Navier-Stokes Equations

The lobe on the right is associated with a rightward motion in the fluid, and the lobe on the left is associated with the leftward motion of the rebound. They are both positive because p(x,t) is a positive scalar quantity. This pressure impulse propagates to the right in the fluid.

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The reader might notice that the viscosity factor, , does not appear in the 4-D Bernoulli's Law. There is a reason. The commutative hypercomplex, analytic treatment makes it unnecessary for an incompressible flow. Go back and review where in the solution process that was eliminated: In the middle of the "Hypercomplex Solution" section, we asserted that we were going to use analytic function theory to solve the quaternion form of the N-S equation (which is also a valid commutative hypercomplex expression). We want the flow field V(Z) to be continuous and single-valued (analytic). Consequently, as for any analytic function, the 4-D scalar Laplacian of V is zero, causing  to drop out. We rationalize this as follows.

In the original, vector-form Navier-Stokes equation, it is the  term that causes a differential flow between different streamlines. It is the term that produces conformal flow lines. When we go to a 4-D, commutative hypercomplex, analytic treatment, it is the mathematical system, itself, that produces the conformal flow lines, making  superfluous. It so happens that laminar flow is analytic in the complex variable sense. Indeed, classical complex function theory has been used since the 1930s to calculate conformal flow over an airfoil shape. See [Kober, 1957] for examples and a long list of references; also, do a Google search for "Joukowski transformation." We should not be surprised at all by our result, here. We do, however, need to check beforehand that our fluid parameters are such that laminar flow is possible. This is indicated by a Reynolds number, which is proportional to (fluid velocity/viscosity), less than about 2,000 (even less near sharp edges). If this is exceeded, then turbulent flow ensues. 

Interpretation of Result

Recall that the Navier-Stokes equations address the reactions of an infinitesimal element of mass to external forces and impulses. Its reaction is qualitatively the same for forces or impulses coming from any direction in 3-D space. Now consider our hypercomplex integrated result,

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It embodies all of the behaviors and characteristics that are addressed by the incompressible Navier-Stokes equations, nothing more or less, yet it appears to be non-isotropic in nature. This conundrum is resolved as follows. V and P are functions of the independent variable Z. Each argument Z can be written in canonical form, as a function of its eigenvalues. There is a group of 4-D orthogonal transformations [see the Hypercomplex Math page] that, when applied asx,y,x,ct coordinate transforms, leave invariant the eigenvalues of Z. Consequently, the entire integrated expression is invariant in form under such transformations, as is the implicit behavior. 

We have a 4-D expression that looks like a Bernoulli's Law, but the original Bernoulli's Law was all-scalar, and the vector form that we wanted to obtain as our objective in this paper is 3-D. Therefore, some interpretation is required. Let us first address Bernoulli's original, all-scalar form and see if we can recover it from the 4-D form. Consider the following special case: Letx,y,z be the usual three-space coordinates (with the Heaviside-Gibbs perspective), with kz in the vertical direction. Let the scalar speed v be in the +x direction. Let g be the acceleration due to gravity (in a vertical direction), and let the displacement be Z = kh in a true vertical direction. We shall also have to convert the scalar v back from a dimensionless form. In these terms,

Up to now, we are still allowing P to be 4-D. But here, all other terms of the equation are scalar, meaning that the equation holds true only with the first (scalar) component of P, 1p, which, if one recalls, is the same p as in the original Navier-Stokes equation. Therefore, in all-scalar terms,

This is the original Bernoulli's Law, as given in most any college introductory physics textbook. This result is significant, but it would be even more useful if we could express it in three dimensions. Indeed, we can do so. Consider the special case (stated in commutative hypercomplex mathematics): 

Here, v, g, and X are 3-D, each with i,j,k components, but their indicated products will be 4-D, with 1,i,j,k components. Therefore, P must be manipulated as a 4-D entity. All of the operations in the above formula must be carried out with commutative hypercomplex rules.We must go "outside of the 3-D box" in order to do calculations. If we are given a scalarp(x,y,z,ct) pressure field in the form of an analytic function, then we must construct its 4-D extension as indicated earlier. If the velocity field is given in the form of an analytic function, and the external force and displacement are given, then we merely calculate P from the above equation, then select its 1-component as the scalar p(x,y,z,ct) pressure field.

Numerical Calculations

All of the above is well and good, but when solving engineering problems, the problem statement usually does not give any part of the field configuration in the form of an analytic function. Typically, we receive only the boundary and initial conditions in the form of numerical data. We must compute the rest.

One might recall that partial differential equations, especially those used to model the behavior of some material substance, typically describe the behavior of some variable, parameter, or physical effect about a point. They are typically derived from a force-energy-momentum-mass balance on an infinitesimal element at a point. If we achieve an integration of the Navier-Stokes equation by whatever means, then the integrated form (characteristic function) will embody the same description of physical effects as the PDE and must be viewed and applied in the same way; i.e., as describing the variation of effects about a point, and not necessarily the macro behavior over all space. That is to say, the behavior of the integrated function about its origin of coordinates describes the qualitative variation of physical effects about any point in the region of validity of the PDE. The region of acceptable approximation of the real, physical effects about a given point might be small, so we might have to do a numerical solution, this time using the characteristic function instead of the PDE. We could use the 4-D constant of integration and the playoff between v and p in the Bernoulli formula to fit together a mosaic of small-area solutions on a grid, quite analogous to what is done in a numerical, finite-element solution of the PDE.

Another way to view the Navier-Stokes equation is that it was developed to describe the immediate, localized reaction of a tiny, incremental element of mass in the fluid field to given external forces and momentum and energy inputs. In physics terms, the integrated result is expressed in body-centered coordinates whose origin moves with the subject particle of mass and whose axes slide parallel to themselves. At any given instant of time and for given local conditions, the integrated result indicates how the particle of mass will move next within the body-centered frame. We continue to emphasize: The integrated result describes an immediate, localized reaction, and says nothing about the long-term motion of a given particle of mass. For that reason, we must do a finite-element-like numerical calculation in order to coordinate the motions and interactions of all the particles, thereby obtaining a view of the overall motion of the fluid. This view explains why there is not any analytic-function solution of the N-S equation that models turbulent behavior in the large. Any "solution" is point-localized.

However, the Bernoulli's Law formula might not be the best choice for use in a numerical solution. Instead, consider the ODE that we integrated to obtain the Bernoulli formula. From it, we can write:

We would use this expression in a finite-difference scheme. Here, dZ can be viewed as an incremental movement on the problem grid as our numerical solution proceeds, and not just a displacement against the constant force associated with g. As we have seen, even when we are working with 3-D quantities, the commutative hypercomplex algebra returns a 4-D result from a product operation, so it is necessary that we carry all results in 4-D terms. In this approach, we would generate at least two four-component numbers V_i, P_i for each 3-D grid point. Starting from a boundary, we could "walk" a solution throughout the problem volume by advancing an increment dZ to a new mesh point, then using the formula to calculate the new pressure P_i and velocity V_i. At the end, on a point-by-point basis, we would extract the 3-D velocity as the i,j,k components of V_i and the scalar pressure as the 1-component of P_i. The unused 4-D components can be viewed as only intermediate data storage registers. Not being a practicing numerical analyst, I leave the details of the scheme to more-experienced specialists. 

In the finite difference scheme as described above, notice that the time step is not arbitrary. The 4-D vector difference equation as shown represents a set of four conditions upon thedx,dy,dx,cdt independent variables. If the dx,dy,dz components are specified, then a unique value for cdt is called for if we are to maintain analyticity.

Yet another way to view the characteristic function solution of the Navier-Stokes equation is as follows. Consider an infinite, uniform, incompressible fluid medium. Let an infinite-magnitude, point impulse be introduced at some arbitrary point in the fluid. The disturbance will assume the functional form of the characteristic function and will move away from the originating point at the characteristic speed for disturbances in the medium. The outwardly-moving disturbance will be radially symmetrical because, as we have shown, the solution is rotationally invariant, given the proper frame of reference.

Conclusions

For more than sixty years, we have had ample illustration that Bernoulli's Law addresses the same laminar flow phenomena as does the Navier-Stokes equation, and that classical analytic function theory can be used to calculate 2-D laminar flow around an airfoil. Here, we have used 4-D analytic function theory to show that under an assumption of laminar flow, the N-S equation integrates directly to a 4-D form of Bernoulli's Law. From this, we can recover Bernoulli's original, all-scalar formula as a special case. Even better, we have a general formula that accommodates 3-D vector values for flow velocity, and the commutative hypercomplex math provides a comprehensive basis for doing calculations. We can use the 4-D Bernoulli's Law in place of the Navier-Stokes equation when doing laminar flow calculations, with potentially great savings in computational expense.

All that aside, possibly the greatest gain is the expanded theoretical insight that we now have about laminar flow in three dimensions.

So, I found the above listed website to be very useful. It puts the equations in a simpler form, but in doing so, converts it into four-dimensions. The problem here, however, is that the idea of dimensions higher than the third are often confusing to the theorist, and cannot be easily comprehended.

The added factor of complex numbers does not help, and makes it all the harder for us to interpret. All equations using this, however, must, in some way, find their way back to the third dimension, in order to be accepted by the Clay Mathematical Institute.

One of these problem solving skills necessary is literally finding all of the variables involved. Another is formulating the variables into workable problems by legitimizing the true effects of the various conditions, beyond the theory behind them. Again I vote for a super computer to calculate things such as this. The problem is telling it what/how to calculate to begin with. It seems you need one just to help you figure out what this data is before hand.

$1,000,000 would be nice, but so would be, having a brain left to spend it.

Agreed, but I plan to expand the equations before going much further.