Navier-Stokes Equations

@Huskie99

Please, no trolling in this forum.

@sapientdust

Yes, everything has been cut and pasted. This will allow theorists to refer to post #so-and-so to comment.

@nameno1had

Yes, there are a lot of variables to be considered, but I believe this is not simply a matter of supercomputers, or else it would have been solved long before now. $1,000,000 is enough to encourage all kinds of brute force methods, but what we need I suppose is skills in problem solving as well.

Another Good Website:

http://universe-review.ca/R13-10-NSeqs.htm

Here's a simplification to Bernoulli's Law:

http://home.comcast.net/~cmdaven/navier.htm

© 2003, 2008, 2011
Clyde M. Davenport
cmdaven@comcast.net
[Updated 12/10/11]

Introduction

[In the following, the initial equation has been slightly modified pursuant to a suggestion by Per Uddholm in order to make it more amenable to real-world applications. Specifically, the constant  multiplier is added on the left, thereby renormalizing the velocity variable V and restoring real-world physical units. Subsequent equations are modified accordingly.] The incompressible Navier-Stokes vector-form equation is a nonlinear partial differential equation of second order, as follows:

where v is a vector representing the velocity of an infinitesimal element of mass at a point in 3-D space, p is the scalar pressure at the same point,  is the mass density at the point and is assumed constant throughout the medium,  is the viscosity of the medium, and g is a constant vector acceleration due to some constant external force on the infinitesimal element, usually taken to be gravity. In other words, the N-S vector equation represents a force-mass-energy-momentum balance about an infinitesimal mass element of the field. The N-S equation addresses the motion of a single, tiny particle of the fluid field, not the overall motion of the fluid. However, it can be used to calculate the flow of incompressible gases and fluids past objects of arbitrary shape, as we shall explain. It is used in fluid dynamics teaching and in engineering as a standard model for turbulence, boundary layer behavior, shock wave formation, and mass transport. Among other things, it is used to calculate the pattern of air flow past airplane wings [The last time that you flew in an airplane, did you realize that your life depended upon this equation holding true?]. It has been studied and applied for many decades. Many different closed-form, series approximation, and numerical solutions are known for particular sets of boundary and initial conditions. 

Our objective, here, is to show that, under laminar flow conditions, the above equation reduces to a simple Bernoulli's Law in 4-D vector form:

where V is the analytic 4-D velocity, P is the 4-D analytic vector pressure field (we shall explain), g is a constant acceleration which we shall allow to be imposed in an arbitrary direction, and Z is a vector representing arbitrary displacement in 4-D space, as we shall explain. We shall show how to recover the traditional scalar Bernoulli's Law, as a special case, from this expression. 

We shall supply the necessary mathematics for interpreting this expression and using it in applications. The informed reader will realize that, if we can do this, then a quantum leap in efficiency and reduction in cost of an enormous array of engineering calculations, from weather patterns to hydraulics to the flight of airplanes, can be made.

We all remember Bernoulli's Law from our introductory physics courses. It was most often illustrated by flow through a constriction in a pipe, as in a Pitot airspeed gage. More significantly, it was also explained as the basis for lift by an airplane wing. The air travels a greater distance over the bulged upper surface than over the relatively flat underside, hence must flow faster over the top. By Bernoulli's Law, this creates a net drop in pressure between bottom and top, hence lift, on the wing. Only much later, when we got to much more advanced courses, did we learn that there is also a complicated set of partial differential equations, called the Navier-Stokes equations, that can be used to calculate the flow of air and the pressure pattern around an airplane wing, consequently the lift. Until now, apparently no has ever said, "Wait a minute - what is the connection between these two formulas?" We intend to elucidate the connection, here.

We shall show, below, that any system of PDEs written in the form of a vector equation, using vector algebra and operators, is only an incomplete statement of some correspondingquaternion expression. The approach that we shall take toward integrating the N-S equation is start with the N-S vector equation, find terms that complete it to its corresponding quaternion expression, and then solve the latter by use of commutative hypercomplex analysis. The hypercomplex system obeys the same axioms, algebraic rules, function theory, and scheme of analysis as the classical complex variables, while treating a 4-D variable. It is based upon a particular commutative group ring with unity. No snake oil is necessary, nor is any applied. In order to understand the following, the reader should review the Hypercomplex Math page before proceeding.

In order to illuminate the argument, we first need to examine a particular, odd feature of vector mathematics that was put there by O.W. Heaviside and J.W. Gibbs at the outset. We begin with a short review of the development of multidimensional algebras and vector analysis, concentrating on those aspects that will be relevant to our argument, here. We urge the reader to follow along, because we shall construct an interpretation and point of view that is not generally seen in the literature. The interested reader may refer to the Hypercomplex Math page for additional supporting references for the following discussion. 

In the 1830s, Sir William Rowan Hamilton set out to create the first multidimensional linear algebra and associated analysis (beyond the complex variables). He wanted to apply it to 3-D problems in optics and mechanics, in much the same way that we use vector analysis, today. As a guide, he had only a few rudimentary concepts from the algebra of complex variables. There was no matrix analysis, group theory, or ring theory at that time. Hamilton initially desired to create an algebra involving multiplication and division over a variable of the formZ=ix+jy+kz, where i,j,k are unit basis vectors and x,y,z are real coordinates. By trial and error, he was unable to do so, because, as we know today, no division algebra exists for three-dimensional numbers. He found that he could create a division algebra over 4-D elements of the form Z=α+ix+jy+kz, which we now know as the quaternion algebra, the only division algebra of order four. He called α the scalar part, and ix+jy+kz the vector part, and neither he nor his immediate successors quite knew what to make of the scalar part. Although Hamilton knew that the basis elements for his new algebra were 1,i,j,k, he could not bring himself to associate the 1 element with the "scalar part" and view the result as a 4-D vector. Apparently, the one thing upon which they were in unanimous agreement was that α could not be a "fourth dimension," neither time nor anything else. They began to treat and think of these components as fundamentally different kinds of things, when in actuality all four coefficients (coordinates) are treated qualitatively the same by the algebra

This is an important insight for our present objectives. Apparently, neither Hamilton nor any of his nineteenth-century successors could quite get their minds around the concept of a four-dimensional space. What would be the "direction" of the supposed "fourth dimension?" How could it possibly be orthogonal to the other three? Because of this bafflement, scientists and engineers of the time steadfastly refused to use quaternion mathematics in their calculations. In the mid-1850s, James Maxwell published four major papers that developed the first formulation of electromagnetic theory, using the clumsy component-by-component calculations of the time. In 1873, he published a treatise [Maxwell, 1873] on electromagnetic theory that included his earlier papers and in which he reformulated all of the fundamental equations in terms of the algebra and notation of quaternions and keeping Hamilton's view that the vector and scalar parts were somehow fundamentally different in nature. This formulation was absolutely rejected out of hand by the scientific community. Instead, they struggled along with a crude, component-by-component means of calculation. In the period 1873-1893, there was an acrimonious, running argument in the scientific literature as to whether quaternion mathematics had a proper place anywhere in science!

There it lay until 1893, when J. W. Gibbs in America and O. W. Heaviside in Britain began to develop and apply what we now know as vector analysis. They based it upon the quaternion algebra, but knew that they could never mention that fact, lest it be rejected instantly. Their notation is basically a modification/shorthand version of the full quaternion notation. Apparently, their thought processes ran something like the following: "Look, we believe that the scalar and vector parts are mathematically fundamentally-different things. The scalar part is 1-D and the vector part is 3-D, so let's just use those two things separately and independently as the basic elements, if you will, and drop all mention of anything that is 4-D. The 4-D objectionists will be left with nothing to argue about." All calculations would appear as separate manipulations in terms of the scalar or vector parts of a quaternion, as if they were independent, and they would never be identified as components of a quaternion. This gave it the desired 3-D look. Heaviside reformulated Maxwell's electromagnetic theory in these terms, and was aided by the circumstance that the dot and cross products involving the del operator with various field variables could be identified with fundamental, physically-measurable electromagnetic field parameters. The subterfuge worked. Scientists and engineers accepted it, and the rest is history.

However, and this is why we have struggled through this tedious chain of events, when Gibbs and Heaviside dropped the full quaternion product in favor of manipulations with the scalar and vector parts, separately, they had to make ad hoc changes to the algebra that are inconsistent with quaternion mathematics. They wanted to assure that no product of two 3-D vectors would ever have a scalar part (i.e., a dreaded "fourth dimension.") They proceeded as follows:

By quaternion rules, if one multiplies two 3-D vectors,

  A = i a₁ + j a₂ + k a₃
  B = i b₁ + j b₂ + k b₃ ,
one obtains:

  AB = -1 (a₁b₁ + a₂b₂ + a₃b₃ )
    + i (a₂b₃ – a₃b₂ )
    + j (-a₁b₃ + a₃b₁ )
    + k (a₁b₂ – a₂b₁ ) .

In vector terminology, this is: AB = - A•B + A×B 

Gibbs and Heaviside simply defined the dot and cross products in this way and proceeded to treat them as if they were entirely unrelated quantities. They wanted to avoid any mention of “quaternion” or “four dimensions.” However, it could occur that one encountered a cross product of, say, i with itself. In order for this to make any sense, they had to arbitrarily set

  i×i = j×j = k×k = 0.

To summarize, the quaternion product rules are:

  ij = k  jk = i  ki = j
  ji = -k kj = -i ik = -j
  ii = jj = kk = -1 ijk = -1 ,

and the corresponding cross product rules are:

  i×j = k  j×k = i  k×i = j
  j×i = -k k×j = -i i×k = -j
  i×i = j×j = k×k = 0 i×j×k = 0 ,

Compare the last line of each. This is the "odd feature" that we alluded to, earlier.

However, when they arbitrarily set certain terms of a product to zero, something was lost. The full quaternion product of two 3-D vectors is

A B = - A·B + A×B,

hence if we go off and do calculations using only the cross product operation, then every time that we do a product, we lose the scalar part (here, denoted as the usual dot product). It is the same with the vector del operator:

.

That is why the Heaviside-Maxwell's equations require the addition of a continuity condition(additional, seemingly-unrelated equation, not generated by the original derivation) to make them consistent. That the resulting system of mathematics worked in practical terms is abundantly testified to by our space-age, technological society, fully undergirded by vector calculations, but is there more insight to be gained?

Indeed, we can see from the above that any typical vector algebra expression, equation, etc., such as the vector N-S equation, must represent only part of a true quaternion expression (i.e., without dot or cross products). There must exist a complementary expression that, when combined with/added to the original will result in a valid quaternion expression. Moreover, the resulting quaternion expression will nearly always allow some consolidation among its components, making it easier to solve. That is the notion that we are pursuing, here. In the following, we shall convert the vector-form Navier-Stokes equation back to a quaternion form, then solve it by use of commutative hypercomplex mathematics. [Aside: It is the author's opinion that if Hamilton, Gibbs, Heaviside, and their nineteenth-century compatriots had not been so abstractly-challenged, there would be no "vector analysis" today, but only quaternion analysis.]

Before we begin, we must make the change to an independent variable that reflects Heaviside's and Gibbs' particular coordinate frame of reference. In the standard C-H notation, we use an independent variable of the form:

which was chosen because of its natural extension of the classical complex variable z=x+iy. Here, we wish to use the Heaviside-Gibbs perspective, which in our notation is:

This represents a simple change of coordinate frames (a rotation + reflection, with determinant -1). Technically, we should carry the primed notation forward, but it merely adds unnecessary distraction. Instead, we shall periodically remind that we are using the Heaviside-Gibbs coordinate perspective.

Complementary Equation

We shall now construct the complementary equation for the vector Navier-Stokes equation. We shall take each term in the N-S equation in turn and construct a complementary expression that completes a valid quaternion expression (i.e., having no dot or cross product terms). Note that, because of what we pointed out above, all terms of the N-S equation are 3-D or less. That is no problem, because the application of an operator is handled exactly like multiplication, and quaternion multiplication is still valid even if one or more components of either or both multiplicands are zero or absent altogether.