Vectorial version of the general Stokes theorem and the Karman vortex street

Our ultimate goal of our bachelor research is the study on the theory of lubrication, which is indispensable in reducing the friction—the most important challenge of the human being in the 21st century. In our paper [14] which is an outcome of our project research we have shown that by the use of chain rule together with differential forms coupled with the general form of Stokes’ theorem, many results in fluid mechanics are made simpler and clearer. In particular, we elucidated the notion of divergence and circulation in the 3-dimensional flow case. Further in the case of 2-dimensional flow by the use of complex analysis, we reestablish the results in that theorem. In this bachelor thesis our main purpose is to prove Theorem 3.3 to the effect that the Cauchy equation of motion implies the Navier-Stokes equation and to correct the missing coefficients in [7]. For this purpose we develop the vectorial version of the general Stokes theorem. From it we deduce the most important unicity theorem from which the equation of continuity follows immediately. In the same vein that the Cauchy equation of motion is a consequence of Newton’s second law of motion follow immediately. We shall start the project of deriving the deepest results in vector analysis by the theory of generalized functions, which are known mainly as distributions. Our stand point is, however, that of [10] which incorporates the Sato hyper-function as a main tool for interpreting the whirl flow.


Introduction
This research is a sequel to our project research whose achievements have appeared in [14], where mathematical foundations of fluid dynamics were laid to some extent.We quote the results from it freely in this thesis.Our main purpose is to prove Theorem 3.3 to the effect that the Cauchy equation of motion implies the Navier-Stokes equation and also corrects the missing coefficients in [7].For this purpose we develop the vectorial version of the general Stokes theorem which itself is of interest in its own.From this we deduce the most important unicity theorem from which the equation of continuity immediately follows.In the same vein that the Cauchy equation of motion is a consequence of Newton's second law of motion follow immediately.The linear relation stated in the proof of Theorem 3.3 given in [7] is rather misleading and we streamline the proof by a linear relation (3.38).In the case of 2-dimensional flows, we revisit the Karman vortex street and reveal an unexpected hidden fact, via the partial fraction expansion of the cotangent function, that the expression (5.76) for the stream function for the Karman vortex street is, in the long run, a consequence of the functional equation for the Riemann zeta-function.

Fluid dynamics and lubrication
The flows are classified as in the following table.
2 Vector analysis Definition 2.1.For a vector-valued function f = , we define its divergence div f and curl curl f (also called rotation rot f ) by and We extend the definition of the divergence to a matrix.Let where p j , 1 ≤ j ≤ n are of C 1 class as in Definition 2.1.Then (2.4) Theorem 2.1.(General Stokes' theorem) Let M be a (k + 1)-dimensional manifold, ∂ M its boundary and let ω be a differential form of degree k in n variables.Then the identity holds true; in more concise form it reads dω(M ) = ω(∂ M ). (2.6) In vectorial form, (2.7) reads ) Corollary 2.1.(Unicity theorem) Suppose n = k + 1 and that we are given the identity for any bounded domain V ⊂ M .Then P = Q (2.11) throughout M .Furthermore, the condition (2.10) may be replaced by (2.12) Proof.Eq. (2.10) amounts to for any bounded domain V .Since the integrand is continuous, if it is non-zero at a point z 0 , say = c > 0. Then in a neighborhood V 0 of the point it is positive and so the integral over V 0 is positive, a contradiction.
By (2.8), the second term on the right of (2.12) is Given (2.12), the general Stokes theorem implies (2.10) and we still have the validity of the corollary, completing the proof.
Remark 2.1.The condition n = k + 1 indicates the saturation of the integrals in (2.10), i.e. they are scalar integrals including the case (2.8) of matrices.In view of (2.4) the latter case amounts to which implies coincidence of components, and so the unicity still holds.
Example 2.1.(Equation of continuity) Let v be the vector field of velocity of the fluid flowing in a domain Ω ⊂ R 3 and let X ⊂ Ω be a bounded closed domain with its boundary ∂ X forming a surface.Let ρ denote the continuous density (distribution) function of the fluid in Ω.Then f = ρv is the vector field describing the flow of mass distribution of the fluid.Suppose there is no source or sink in X.Then since Q(∂ X) indicates the rate of decrease of all masses in X, we have

14)
By Corollary 2.1, the left-hand side is Since dρ dt is continuous, we may change the integration and differentiation on the right of (2.14) and we obtain Hence, by Corollary 2.1

the mass and F is the sum of all external forces acting on the fluid volume and it consists of body force F b and surface force F s . The body forces act on the center of mass of the fluid volume and we assume it to be the gravitational force
and the surface force is given by stress dyadic Corollary 2.1 implies the Cauchy equation of motion We state well-known special cases given in Table 3 of the general Stokes theorem in [14].
be the position vector of a particle which is a function in time t, i.e. all components are functions in t.Hence the velocity is where we write In many cases, one defines the acceleration as the derivative of the velocity with respect to time, i.e.
However, in fluid dynamics it is customary to follow Euler to assume that the velocity components are functions of the position components as well as that of the time.By the chain rule, Therefore, the acceleration a = a(x, y, z,t) is to be understood as We confine ourselves to the isotropical fluids, i.e. those which behave in the same way for all directions.where in the notation p i j , the subscripts i, j take on values x, y, z and the outer subscript indicates the direction of stress and the inner one indicates the plane on which stress acts.
Let Ṡ denote the strain rate dyadic given by Note that Ṡ is a symmetric matrix.

.31)
where µ indicates the viscosity of the fluid and E is the identity matrix of degree 3. Or in terms of components Proof.We write Incorporating (3.42) and (3.45), we find that the diagonal of A is

.48)
This completes the proof.
Theorem 3.2.The Navier-Stokes equation for general flow reads Note that (3.50) corrects [7, p. 198] in which there is missing the coefficient 1 3 .Also the statement on [7, p. 181] to the effect that one can consider the most general form of a linear relation between a stress and rate of strain as Substituting these equalities in (2.22) proves (3.50), thereby proving the theorem.
In case the mass force is negligible, the Navier-Stokes equation for an incompressible flow reads where M is the inertance of mass, R is the viscous resistance of the dashpot and K is the spring stiffness.
Introducing the new parameters • Electrical circuits The electric current i = i(t) flowing an electrical circuit which consists of four ingredients, electromotive-force e = e(t), resistance R, coil L and condenser C satisfies

Appearance of the Reynolds number
According to the size of the Reynolds number, there appear different stages of a steady flow, cf.e.g.[2].where the last expression is valid for u ̸ = 0.
Example 5.1.(2-dimensional flow of a vortex) Consider the complex potential given by w = − Γ 2πi log z, where Γ > 0 is a constant.Putting w = u + iv and z = re iϑ , we have by (5.59) (5.60) The stream line ψ = const.is r = const., i. e. the concentric circles around the origin.This is the flow (in the positive direction) around the vortex at the origin as we see presently.The radiation velocity v(r), the tangential velocity v(ϑ ) are given respectively by Hence, in our case, by (5.60) .61) so that the induced velocity caused by the vortex is only the circulation v(ϑ ) around the origin.The constant Γ turns out to be the circulation and is called the intensity of the flow.Apparently, if the vortex is at the point α, the complex potential is given by w = − Γ 2πi log(z − α). (5.62)

The Karman vortex row
First consider the case of a vortex row in which there are infinitely many vortices with the same intensity and rotating in the positive direction at distance a apart.We consider the 2n + 1 vortices with the central one at the origin.Then by (5.62) and the principle of superposition, the complex potential is ) . (5.63) Lemma 5.1.The limit of (5.63) ) .
(5.64) can be expressed as (5.65) Proof.We invoke the partial fraction expansion of the cotangent function valid for all z ∈ C save for multiples of π: Since the series is uniformly convergent in any domain not containing multiples of π, we may integrate the first equality of (5.66) term by term from 1 to z to obtain ) .
(iv) The product representation for the sine function which is a proper form of (5.68).
With (5.65) in mind, we may consider the Karman vortex street which is the row of two rows of infinite vortices with the same intensity but with the opposite rotation.We express them as two rows above and below the real axis at the height h and each vortex in the one row lies in the middle of the other row.The complex potential of the above vortex row is given by ) , and that of the lower row is ) . ) ) + const.

. 17 )
which is called the equation of continuity of fluid.Example 2.2.(Cauchy equation of motion) In this example we show that Newton' second law of motion entails the Cauchy equation of motion (2.22).We consider a flow in a domain Ω ⊂ R 3 and let V ⊂ Ω be a bounded closed domain with its boundary surface A = ∂V .Let ρ denote the continuous density (distribution) function of the fluid in Ω.Then Newton' second law of motion reads Ma = F,(2.18)

2 (a 11 − 11 −
.36) and A = (a i j) 1≤i, j≤6(3.37)and consider the linear relationP 1 = A Ṡ1 + b. (3.38)Stokes proved that a 11 = a 44 = a 66 := a, (3.39) say,a 22 = a 33 = a 55 = 1 a 14 ) , (3.40) b 1 = b 2 = b 3 := b, a 14 ) = µ (3.42)and that all other constants are zero.It follows that A is diagonal.Recall that the pressure p at a point in a fluid is, by definition, the negative of the arithmetic mean of the three normal stresses p i j acting on three mutually perpendicular surface elements:p = − 1 3 (p xx + p yy + p zz ) .(3.43) Now comparing the traces of P and Ṡ, we obtain p xx + p yy + p zz = a (ε xx + εyy + εzz ) + 3b = a∇ • v + 3b, (3.44) in which Stokes found that a = 2µ.(3.45)Since the left-hand side is −3p, it follows that b


is the mass force and ν = µ ρ is the kinematic viscosity, which is the ratio of the viscosity to the density of the fluid, and p is the pressure given by (3.43) above.In component form, (3.49) reads

.55) Theorem 3 . 4 .
Newton's second law of motion up to Stokes' theorem entails the Cauchy equation of motion, which in turn up to Theorem 3.1 entail the Navier-Stokes equation.This follows from Theorem 3.3 and Example 2.2. 4 Examples of the second-order systems • Newton's equation of motion (cf.[8]) M d 2 y dt 2 + R dy dt + Ky = e(t) = F, (4.56)

Definition 5 . 1 .where
log sin z = log z + ∑ ′ log(z − πn)e The Hurwitz zeta-function is defined as the perturbed Dirichlet series ζ (s, a) = Re s > 1 and a is a constant Re a > 0. Its counterpart is the polylogarithm function l s (a) = The special case a = 1 of the Hurwitz zeta-function or the polylogarithm function is the celebrated Riemann zetafunction These functions can be continued meromorphically over the whole plane via their symmetry-functional equations.Lemma 5.2.[11, Theorem 5.5] The following formulas are all equivalent.(i) The asymmetric form of the functional equation for the Riemann zeta-function: The functional equation for the Hurwitz zeta-function (or the Hurwitz formula) (0 < a < 1)

Theorem 5 . 1 .Remark 5 . 1 .
The expression for the complex potential of the Karman vortex street w is, in a long run, is a consequence of the functional equation for the Riemann zeta-function, and a fortiori, such is the expression for the stream function ψ, being the imaginary part of (5.75), from Lemma 5.1 and Lemma 5.2.It is known due to Karman that the vortex street is stable when sin πh a = 1 or h a ≈ 0.281.Let ν be the kinematic coefficient of viscosity of the fluid defined above.Let r be the radius of the cylinder and let U be the uniform speed of the flow.Then the Reynolds number R e is defined byR e = 2rU ν .(5.77)It was shown by Roshko (1953) that the Karman vortex street arises for R e ≥ 40.In most of the textbooks on fluid mechanics, one finds the following argument: Since log(z − ka) + log(z + ka) = log(z 2 − ka 2 ), it follows, after slight transformation, that w n = − Γ 2πi

Table 1 :
Classification of fluid flows

Table 3 :
Special cases of general Stokes' theorem

Table 4 :
Types of flows according to the size of the Reynolds number We are mainly concerned with the Karman vortex street introduced below and the contents are partly modified and essentially strengthened form of [4, Example 26, §7].Recall the definition of the complex logarithm function log w = z = x + iy = log |w| + i arg w = log |w| + i arctan