New linear theory of hydrodynamic instability of the Hagen -Poiseuille flow and the blood swirling flows formation

Aims This paper deals with solving of a century-old paradox of linear stability for the Hagen-Poiseuille flow. A new mechanism of dissipative hydrodynamic instability has been established herein, and a basis for the forming of helical structural organization of bloodstream and respective energy effectiveness of the cardiovascular system functioning has been defined by the authors. Materials and methods Theory of hydrodynamic instability, Galerkin’s approximation. Results A new condition Re > Rethmin ≈ 124 of linear (exponential) instability of the Hagen-Poisseuille (HP) flow with respect to extremely small by magnitude axially-symmetric disturbances of the tangential component of the velocity field is obtained. The disturbances necessarily shall have quasi-periodic longitudinal variability along the pipe axis that corresponds to the observed data. Conclusion We show that the obtained estimate of value of Rethmin corresponds to the condition of independence of the main result (on the linear instability of the HP flow when Re > Rethmin) from the procedure of averaging used in the Galerkin approximation. Thus, we obtain the possible natural mechanism for the blood swirling flows formations observed in the aorta and the large blood vessels. Keywords Instability ≈ Helicity ≈ Structure ≈ Bloodstream Imprint Sergey G. Chefranov, Alexander G. Chefranov. New linear theory of hydrodynamic instability of the Hagen-Poiseuille flow and the blood swirling flows formation; Cardiometry; No.1; November 2012; p.24-30; doi:10.12710/cardiometry.2012.1.2430 Available from:

The problem of defining mechanisms of hydrodynamic instability for the Hagen-Poiseuille (HP) flow is of great fundamental and application significance, starting from the famous Hagen’s researching of blood flows in the pipe ¹ .

So, currently, it is decided [1-4] that the HP flow is exponentially stable with respect to

VR extremely small by magnitude disturbances for any large Reynolds number Re = m—, where ν

V max - maximal (near axis) velocity of the HP flow in the pipe of radius R , and v - coefficient of kinematic viscosity. Such a conclusion of the linear theory of hydrodynamic stability is based on the traditional consideration of pure periodic spatial variability of disturbances along the pipe axis and contradicts to real experimental data and observations in different technical and biology systems.

In [5], it is shown that the conclusion on linear stability of the HP flow needs clarification since if instead of periodic to consider conditionally-periodic (quasi-periodic) disturbances, then already for finite Re it might happen linear (exponential, not algebraic [6, 7]) instability of the HP flow.

In the present paper, we further develop the representation [5] within the framework of the new theory of linear instability of the HP flow. Meanwhile, contrary to [5] in particular we show a possibility of getting a threshold by Re condition of linear instability of the HP flow which does not depend on the procedure of averaging when using Galerkin approximation (necessary because of consideration of longitudinal quasi-periodicity of disturbances).

Materials and methods

Let’s consider evolution in time of axially symmetric extremely small by magnitude hydrodynamic disturbances of the tangential component of the velocity field V _ϕ in the cylindrical system of coordinates ( z,r, ф ) :

∂ V        ∂ V          V

* + V оz ( r ) * = v ( A V , - ^ ) (1) d t           d z             r

Where:

radius R

V0z(r)=Vmax(1-   ) , V = V = 0 - main (undisturbed) HP flow along the pipe of z max R 2 r

∂p having in it a constant longitudinal pressure gradient = const , when ∂z

R ² ∂ p

V _max =     ⋅    ⁰ with constant density ρ of the uniform fluid. In (1) ∆ is Laplace operator.

4ρν ∂z

Due

∂Vϕ to assumed axial symmetry of the extremely small disturbances V ( i.e. since ϕ =0, ϕ         ∂ϕ

∂p there is no derivative for small disturbance of the pressure field p in the right-hand side of ∂ϕ

(1)). Meanwhile, (1) allows closed description of evolution of pure tangential disturbances of the HP flow.

Let’s find solution of equation (1) in the following form

N

V ϕ = V max ∑ A n ( z , t ) J 1 ( γ 1, n r ) (2)

n=1                 R which automatically meets boundary conditions of finiteness of Vϕfor r = 0 and non-slipping

V _ϕ = 0 for r = R on the hard pipe boundary since J ₁ is the Bessel function of the first order, and

γ _{1, n} are zeroes of that function ( n = 1,2,.. ).

Using the feature of orthonormality of Bessel functions and a standard averaging procedure in the Galerkin approximation (see [2]), one gets in the dimensionless from (1), (2) the following closed system of equations for the functions A_n ( z , t ) :

2 N

^∂ A^m + γ 1², m A m -^∂ A 2 ^m + Re ∑ p nm ^∂ Aⁿ = 0 (3)

∂τ            ∂x       n=1     ∂x z      tν

In (3), x = , τ =     , m = 1,2,.., N , and coefficients p are as follows:

R     R2                                nm p = δ -         dyy3J (γ y)J (γ y) (4)

nm nm    J22 (γ1,m ) 0         1 1,n 1 1,m where J2 is a Bessel function of the second order, , δnm is Kronekker’s symbol (δnm = 1for n = m and δnm = 0 if n ≠ m ). Obviously that p12 ≠ p21 in (4) due to the presence of a factor before the integral in (4) (since J2(γ1,1) ≠ J2(γ1,2)).

Let’s limit ourselves by the case of N = 2 . For amplitudes A ₁ and A ₂ ,corresponding to different modes of radial variability V _ϕ in (2), we consider to have different periods of variability along the pipe axis when the next presentation takes place:

A ₁( x , t ) = A ₁₀ e ^{λτ + i 2 πα x} , A ₂( x , t ) = A ₂₀ e ^{λτ + i 2 πβ x} (5)

where α ≠ β contrary to the usual (see [2]) consideration of the problem of stability of the HP flow in the linear approximation by amplitudes of disturbances. Meanwhile, the value of λ = λ ₁ + i λ ₂ in (5) assumes and defines the same (synchronous) character of dependency of functions A ₁ and A ₂ on time. Substituting (5) in (3) (for N = 2 ) leads to the following system

( λ + γ ₁ ² _,1 + 4 π ² α ² + i Re p ₁₁ ⋅ 2 πα ) A ₁₀ e^{i 2 πα x} + i Re p ₂₁2 πβ A ₂₀ e^{i 2 πβ x} = 0 (6) i Re p ₁₂2 πα A ₁₀ e^{i 2 πα x} + ( λ + γ ₁ ² _,2 + 4 π ² β ² + i Re p ₂₂ ⋅ 2 πβ ) A ₂₀ e ^{i 2 πβ x} = 0 (7)

System (6), (7) admits exact solution for constant coefficients A ₁₀ and A ₂₀ only in the case when α = β and in (5) functions A ₁ and A ₂ have the same pure periodic character of variability along the pipe axis. It is not difficult to check that from the condition of solvability of uniform system (6), (7) there may be obtained the well-known conclusion [1-4] on linear stability of the HP flow since in that case it is found that for any Re , λ ₁ <  0 .

Results

Considering the quasi-periodic variability of V _ϕ along the pipe axis for α ≠ β in (5), we use Galerkin approximation to solve the system (6), (7). Meanwhile, let’s average (6) multiplying (6) by the function e ^{- i 2 πγ 1 x} and integrating over x in the limits from 0 to 1/ γ ₁ (i.e. applying to

1/| γ 1 |

(6) an

operation of | γ ₁ | ∫ dx , where | γ ₁ | is the modulus of γ ₁ ). The equation (7) is averaged 0

applying to (7) the same as in (6) operation of averaging but with replaced in it γ ₁ by γ ₂ , where in the general case γ ₁ ≠ γ ₂ .

The solvability condition of the system of the equations obtained from (6), (7) after the specified above averaging is the following dispersion equation for λ :

λ ² + λ ( a + b ) + ab + gc = 0 (8)

where complex value 1 = 21 + i22 is uniquely defined by the following coefficients: a = /121 + 4n2a2 + iRe2nopu = a1 + ia2,                  b = /122 + 4n2P2 + iRe2nPp22 = b1 + zb2, c = 4n2aPRe2 p12p21 and g = 2a 1P

.

1 2 p I 1 a

W m l

Here, in g, we have values of elementary integrals: I_{m a} = | Y _m | J dxe ^² " ^{( a-Y} - ) and

1/| Ym\

I m p = | Y m | J dxe 2 ” ^{( p-Y m} ) , where m = 1,2. From (8), one can obtain (see[5]) the condition of the 0

linear instability of the HP flow when in (8) 2 1 >  0 for some Re >  Re th . A result in that case is significantly depending on the value of g , defined by the way of averaging of the system (6), (7) on the base of Galerkin approximation.

So, if we change the averaging procedure (applying operation | Y 1 | J d % e - ^{z 2 nY 1 x} already to the 0

equation (7), not to (6), as it was done above; and vice versa, we apply to (6) the operation of averaging applied above for averaging of the equation (7)), then, in the dispersion equation, value of g is replaced by 1/ g .

We require that the conclusion on the stability of the HP flow should not depend on the pointed difference in the averaging procedure conducting that is possible only when g ² = 1 . This equation for g has two roots g = 1 and g = - 1 . For g = 1 the conclusion on the stability of HP the flow exactly coincides with the case of the pure periodic disturbances when a = P .

Let’s consider the second case of g = - 1 , and show that meanwhile the linear instability of the HP flow is possible already for finite value of the threshold Reynolds number Re _th .

Actually, for g = - 1 from (8) it follows that 2 1 >  0 when

~        F

^Re >  ^Re * = nD ⁽⁹⁾

where F = ( a 1 + b 1 ) Jap ^ , D = ( a 1 + b 1 ) ² aP p₁₂p ₂₁ - a 1 b 1 ( p 11 a - p ₂₂ P ) ² .

Obviously, for realization of the linear exponential instability of the HP flow D >  0 is necessary, that is trivially to be met when ( p 11 a - p ₂₂ P ) ² <  4 aP p ₁₂ p ₂₁ .

Theory of cardiovascular system performance.

Cardiometry basics.

If to introduce a parameter p _ —, defining the ratio of periods in (5), then from the pointed inequality providing positiveness of D in (9), it follows that the following holds x< p < x + (10)

where    x _±

_ p_u p 22 + 2 P i2 p 21 ± ²V P i2 p 21 ⁽ Ри p 22 + p 21 ^p ”⁾

2 p 1²1

,

i.e. according to (4),

x ₊ _ 1.739.., x - _ 0.588.. .

On the other hand, from the condition

g _ - 1 it follows that the following inequality holds

g _

i ^

(1 - e ^{I Y 1}' )(1

2 па

i

- e Y2 )(а - Y1)(- - Y2) _

2пв        2na ii

(1 - e ^ )(1 - e ^{Y I} )( — - Y 1 )( a - Y 2 )

- 1 (11)

In particular, equation (11) is satisfied when — _ а+ 1 / 1 | n and — ⁼ а+ I Y 2 I m , where m , n -

I Y I n are any integers having the same sign since with necessity then holds ¹ _   >  0 .

¹ Y 2 ।     m

а

Meanwhile from (11), it follows that the following relations defining the value p _ — depending on the values of m,n and signs of /1,/2 hold:

p _

1+

B

B _ B ±

-^j — + —--1 ±3 1 1 + A + A (12)

n I Y 1 1 m I Y 2 I         < m n

Obviously, p from (12) shall meet inequality (10). In particular, for m _ n _ 1 value of p (when B _ B- and /i > 0,Y2 > 0) is p ~ 1.58... Since Re th in (9) is a function of —and p , for the pointed value of p , meeting inequality (10), from (9), we can get that minimal value Re th mm ~ 124 is reached in the proximity of — ~ 0.5 •

Discussion and conclusions

Thus, it has been found the possibility of the linear (exponential) instability of the HP flow already for Re > Re th min = 124, that does not contradict to the well-known estimates of the guaranteed stability of the HP flow obtained from the energy considerations (see [1]) for Re < 81. Obviously, an exponential growth of V after reaching of some finite values shall be replaced by a new non-linear mode of evolution in which all components of velocity and pressure are already mutually cross-linked. This growth also produces the spiral type of the resulting flow. Indeed, for the flow of blood, the arising of spiral structure is observed in the aorta in a wide range of Re number and not only for very high values of the latter [8, 9]. The results of this paper are also published in [10].

Statement on ethical issues

Research involving people and/or animals is in full compliance with current national and international ethical standards.

Conflict of interest

None declared.

Author contributions

All authors contributed equally to this work. S.G.C. read and met the ICMJE criteria for authorship. All authors read and approved the final manuscript.