B+\sqrt[3]{-c_4\lambda s+C_1}$, where B is a constant. It can be verified that near $s=s_0$Eq. (14) gives $\mathrm{\Phi }\left(s\right)=-\lambda$ from the tensor product term. Furthermore the spatial solution is given by,

$$\begin{array}{l}-4c_4{F}_5^3\left(y_1^2+y_2^2+y_3^2\right)\left(\frac{\partial F_5}{\partial y_1}-\frac{\partial F_5}{\partial y_2}\right)\frac{{\partial }^2{F}_5}{\partial {y_3}^2}+2c_4{F}_5^2\left(\frac{\partial F_5}{\partial y_2}\right)\left(y_1^2+y_2^2+y_3^2\right){\left(\frac{\partial F_5}{\partial y_1}\right)}^2\\ +(-2c_4{F}_5^2\left(y_1^2+y_2^2+y_3^2\right){\left(\frac{\partial F_5}{\partial y_2}\right)}^2-4c_4{F}_5^3\left(y_1-y_2\right)\frac{\partial F_5}{\partial y_2}\\ -\left(4y_1^2+4y_2^2+4y_3^2\right)\left(\frac{\partial F_5}{\partial y_3}\right)\left(c_4{F}_5^2\frac{\partial F_5}{\partial y_3}-1/4\right))\frac{\partial F_5}{\partial y_1}\\ +\left(4y_1^2+4y_2^2+4y_3^2\right)\left(\frac{\partial F_5}{\partial y_3}\right)\left(c_4{F}_5^2\frac{\partial F_5}{\partial y_3}-1/4\right)\frac{\partial F_5}{\partial y_2}\\ =0\end{array}$$E46

In Appendix 2, the general solution for $F_5$ PDE associated with Case 1 is,

$$F_5\left(y_1,y_2,y_3\right)=\begin{array}{l}G\left(y_1,y_2\right)\sqrt{B}[-m/2+3/2m(csc(1/4\frac{2^{2/3}{y}_3}{\sqrt{-G\left(y_1,y_2\right)B{2}^{2/3}}}\\ {+C/2\sqrt{6}\sqrt{m}))}^2]\end{array}$$

Using the continuity equation a specific form for the surface $y_3=F\left(y_1,y_2\right)$ emerges. Substitution of $F_1,F_2$ and $F_5$ (on the surface $y_3=-F\left(y_1,y_2\right)$ into the continuity equation gives a surface to be described below, Differentiating $v_3$ wrt to $y_3$ gives,

$$\begin{array}{c}\frac{\partial v_3}{\partial y_3}=-3/4\frac{G\left(y_1,y_2\right)\sqrt{B}{m}^{3/2}{2}^{2/3}\sqrt{6}}{\sqrt{-\mathit{BG}\left(y_1,y_2\right)2^{2/3}}}{\left(csc\left(\left(1/4\frac{2^{2/3}{y}_3}{\sqrt{-\mathit{BG}\left(y_1,y_2\right)2^{2/3}}}+C/2\right)\sqrt{6}\sqrt{m}\right)\right)}^2\times \\ cot\left(\left(1/4\frac{2^{2/3}{y}_3}{\sqrt{-\mathit{BG}\left(y_1,y_2\right)2^{2/3}}}+C/2\right)\sqrt{6}\sqrt{m}\right)\end{array}$$

Taking the limit as m approaches 0 and consequently C and A approaching zero gives, where the continuity equation has been used to set $\frac{\partial v_3}{\partial y_3}=-\frac{\partial v_1}{\partial y_1}-\frac{\partial v_2}{\partial y_2}$,

$$4\frac{B^{3/2}{\left(G\left(y_1,y_2\right)\right)}^2\sqrt[3]{2}}{{y_3}^3}=-\left(2y_1+2y_2\right)\left({y_3}^2+A\right)\left(y_1{y}_2+A\right)$$

Solving algebraically for $y_3$ gives two roots and setting the result to $y_1^2-y_2^2$, a saddle surface form, gives an equation which can be solved for $G\left(y_1,y_2\right)$, giving,

$$\frac{\sqrt[5]{2}\sqrt[5]{B^{3/2}{\left(G\left(y_1,y_2\right)\right)}^2\sqrt[3]{2}{y_1}^4{y_2}^4{\left(y_1+y_2\right)}^4}}{y_1{y}_2\left(y_1+y_2\right)}={y_1}^2-{y_2}^2$$

G is solved for and is exactly,

$$G\left(y_1,y_2\right)=\pm 1/2\frac{\sqrt[6]{2}\sqrt{\sqrt[3]{2}{B}^{3/2}{y}_1{y}_2\left(y_1-y_2\right)}{\left(y_1+y_2\right)}^3{\left(y_1-y_2\right)}^2}{B^{3/2}}$$

Substituting $G\left(y_1,y_2\right)$ into the expression for $F_5$ gives,

$$F_5=\pm y_1{y}_2{\left(y_1-y_2\right)}^3{\left(y_1+y_2\right)}^4$$

with plot in Figure 2a and b.

Figure 2.

Plots of oscillations at boundary of cell in star and unstarred variables. (a) Sinusoidal velocity at boundary of cell for v3. (b) Different perspective for sinusoidal velocity at boundary of cell for v3. (c) Sinusoidal velocity at boundary of cell for v3 in terms of y_i coordinates, $\frac{-y1y2(y1-y2){}^3(y1+y2){}^4}{3\times 10{}^{35}}$. (d) Sinusoidal velocity at boundary of cell for v3 in terms of x_i coordinates, $\frac{-xy(10000x-10000y){}^3(10000x+10000y){}^4}{3\times 10{}^{35}}$.

Recalling the transformations to $\eta$ variables, we now return to star variables for the original PNS system. These are shown in Figure 2d in star variables. Note that the plot in Figure 1c, shows the range in y. In Eq. (8), the ${\mathcal{G}}_{\delta 4}$ term consists of the expression which has implicitly been set to zero,

$${\mathrm{\Lambda }}_z=-{\delta }^3{v}_3\frac{\partial v_3}{\partial y_3}{F}_z+{\delta }^2\overrightarrow{v}\cdot \nabla \left(v_3{F}_z\right)=0$$

Substituting $v_3=c_4{F}_5{\left(s-s_0\right)}^{1/3}$ into this equation results in approximately zero as $m\to 0$ as $F_z$ is unbounded at corners and $v_3$ is zero at the corners. In Figure 3c$\kappa$ for $\overrightarrow{u}=u_x\overrightarrow{i}+u_y\overrightarrow{j}+u_z\overrightarrow{k}$ is shown after transforming from unstarred variables to starred ones. One can note that cancelation of oscillations will occur in a finite Lattice for $\kappa$ at the wall of adjacent cells since there the sinusoid is of equal height everywhere on [−1, 1]. Oscillations at infinity can occur as the Lattice dimension approaches infinity. A cusp-like bifurcation in vorticity field occurs indicating that there is a singularity upon using the correct definition of vorticity. This is shown in Figure 3d. Following the same recipe as above for case 1 $v_1,v_2$ smooth with $v_3$ oscillating at wall and a blowup in acceleration in time, case 2 has a new term now for $v_1+v_2$ in the continuity equation. Using this new right hand side expression it follows after some calculations that a spatially singular $F_5$ is given as,

Figure 3.

a. Plot of spatial blowup at center of cell for case 2, b. pressure function for case 1 and 2, c. Liutex magnitude difference $\kappa$ defined on the saddle surface $z=x^2-y^2$, for temporally non-smooth $u_z$ case 1 and d. the existence of a vortex is found. (a) Spatial blowup at center of cell for no-finite time blowup. Excluding the origin along main principle axis cusp bifurcations are observed. (b) Pressure function in a given cell comprising of linear and nonlinear parts for both cases 1 and 2 for u_z. (c) Liutex magnitude difference κ. (defined on the saddle surface $z=x^2-y^2$, for temporal non-smooth u_z). (d) Top view of the real vector field for $\mathit{\omega }=(\mathit{\eta },\mathit{\xi },\mathit{R}+\mathit{\epsilon })$ in terms of pure rotation, shearing and stretching. Here there is a cusp-like bifurcation along main principle axis in vorticity field. It was also evident that about y = x plane the field was parabolic. A non zero vorticity shows the existence of a vortex.

$$F_5=\frac{-{y_1}^3-{y_1}^2{y}_2+y_1{y_2}^2+{y_2}^3}{{\left({y_1}^2-{y_2}^2\right)}^2{y_1}^3{y_2}^3}$$

with plot of vorticity in Figure 3a. Finally the solutions for $v_1,v_2$ and $v_3$ in case 1 can be verified to satisfy the continuity equation if and only if $m\to 0$ for any $\eta$ arbitrarily small and positive and for case 2 the solutions for $v_1,v_2$ and $v_3$ can be verified to satisfy to arbitrary small precision the continuity equation if and only if $m\to 0$ and $\eta$ is arbitrarily small and positive. Both of these hold on a saddle surface. Summarizing, for non smooth inputs $v_1$ and $v_2=-v_1,\mathrm{\Phi }\left(s\right)=0$ and $f_0\left(s\right)\ne 0$ (a general function of s) gives $v_3$ a no finite time blowup in s, on the other hand for smooth inputs $v_1,v_2=v_1$ and $f_0\left(s\right)=1,\mathrm{\Phi }\left(s\right)\ne 0$ (see Appendix 4) gives a finite time blowup in s for the derivative of $v_3$ wrt to s. Solving for pressure for case 1 and then for case 2 velocities separately and thereby equating the second derivatives of pressure for each expression with respect to $y_3$ gives a new PDE for pressure. The plot is shown in Figure 3b. Here it can be seen that there is a max point and on the crests of the distribution function, the pressure is linear in $y_1$ and $y_2$. On the curved portions there will be finite time blowup starting with the first derivative wrt to s as $\mathrm{\Phi }\left(s\right)\ne 0$. The form of the solution for pressure P associated with the non-blowup is $P=R\left(s\right)\left({\mathit{Ay}}_1+{\mathit{By}}_2+C\right)$.

4. General solution with no restrictions on forcing and spatial velocities

Here there are no assumptions made on forcing and spatial velocities as being separable in space and time (Figure 4). PNS is made of the following component parts $C_i$ and is given by Eq. (47) (the same as Eq. (58) in Appendix 3- written in terms of special terms ${\mathrm{\Phi }}_1$ here).

Figure 4.

Top view of imaginary vector field-perpendicular along y = −x.

$$C_1+C_2=C_3$$E47

$$\begin{array}{c}{C}_1=\left(\frac{\partial v_3}{\partial s}\right){\left(v_3\right)}^2\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^3{v}_3}{\partial {y_3}^2\partial s}-{\left(v_3\right)}^2\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right){\left(\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\right)}^2-\\ \left(-2\left(\frac{\partial v_3}{\partial y_3}\right)v_3\frac{\partial v_3}{\partial s}+{\mathrm{\Phi }}_1+\mathrm{\Phi }\left(s\right)\right)\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\end{array}$$E48

$$C_2=\begin{array}{l}[\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{\partial {\mathrm{\Phi }}_1}{\partial y_3}+3\left(\frac{\partial v_3}{\partial s}\right)v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^2{v}_3}{\partial {y_3}^2}\\ -\left(2\frac{\partial v_3}{\partial y_1}-2\frac{\partial v_3}{\partial y_2}\right)v_3\left(\frac{\partial v_3}{\partial y_1}\right)\frac{{\partial }^2{v}_3}{\partial y_1\partial s}-\left(2\frac{\partial v_3}{\partial x}-2\frac{\partial v_3}{\partial y_2}\right)v_3\left(\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^2{v}_3}{\partial y_2\partial s}\\ +2v_3{\left(\frac{\partial v_3}{\partial y_1}\right)}^2\left(\frac{{\partial }^2{v}_3}{\partial y_1\partial s}+\frac{y_{3}N}{y_1^2+y_2^2+y_3^2}\right)-2v_3{\left(\frac{\partial v_3}{\partial y_2}\right)}^2\left(\frac{{\partial }^2{v}_3}{\partial y_2\partial s}-\frac{\partial }{\partial s}\left(\frac{2\left(y_2-y_1\right)v_3}{y_1^2+y_2^2+y_3^2}\right)\right)\\ +3{\left(\frac{\partial v_3}{\partial y_3}\right)}^2\left(\frac{\partial v_3}{\partial s}\right)\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)]\frac{\partial v_3}{\partial s}\end{array}$$E49

$$\begin{array}{c}{C}_3=\left[2\frac{\left(-2y_1+2y_2\right)\frac{\partial v_3}{\partial s}+2y_3\left(\frac{\partial v_1}{\partial s}-\frac{\partial v_2}{\partial s}\right)}{{y_1}^2+{y_2}^2+{y_3}^2}-2\frac{{\partial }^2{v}_3}{\partial y_2\partial s}+2\frac{{\partial }^2{v}_3}{\partial y_1\partial s}\right]\times \\ v_3\left(\frac{\partial v_3}{\partial y_1}\right)\left(\frac{\partial v_3}{\partial s}\right)\frac{\partial v_3}{\partial y_2}\end{array}$$E50

where ${\mathrm{\Phi }}_1$ is given as,

$${\mathrm{\Phi }}_1=-\left(\frac{\partial v_3}{\partial y_3}\right)v_3\frac{\partial v_3}{\partial s}-v_3^2\frac{{\partial }^2{v}_3}{\partial y_3\partial s}-\frac{1}{2}\mathrm{\Phi }\left(s\right)$$E51

It is immediate that Eq. (47) gives as solution the same form as in the main section of this paper, that is the solution of Eq. (45) multiplied by $F_5\left(y_1,y_2,y_3\right)$. The expression for ${\mathrm{\Phi }}_1$ is determined by solving for $\overrightarrow{F}\cdot \nabla v_3^2+\nabla v_3^2\cdot \frac{\partial \overrightarrow{b}}{\partial s}$ in Eq. (4) when $\delta \approx 1$. Simplifying Eq. (47) using Eq. (51) and using the definition of the Liutex part of vorticity in Eqs. (27) and (28) and solving algebraically for $N=\frac{\partial v_1}{\partial s}-\frac{\partial v_2}{\partial s}$ in Eq. (48) the problem is simplified to,

$$\begin{array}{l}-\frac{\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\left(-1/4\mathrm{\Phi }\left(s\right)\left({y_1}^2+{y_2}^2+{y_3}^2\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial s}+{\left(\frac{\partial v_3}{\partial s}\right)}^2\left(v_3\left({y_1}^2+{y_2}^2+{y_3}^2\right)\frac{{\partial }^2{v}_3}{\partial {y_3}^2}+2v_3\left(-y_2+y_1\right)\frac{\partial v_3}{\partial y_2}+{\left(\frac{\partial v_3}{\partial y_3}\right)}^2\left({y_1}^2+{y_2}^2+{y_3}^2\right)\right)\right)}{y_3\left(\frac{\partial v_3}{\partial s}\right)\left(\frac{\partial v_3}{\partial y_1}\right)v_3\left(\frac{\partial v_3}{\partial y_1}-2\frac{\partial v_3}{\partial y_2}\right)}\\ -\frac{\partial v_1}{\partial s}+\frac{\partial v_2}{\partial s}\\ =0\end{array}$$E52

This PDE is separable as $v_3=F_7\left(y_1,y_2,y_3\right)F_4\left(s\right)$, where,

$$\begin{array}{c}4F_7^3\left(\frac{{\partial }^2{F}_7}{\partial {y_3}^2}\right){y_1}^2+4F_7^3\left(\frac{{\partial }^2{F}_7}{\partial {y_3}^2}\right){y_2}^2+4F_7^3\left(\frac{{\partial }^2{F}_7}{\partial {y_3}^2}\right){y_3}^2+4F_7^2{\left(\frac{\partial F_7}{\partial y_3}\right)}^2{y_1}^2+\\ 4F_7^2{\left(\frac{\partial F_7}{\partial y_3}\right)}^2{y_2}^2+4F_7^2{\left(\frac{\partial F_7}{\partial y_3}\right)}^2{y_3}^2+8F_7^3\left(\frac{\partial F_7}{\partial y_2}\right)y_1-8\left(F_7^3\right)\left(\frac{\partial F_7}{\partial y_2}\right)y_2-\\ \left(\frac{\partial F_7}{\partial y_3}\right){y_1}^2{c}_4-\left(\frac{\partial F_7}{\partial y_3}\right){y_2}^2{c}_4-\left(\frac{\partial F_7}{\partial y_3}\right){y_3}^2{c}_4=0\end{array}$$

We have used the following equalities,

$$\frac{{\partial }^2{v}_1}{\partial y_3\partial s}=\frac{{\partial }^2{v}_3}{\partial y_1\partial s}+\frac{y_{3}N}{y_1^2+y_2^2+y_3^2}$$

$$\frac{{\partial }^2{v}_2}{\partial y_3\partial s}=\frac{{\partial }^2{v}_3}{\partial y_2\partial s}-\frac{\partial }{\partial s}\left(\frac{2\left(y_2-y_1\right)v_3}{y_1^2+y_2^2+y_3^2}\right)$$

where N is defined by,

$$N=\frac{\partial v_1}{\partial s}-\frac{\partial v_2}{\partial s}$$

5. Discussion and conclusion

Comparing solutions for Eqs. (42)–(46) it is evident that the former one has as solution for $u_z$ a time component that is smooth(except at infinity) opposite to that of the smooth force and spatially non-smooth $y_1$ and $y_2$ velocities whereas in the latter equations $u_z$ has a finite time blowup(with the first derivative and higher) for $f_0,f_1$ and $y_1$ and $y_2$ smooth velocity function inputs. (see Eq. (45)) [4]. Eq. (47) is the full non-separable reduced PDE. Oscillations of arbitrary height can occur at spatial infinity. For (PNS) it is shown that there exists a vortex in each cell of the lattice associated with ${\mathbb{T}}^3$ using the decomposition of pure rotation(Liutex), antisymmetric shear and compression and stretching. A cusp bifurcation for vorticity shows the birth and destruction of vorticies. Here it is known that streaklines can be used to give an idea of where the vorticity in a flow resides. The question of no-finite time blowup for the new eqs. [1] replacing the Navier Stokes equations is left for future study.

A. Appendix 1

Eq. (1) including forcing terms and the three associated velocities $v_1,v_2$ and $v_3$,

$$L=L_1+2\left(L_2+L_3\right)\frac{\partial v_3}{\partial s}=0$$E53

$$\begin{array}{c}{L}_1=v_3^3{\left(\frac{\partial v_3}{\partial s}\right)}^2\mu \left(-1+\delta \right)\frac{{\partial }^3{v}_3}{\partial y_3\partial {y_1}^2}+{\left(v_3\right)}^3{\left(\frac{\partial v_3}{\partial s}\right)}^2\times \\ \mu \left(-1+\delta \right)\frac{{\partial }^3{v}_3}{\partial y_3\partial {y_2}^2}+{\left(v_3\right)}^3{\left(\frac{\partial v_3}{\partial s}\right)}^2\mu \left(-1+\delta \right)\frac{{\partial }^3{v}_3}{\partial {y_3}^3}+\\ {\left(v_3\right)}^5\left(\frac{\partial v_3}{\partial s}\right)\left(\frac{{\partial }^3{v}_3}{\partial {y_3}^2\partial s}\right)\delta \rho -{\left(\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\right)}^2{\left(v_3\right)}^5\delta \rho -\\ \left(-2{\left(v_3\right)}^3\left(\frac{\partial v_3}{\partial t}\right)\left(\frac{\partial v_3}{\partial y_3}\right)\delta +2{\left(v_3\right)}^3\delta \left(F_{T1}+\frac{\partial v_1}{\partial s}\right)\frac{\partial v_3}{\partial y_1}+\right.\\ \left.2{\left(v_3\right)}^3\delta \left(F_{T2}+\frac{\partial v_2}{\partial s}\right)\frac{\partial v_3}{\partial y_2}-\delta +1+\frac{1}{2}{v}_3^2\mathrm{\Phi }\left(s\right)\right)v_3\rho \frac{{\partial }^2{v}_3}{\partial y_3\partial s}\end{array}$$E54

where $\mathrm{\Phi }$ is given by Eq. (19).

$$\begin{array}{l}{L}_2=\left[\frac{1}{2}\left(-1+\delta \right)\left(\delta v_1-1\right)\frac{\partial v_3}{\partial s}+v_3\delta \rho \left(F_{T1}+\frac{\partial v_1}{\partial s}\right)\right]v_3^3\frac{{\partial }^2{v}_3}{\partial y_3\partial y_1}+\\ v_3^3(\frac{1}{2}\left(-1+\delta \right)\left(\delta v_2-1\right)\frac{\partial v_3}{\partial s}+\\ \left.v_3\delta \rho \left(F_{T2}+\frac{\partial v_2}{\partial s}\right)\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial y_2}+\frac{1}{2}\left(-2+3\left(\rho +\frac{2}{3}\right)\delta \right)\left(\frac{\partial v_3}{\partial s}\right)\\ v_3^4\frac{{\partial }^2{v}_3}{\partial {y_3}^2}+v_3^4\left(\frac{\partial v_3}{\partial t}\right)\left(-1+\delta \right)\frac{{\partial }^2{v}_3}{\partial {y_1}^2}+v_3^4\left(\frac{\partial v_3}{\partial s}\right)\left(-1+\delta \right)\frac{{\partial }^2{v}_3}{\partial {y_2}^2}+v_3^4\left(\frac{{\partial }^2{v}_1}{\partial y_3\partial s}\right)\left(\frac{\partial v_3}{\partial y_1}\right)\delta \rho \end{array}$$E55

$$\begin{array}{c}{L}_3=v_3^4\left(\frac{{\partial }^2{v}_2}{\partial y_3\partial s}{v}_2\right)\left(\frac{\partial v_3}{\partial y_2}\right)\delta \rho +\frac{1}{2}[\left(-1+\left(3\rho +1\right)\delta \right){\left(\frac{\partial v_3}{\partial y_3}\right)}^2+\left(-1+\delta \right)\\ \left({\left(\frac{\partial v_1}{\partial y_1}\right)}^2+2\left(\frac{\partial v_1}{\partial y_2}\right)\frac{\partial v_2}{\partial y_1}+{\left(\frac{\partial v_2}{\partial y_2}\right)}^2\right)]{\left(v_3\right)}^3\frac{\partial v_3}{\partial s}+\\ \left(\left({\left(v_3\right)}^3\delta \left(F_{T1}+\frac{\partial v_1}{\partial s}\right)\frac{\partial v_3}{\partial y_1}+{\left(v_3\right)}^3\delta \left(F_{T2}+\frac{\partial v_2}{\partial s}\right)\frac{\partial v_3}{\partial y_2}-1+\delta \right)\right.\frac{\partial v_3}{\partial y_3}+\\ {\left(v_3\right)}^4\delta \left.\left(\left(\frac{\partial v_3}{\partial y_1}\right)\frac{\partial F_{T1}}{\partial y_3}+\left(\frac{\partial v_3}{\partial y_2}\right)\frac{\partial F_{T2}}{\partial y_3}\right)\right)\rho \end{array}$$E56

B. Appendix 2

Solving Eq. (52) by assuming,

$$\begin{array}{c}{F}_5\left(y_1,y_2,y_3\right)=c_{4}P\left(\frac{2^{2/3}{y}_3}{2\sqrt{-\mathit{BG}\left(y_1,y_2\right)2^{2/3}}}+C\mathrm{,0,0}\right)G\left(y_1,y_2\right)sech{\left(F\left(y_1,y_2\right)+y_3\right)}^2\\ \end{array}$$E57

gives a product of two factors of equations, one of which has solution on an arbitrary surface $y_3=-F\left(y_1,y_2\right)$ and the other on an $ϵ$ ball containing this arbitrary surface.

Substituting Eq. (57) into Eq. (52) and solving algebraically for the second derivative of $v_3$ wrt to $y_3$ and the result was set equal to $v_3^2$. The resulting equation is checked to see if it has a Weierstrass P function as solution, and it does by solving for it using the pdsolve command. In applying the Geometric Algebra method used in [6] which this work is based on, it was checked that the term ${‖\nabla u_z‖}_2$ approaches zero and is an element of the Schwartz class for any constant C and in particular for arbitrarily large values of $y_3$. It was assumed that $v_3$ is in the form $v_3=F_4\left(s\right)G\left(y_1,y_2\right)U\left(y_3,s\right).\left(\text{Also}{v}_3=\frac{U\left(y_3,s\right)F\left(s\right)}{\sqrt[3]{s-s_0}\sqrt{G\left(y_1,y_2\right)}}\right.$ solves the problem in question. Note the modular form $\left.U\left(y_3,s\right)=\frac{2^{2/3}}{\sqrt[6]{G\left(y_1,y_2\right)}}P\left(1/2\frac{2^{2/3}{y}_3}{\sqrt{-{\left(G\left(y_1,y_2\right)\right)}^{2/3}{2}^{2/3}}}+F_1\left(s\right)\mathrm{,0,0}\right)\right)$

C. Appendix 3

The following equation is similar to Eq. (29) with the exception that it is expressed more generally in terms of $v_1$ and $v_2$,

$$B_1+2\frac{\partial v_3}{\partial s}\left(B_2+B_3\right)=B_4$$E58

$$\begin{array}{c}{B}_1=\left(\frac{\partial v_3}{\partial s}\right){\left(v_3\right)}^2\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^3{v}_3}{\partial {y_3}^2\partial s}-{\left(v_3\right)}^2\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right){\left(\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\right)}^2-\left(2\frac{\partial v_3}{\partial y_1}-2\frac{\partial v_3}{\partial y_2}\right)\times \\ \left[-\left(\frac{\partial v_3}{\partial y_3}\right)v_3\frac{\partial v_3}{\partial s}+v_3\left(F_{T_1}+\frac{\partial v_1}{\partial s}\right)\frac{\partial v_3}{\partial y_1}{v}_3+v_3\left(F_{T_2}+\frac{\partial v_2}{\partial s}\right)\frac{\partial v_3}{\partial y_2}+\frac{1}{2}\mathrm{\Phi }\left(t\right)\right]\frac{{\partial }^2{v}_3}{\partial y_3\partial s}\end{array}$$E59

$$\begin{array}{c}{B}_2=v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\left(F_{T_1}+\frac{\partial v_1}{\partial s}\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial y_1}+v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\left(F_{T_2}+\frac{\partial v_2}{\partial t}\right)\frac{{\partial }^2{v}_3}{\partial y_3\partial y_2}+\\ \frac{3}{2}\left(\frac{\partial v_3}{\partial s}\right)v_3\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\frac{{\partial }^2{v}_3}{\partial {y_3}^2}\end{array}$$E60

$$\begin{array}{c}{B}_3={\left(\frac{\partial v_3}{\partial y_1}\right)}^2\left(\frac{{\partial }^2{v}_1}{\partial y_3\partial s}\right)v_3-{\left(\frac{\partial v_3}{\partial y_2}\right)}^2\left(\frac{{\partial }^2{v}_2}{\partial y_3\partial s}\right)v_3+[\frac{3}{2}{\left(\frac{\partial v_3}{\partial y_3}\right)}^2\frac{\partial v_3}{\partial s}+(\left(F_{T_1}+\frac{\partial v_1}{\partial s}\right)\frac{\partial v_3}{\partial y_3}+\\ v_3\frac{\partial F_{T_1}}{\partial y_3})\frac{\partial v_3}{\partial y_1}+\left(\left(F_{T_2}+\frac{\partial v_2}{\partial s}\right)\frac{\partial v_3}{\partial y_3}+v_3\frac{\partial F_{T_2}}{\partial y_3}\right)\frac{\partial v_3}{\partial y_2}]\left(\frac{\partial v_3}{\partial y_1}-\frac{\partial v_3}{\partial y_2}\right)\end{array}$$E61

$$B_4=\left[2\frac{\left(-2y_1+2y_2\right)\frac{\partial v_3}{\partial s}+2y_3\left(\frac{\partial v_1}{\partial s}-\frac{\partial v_2}{\partial s}\right)}{{y_1}^2+{y_2}^2+{y_3}^2}-2\frac{{\partial }^2{v}_3}{\partial y_2\partial s}+2\frac{{\partial }^2{v}_3}{\partial y_1\partial s}\right]v_3\left(\frac{\partial v_3}{\partial y_1}\right)\left(\frac{\partial v_3}{\partial s}\right)\frac{\partial v_3}{\partial y_2}$$E62

D. Appendix 4

The second term in Eq. (14) involving the tensor product expression for the solution of $v_3$ given in terms of $F_5$ is independent of s $\left(\mathrm{\Phi }\left(s\right)=H_2\left(y_1,y_2,y_3\right)\right)$, and is given as,

$$\begin{array}{l}1/3\frac{{\left(G_2\left(y_1,y_2,y_3\right)\right)}^2\left(\sqrt[3]{s-s_0}\frac{\partial }{\partial y_1}{G}_2\left(y_1,y_2,y_3\right)+\sqrt[3]{s-s_0}\frac{\partial }{\partial y_2}{G}_2\left(y_1,y_2,y_3\right)+\sqrt[3]{s-s_0}\frac{\partial }{\partial y_3}{G}_2\left(y_1,y_2,y_3\right)\right)}{\sqrt[3]{s-s_0}}\\ =1/3{\left(G_2\left(y_1,y_2,y_3\right)\right)}^2\left(\frac{\partial }{\partial y_1}{G}_2\left(y_1,y_2,y_3\right)+\frac{\partial }{\partial y_2}{G}_2\left(y_1,y_2,y_3\right)+\frac{\partial }{\partial y_3}{G}_2\left(y_1,y_2,y_3\right)\right)\end{array}$$E63

The surface integral in Eq. (14) involving the pressure terms Q is either zero or non-zero depending on if $\mathrm{\Phi }\left(s\right)=0$ which occurs when $v_1=-v_2$ or $\mathrm{\Phi }\left(s\right)\ne 0$ which occurs when $v_1=v_2$. In general we have the term $K_1-\mathrm{\Phi }\left(s\right)={\lambda }_1$, where ${\lambda }_1\ne 0$. The term $K_1-\mathrm{\Phi }\left(s\right)$ is associated with taking the gradient of the extended expression ${\nabla }_D^{-1}\left({\mathcal{G}}_{\delta 1}+{\mathcal{G}}_{\delta 2}+{\mathcal{G}}_{\delta 3}+{\mathcal{G}}_{\delta 4}\right)=\overrightarrow{r}$.