Incompressible Navier-Stokes equations reduce to
Bernoulli's Law

© 2003, 2008, 2011

Clyde M. Davenport

cmdaven@comcast.net

[Updated 12/10/11]

Introduction | Complementary Equation | Quaternion Form | Hypercomplex Integration |

Interpretation of Result | Numerical Calculations | Conclusions |

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 where 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 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 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
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 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, In vector terminology, this is: Gibbs and Heaviside simply 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
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 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 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 which was chosen because of its natural extension of the classical complex variable 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 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. Top The first term in the N-S equation is a partial derivative,
, where 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 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 We could just for whatever independent variables
[Aside: We can always do a similar vectorization of a scalar field All that being said, we can assume that 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 Quaternion Form 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 The reader might notice that some of the elements of the quaternion equation are not 4-D, for example the del operator and Hypercomplex Integration 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
Also, recalling the definition of the quad operator, we expand the second and third terms as follows:
The "
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, 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 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 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
where 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, The velocity disturbance moving through the fluid generates a related pressure disturbance given by the P( , where 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 _______________________________ The reader might notice that the viscosity factor,
, does not appear in the 4-D Bernoulli's Law. There is a reason. 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, 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. 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 Up to now, we are still allowing 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, 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. 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 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 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 to a new mesh point, then using the formula to calculate the new pressure P and velocity _{i}V. At the end, on a point-by-point basis, we would extract the 3-D velocity as the _{i}i,j,k components of V and the scalar pressure as the _{i}1-component of P. 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_{i}In the finite difference scheme as described above, notice that 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 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. Top
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