Note on the Stability Criteria for a New Type of Helical Flows

Note on the stability criteria for a new type of helical flows

Sergey V. Ershkov

Affiliation: Plekhanov Russian University of Economics, Scopus number 60030998, e-mail: [email protected]

In this paper, we proceed exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids with variable (spatially dependent) coefficient of proportionality  between velocity and the curl field of flow. Meanwhile, the system of Navier-Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. The main motivation of the current research is the exploring the stability of previously obtained helical flows. Conditions for the stability criteria of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered. As it has been formulated before, the spatial part of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained.

Keywords: Navier-Stokes equations, non-stationary helical flow, Bernoulli-function.

1 1. Introduction, stability criteria, equations of motion for helical flows.

In fluid mechanics (especially, in hydrodynamics), for distinguishing between the different states of fluid flow one must consider how the fluid would be reacting to a negligible variation of the initial state [1]. These distortions will relate to the physical properties of the system, such as velocity and pressure (in case of incompressible flow). James Clerk Maxwell expressed the qualitative concept of stable and unstable flow well enough when he said [2]:

“When an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be (Q1) stable but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the system is said to be unstable. ”

We shall consider the aforementioned statement as the stability criteria (Q1) for testing of various fluid flows (for a stable flow or for an unstable flow), which have a clear natural basis for the appropriate validation. That means that for a stable flow, any infinitely small variation, which is considered as disturbance, will not have any noticeable effect on the initial state of the system and will eventually die down in time [1]. For a fluid flow to be considered stable it must be stable with respect to every possible disturbance. This implies that there exists no mode of disturbance for which it is unstable [2]. On the other hand, for an unstable flow, any variations will have some noticeable effect on the state of the system which would then cause the disturbance to grow in amplitude in such a way that the system progressively departs from the initial state and never returns to it [1].

In [3] we have obtained exact solution of the system (1)-(2) in a form of helical or Beltrami flow below (only numeration of resulting formulae will be mentioned here):    (x, y, z, t)  (x, y, z)u(x, y, z, t)

2  here we denote u is the flow velocity, u = {U, V, W} (notation u or u means a vector field), the curl field  = (∇×u) (which means the vorticity of the fluid flow);  is variable parameter (which differs from the case  = const, considered in [4-5]).

Besides, the aforesaid exact solution {u, p} of the system (1)-(2) has been obtained in a form of helical or Beltrami flow (3) which should conserve the Bernoulli-function to be the invariant of the aforementioned system:

1  B  (u 2 )  p    const 2

Condition above is the essential point of the helical solutions {u, p} of the system (1)-(2), which were obtained in our previous work [3].

2. Stability criteria for the obtained solution (new type of helical flows).

Applying of the stability criterion (Q1) for the pressure field p is the most obvious. Indeed, the pressure field of the flow should be determined via Bernoulli-function (11) in [3], so it should be stable if we prove that the components of velocity field {U, V, W} are stable insofar. The same conclusion, according to the formulae (18) in [3]

1    p     {u 2} , u  {U, V , W}, 2

       x  W     U ,        z 

2              2 2 2 2   x  2 V   U (t 0 )  V (t 0 )  W (t 0 )exp 2  (t  t 0 )  1 2  U ,             z   3 should be achieved with respect to the component W of flow velocity. Indeed, it is obtained to be linearly depended on the component U of the flow velocity.

So, we should only consider the stability of the components U and V - should they be stable or unstable for the chosen class of solutions? (if yes, we should point out the proper conditions for the aforementioned stability).

Let us further consider the partial solution (A.8) in [3] for helical flows:

  t  t    0  U U(t 0 ) 1 tan  cos exp           2 

As for the component U of the aforementioned solution, let us deviate it in regard to the initial data of this component U(t₀) << U(t₀) as below:

    t  t     0   U   U(t 0 )  U(t 0 )   1 tan   cos  exp             2  

- but, according to restriction for such the solution (see formulae (18) in [3])

2       2   z     U 2      U(t )  U(t )   V 2 (t )  W 2 (t )  exp  2  2 (t  t ) 2 2  0 0  0 0  0                         x    z        

Taking into account that for solution (A.8), which has been previously constructed in [3], the component U(t) can be considered as a class of perturbation [6] decreased exponentially (in accordance with the aforementioned restriction) if t is going to infinity ∞ (see Fig.1), we can come to a reasonable conclusion as below

2        2   z    U(t 0 )   U 2     U 2 (t ) 1    V 2 (t )  W 2 (t ) С 2 (t) 2 2  0   U(t )  0 0               0               x    z         4 where C² (t) equals to the appropriate positive constant at each moment of time t. The last inequality means that, according to the stability criterion (Q1), any infinitely

small variation of the initial data of component U(t₀), which is considered as a negligible disturbance, will not have any noticeable effect on the initial state of the system and will eventually be decreasing down in time. So, we have proved that component U of the aforementioned solution is to be stable.

As for the component V which corresponds to the aforementioned solution for U(t) (A.8) via formulae (18) in [3], let us deviate it also in regard to the initial data of this

component V(t₀) << V(t₀) as below:

 2       2          x   V 2  U 2 (t )   V (t )  V (t )   W 2 (t )exp  2  2(t  t )  1   U 2 ,   0  0 0  0   0   2                  z    2        2       V (t 0 )     x  V 2  U 2 (t )  V 2 (t )1    W 2 (t ) exp  2  2(t  t )  1   U 2 ,   0 0    0   0   2   V (t 0 )                   z   2              2 2 2 2 2   x  2 V  U (t 0 )  V (t 0 )  W (t 0 )exp 2  (t  t 0 )  1 2  U              z   2  2V (t 0 )V (t 0 )exp 2  (t  t 0 ),

where we can see at the right part of the last expression for V ² the additional difference

2 2V(t 0 )V(t 0 )exp  2  (t  t 0 )

with respect to the expression for the unperturbed magnitude of V² .

5 It means that, according to the stability criterion (Q1), any infinitely small variation of the initial data of component V(t₀), which is considered as a negligible disturbance, will not have any noticeable effect on the initial state of the system and will eventually be decreasing down in time. So, we have proved that component V of the aforementioned solution should be stable insofar.

3. Discussion.

The main motivation of the current research is the exploring the stability of previously obtained new type of of non-stationary helical flows for incompressible 3D Navier-Stokes equations [3]. We proved that at least one sub-class from previously presented non-stationary helical solutions [3] should be stable: - e.g., component U of the flow velocity (A.8) as well as corresponding other components of solution via formulae (18). Condition (A.4)

(where I = (1 second)²)

2 2       I        4   0     2   z    x   as one of the existence and stability criteria for the exact solution of the aforementioned type of flows is established. As for other examples of exact non-stationary flows [7] with invariant Bernoulli- function in the whole space (the Cauchy problem), we should refer in this respect to the modern researches [8-9] or, for example, we should recall the results of comprehensive article [10] (which reports the alternative example of using the Bernoulli-invariant to obtain analytical non-stationary solutions of the Navier-Stokes system of equations). So, in the previous work [3] we have proved that the aforementioned solutions exist, when appropriate initial and boundary conditions are provided. Moreover, we suggested in [3] the appropriate form of the exact solution in case of the constant Bernoulli-function in the whole space (the Cauchy problem).

6 But in the current research we have proved additionally the stability of one of sub- classes of the suggested solutions. According to our understanding, there will no principal problem to prove in the future researches (according to the general ansatz in Section 2 of this paper) the stability of other non-stationary helical flows of a new type, suggested in [3]. But we should check the validity of inequality

2        z  U 2 (t )     U 2 (t ) V 2 (t ) W 2 (t ) exp  2  2(t  t ) , 0 2 2  0 0 0   0 0                    x    z         in regard to the each case of partial flows (this is the basis for existence and for real meanings of physical components for such a type of flows). We have considered these questions here in Section 2 for the component U of the flow velocity as well as for other components of solution (see formulae (18) in [3]). As for the proper restriction at choosing the initial conditions for non-stationary helical flows of a new type, suggested in [3], it should obviously be satisfying to the aforementioned inequality as below

2        z  U 2 (t )     U 2 (t )  V 2 (t )  W 2 (t ) exp  2  2(t  t ) ,  0 2 2  0 0 0   0 0                    x    z        

2 2    2    2 2   U (t 0 )    V (t 0 )  W (t 0 )   x    z 

4. Conclusion.

The main motivation of the current research is the developing of the investigation in regard to the stability of non-stationary helical flows for incompressible Newtonian

7 fluids, the previous types of which were successfully investigated in [4-5] ( = const) and [3] ( = (x, y, z)), here  is the coefficient of proportionality between velocity and the curl field of flow. Meanwhile, the system of Navier-Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. We have pointed out at least one sub-class from previously presented non-stationary helical solutions [3] which should be stable; conditions for the stability criteria of the exact solution for the aforementioned type of flows are obtained, for which non- stationary helical flow with invariant Bernoulli-function is considered. Besides, stability criterion for the components of velocity field as well as pressure field is valid only under special conditions, which restrict the choosing of the form of variable coefficient  (x, y, z) (among them, there is additional restriction /y = 0).

Conflict of interest

Author declares that there is no conflict of interests regarding publication of article.

Acknowledgements

Author is thankful to Dr. A.A.Kurkin with respect to the fruitful discussions during preparing of the author for defending his PhD-degree at the end of December 2018 in Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Russia. Also author appreciates critical but excessive remark which was made (during author’s defense his PhD-degree) by Dr. O.A.Druzhinin from Institute of Applied Physics of Russian Academy of Science, Nizhny Novgorod, who is one of the members of the PhD-council where author’s defense has taken place. Namely, Dr.Druzhinin claimed that suggested new non-stationary helical solution [3] would likely not be

8 stable at all; it was the sufficient motivation for author to write the current note to argue against the critical remark from Dr. Druzhinin.

References:

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Fig.1. Schematically presented, the upper restriction for component of solution U(t), here we designate x = t just for the aim of presenting the plot of solution