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.
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 pwill have to have a vector form in order to have the proper co-dependence with velocity.
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 functionp(x,y,z), even allowing some of the independent variables to be missing. If we make the substitutions...
...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
[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 fieldv from the Bernoulli's Law. In conclusion, the term can be assumed to be a quaternion expression as is. No complementary term is needed.
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.
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 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.
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.
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.
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.
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 pare 3-D and 1-D scalar, respectively. We showed how to analytically extend a scalar functionp(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:
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 otheranalytic function,
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:
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 theHypercomplex 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:
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:
The commutative hypercomplex mathematics tell us how to interpret these expressions. We can even break them down into 4-D vector functions of the form1u(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.
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 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.