© 2003, 2008, 2011
Clyde M. Davenport
|Introduction||Complementary Equation||Quaternion Form||Hypercomplex Integration|
|Interpretation of Result||Numerical Calculations||Conclusions|
[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. Top
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. Top
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. Top
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 corresponding quaternion 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. Top
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 form Z=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. Top
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! Top
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. Top
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
AB = -1 (a1b1 + a2b2 + a3b3 )
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:
and the corresponding cross product rules are:
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? Top
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.] Top
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. Top
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. Top
The first term in the N-S equation is a partial derivative, , where v is the 3-D velocity. We first note that, in quaternion notation,
The parenthetical quantity is a part of a 4-D gradient operation,
Consequently, the complementary part for the term is , where the latter is a quaternion operation. Note that, although the quad operator is 4-D and v is 3-D, the operation is performed like a quaternion multiplication, hence is valid. Top
The second term of the N-S vector equation is . The obvious complementary part is , the sum of the two parts then yielding , a quaternion expression.
Recall that quaternion multiplication is noncommutative, and note that we are maintaining the proper left-right orientation of all operators and variables, hence we are maintaining quaternion algebraic rules. The quaternion algebra is also associative and distributive. Top
The third term of the N-S vector equation is . This brings up a different kind of problem because, looking ahead, we are going to integrate once and obtain p as a free-standing entity, without further specification of its functional form. However, we must remember that the original Bernoulli's Law was developed to show the co-dependent relationship between speed and pressure in a flowing medium. If the pressure was specified at a given point, then the corresponding speed could be calculated from Bernoulli's Law; conversely, if the speed at a given point was specified, then the pressure could be calculated. The formula was expressed in all-real terms. Here, we will have a vector velocity v, rather than scalar speed, consequently p will have to have a vector form in order to have the proper co-dependence with velocity. Top
We could just assume that a real, analytic function p(x,y,z) can be analytically continued into 4-D hypercomplex form, and work with the vector Bernoulli's Law and numerical values without ever having to know its precise analytical form. However, we can actually show that this is a good assumption, in concrete terms. Suppose that we are given a scalar (real) analytic function p(x,y,z), even allowing some of the independent variables to be missing. If we make the substitutions Top
for whatever independent variables x,y,z,ct that are present, then we have a hypercomplex-valued, analytic function p(Z) that properly subsumes the original scalar function. The original scalar coordinates x,y,z,ct are still present exactly as they were, but we have analytically continued the function into four dimensions. This works even if we start with a function of only one independent variable, say p(x). Moreover, we have preserved the form of the function, and the commutative hypercomplex mathematics always tell us how to interpret and manipulate the extended form. Top
[Aside: We can always do a similar vectorization of a scalar field p(x,y,z) by use of classical vector operations. We merely need to construct a unit vector for each point of the field, oriented in a direction opposite to increasing gradient. A real pressure field is single-valued and differentiable. Therefore, the vectorized field P(x,y,z) is:]
All that being said, we can assume that p can always be represented in an analytic, 4-D vector form. If the vector field v is given, then the corresponding vector p field can be calculated from the Bernoulli's Law formula. Conversely, if a 4-D scalar p(x,y,z,ct) field is given, then we know how to construct its 4-D vector extension. Having that, we can calculate the vector field v from the Bernoulli's Law. In conclusion, the term can be assumed to be a quaternion expression as is. No complementary term is needed. Top
The fourth term of the N-S vector equation is . The del-squared operator is a scalar operator. We note that
Therefore, the necessary complementary term is:
The fifth and last term of the N-S vector equation is . Both elements are constant, being a scalar and g being a 3-D acceleration which we intend to allow being imposed in any direction, and as such their product is a valid quaternion expression. No complementary term is needed. This concludes our derivation of the complementary terms. Top
Now we can summarize our findings and show the N-S equation, the complementary vector equation, and their sum, which is the associated quaternion equation, as follows:
Remember that, in the quaternion equation, we are assuming that p will be treated as a 4-D analytic (vector) function, rather than a scalar function, and that we showed earlier how to construct it, if given a scalar function as part of the initial conditions. Top
The reader might notice that some of the elements of the quaternion equation are not 4-D, for example the del operator and g. 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. It is a valid quaternion expression because we eliminated the dot and cross products. Top
The quaternion-form N-S equation,
is also a valid commutative hypercomplex equation, because every element and operation has an equivalent interpretation in the latter system. From this point forward, we shall treat it as such, and solve it by means of commutative hypercomplex functional analysis techniques. Are we entitled to do this? Yes, as long as we are consistent throughout, because we can verify the result by substitution into the original N-S equation. We do not use quaternion functional analysis because a classical function theory for a quaternion variable does not exist, as a consequence of the noncommutativity of quaternion multiplication. Top
The commutative hypercomplex mathematics is a system that obeys the axioms of the classical complex variables, including the function theory, and behaves in all ways like the classical complex analysis, while treating a 4-D independent variable. The algebra has much of the notation and appearance of quaternions, the main difference being that quaternion multiplication is noncommutative. Refer to the Hypercomplex Math page for details. In this system of mathematics, the vector Bernoulli's Law as given earlier has a rational and consistent interpretation in the same way as would a classically-complex expression, as we shall show. Top
We shall now convert the Navier-Stokes PDE to an ODE by use of the 4-D Cauchy-Riemann conditions. In doing so, we shall analytically continue the dependent variables v and p into 4-D. At the end, we shall extract the lower-dimensional solution. Recall that, to this point,v and p are 3-D and 1-D scalar, respectively. We showed how to analytically extend a scalar function p(x,y,z,ct) to a 4-D vector function. Here, we are going to be integrating in terms of a 4-D variable Z=1ct+ix+jy+kz, which analytically continues the results into 4-D. Therefore, to emphasize the enlarged problem, we write the broadened variables with capital letters: Top
Also, recalling the definition of the quad operator, we expand the second and third terms as follows:
The "1" element is just that - the unity element. Here, it can be explicitly displayed or not, as desired. Now, as consequences of the 4-D Cauchy-Riemann conditions, for V,P, or any other analytic function, Top
Remember that we are using the Heaviside-Gibbs coordinate perspective. Folding all of this back into the broadened N-S equation, we arrive at a dramatically simplified ODE expression:
In the process of making this conversion, we have introduced the Cauchy-Riemann conditions, so that when we integrate, our results will automatically be analytically continued into 4-D. We are operating under axioms and functional behavior exactly like that for real or classically-complex variables, so without further ado, we integrate by inspection to get: Top
This is our result in 4-D terms, from which we shall extract special cases for the traditional scalar Bernoulli's Law and a 3-D vector form. All of this leadup may have seemed obscure, and the reader might have difficulty in believing the result, but a closer reading of the Hypercomplex Math page can verify that everything that we have done is valid. If it were not, then it would be quite a coincidence that after letting logic take us where it will, we arrived at a conclusion that, upon reflection, makes great intuitive sense, because both Bernoulli's Law and the incompressible Navier-Stokes equations deal with laminar flows of incompressible liquids or gases.
The result of integrating the vector N-S equation has produced an atypical characteristic function. There is not a single function of the 4-D coordinates, f(Z), but two: V(Z) and P(Z). The characteristic function reveals the exact relationship between V and P, and how they must interact and play off of each other in a dynamic, incompressible-flow situation. For example, if we are given the velocity field in the form of an analytic function V(Z) (or enough information to construct it by use of the 4-D Cauchy-Riemann conditions), then the pressure field P(Z) is: Top
Note that all elements are manipulated by use of the same axioms and functional rules as for the real or complex variables. Conversely, if we are given a 4-D pressure field P(Z) (or enough information to construct it by use of the 4-D Cauchy-Riemann conditions), then the velocity field V(Z) is: Top
The commutative hypercomplex mathematics tell us how to interpret these expressions. We can even break them down into 4-D vector functions of the form 1u(x,y,z,ct)+iq(x,y,z,ct)+jw(x,y,z,ct)+ks(x,y,z,ct). Although every element in these expressions can be written in 4-D vector form, we do not use classical vector algebra when manipulating them. Instead, we use the rules and function theory of the commutative hypercomplex mathematics, which are the same as for the classical complex variables, with a few, minor notational differences. Top
We can give a simple illustration involving the above analytical expressions. Let us set up a greatly-simplified, one-dimensional problem. Suppose that we have an infinite half space filled with an incompressible fluid lying along the +x-direction and bounded on the left by a rigid plate boundary lying in the yz-plane. Further suppose that this done in a weightless environment, so that we need not take gravity into account. Now let us apply a uniform mechanical impact to the boundary plate, in the +x-direction, that causes an impulse reaction (movement) of the plate with the form
v(x,t) = (x-ct)exp[-(x-ct)2],
where t is time and c is the speed of sound in the fluid. It is an initial condition that we merely impose. It has the following shape, with x as the horizontal axis:
The impact causes a small movement of the boundary plate to the right (in the +x-direction) followed by a rebound movement back to its original position. This boundary movement applies the same action to the fluid that is in contact with the boundary. Provided that we do not strike the plate so sharply that cavitation occurs on the rebound, the impulse propagates into the fluid at the speed of sound in the fluid, c, keeping its shape and moving along the +x-direction. It is a uniform, planar disturbance that might be described as a shock wave.
The velocity disturbance moving through the fluid generates a related pressure disturbance given by the P(Z) equation, above. The latter reduces to:
where p0 is the uniform background pressure in the fluid. This has the following form:
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.
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. Top
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. Top
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,
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 as x,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. Top
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: Let x,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, Top
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, Top
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): Top
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 scalar p(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. Top
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. Top
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. Top
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. Top
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 Vi, Pi 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 Pi and velocity Vi. At the end, on a point-by-point basis, we would extract the 3-D velocity as the i,j,k components of Vi and the scalar pressure as the 1-component of Pi. 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. Top
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 the dx,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. Top
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. Top
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