The problem of spontaneous swirling was considered in [1][2][3][4][5][6][7] and is as follows: can rotary motion occur in the absence of external source of rotation, i.e., under conditions where motion without rotation is realizable?A more rigorous formulation of this problem was given by Lugovtsov [7]. The proposed formulation ensures a close control of the kinematical flux of the axial component of the angular momentum, which eliminates inflow of the rotating fluid in the flow region.The occurrence of rotary motion is regarded as a bifurcation of the initial axisymmetric flow due to the loss of stability against swirling flow (not necessarily rotationally symmetric).At present, examples of the occurrence of spontaneous swirling [1][2][3], including swirling in MHD flow [5, 6], have been given. However, as was shown in [4, 7], the available examples do not satisfy the more rigorous requirements formulated in [7]. Thus, the question of the possibility of spontaneous swirling remains open.The proof that spontaneous swirling is impossible, if this statement is valid, involves significant difficulties and can hardly be obtained in a fairly general form. To prove the existence of this phenomenon, it is sufficient to find at least one example. To narrow the region of search for such an example, it is of interest to consider transition of axisymmetric to rotationally symmetric flow or a plane analog of this transition, i.e., the occurrence of a spontaneous cross (normal) flow which is independent of the transverse coordinate in the case of an initial plane-parallel flow [7].Lugovtsov [7] showed that the bifurcation axisymmetric flow-rotationally symmetric flow (and the corresponding plane analog of this transition) does not take place for a compressible fluid with a variable viscosity coefficient. In the case of the plane analog, this statement is also valid for a conductive fluid moving in the presence of a magnetic field, irrespective of the character of connectedness of the flow region.Such a general result is difficult to obtain for axisymmetric flows in the presence of a magnetic field. In this case, as was noted by Lugovtsov and Gubarev [7], swirling flows can occur which are maintained by electromagnetic forces, and the formulation of the problem of spontaneous swirling requires refinement.Below, we consider axisymmetric flow of an incompressible conductive fluid. It will be shown that axisymmetric spontaneous swirling is impossible if the meridional section of the flow region is simply connected.The equations describing such flows have the following form in conventional notation: 0Zv 1H'ere f = 0, w(r, z,t)).(2)(fr(r, z,t), O, fz(r, z,t)) are forces that maintain the initial axisymmetric flow; v = (u(r,z,t), Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090.
532.516References [1][2][3][4] proposed a hypothesis on the possibility of the appearance of spontaneous swirling of an axisymmetric flow in the absence of explicit sources of rotation, when axisymmetric flow without rotation is clearly possible. The replacement of the symmetry type (axial symmetry/rotational-axial symmetry) is associated with bifurcation of the starting regime at some Reynolds number, when the equation for the rotational component admits nontrivial solutions, corresponding to the stable regime, and the regime without rotation becomes unstable. In the steady-state flow an initial disturbance is completely forgotten, and the intensity of the rotation is independent of the initial disturbance (however the direction of rotation is determined by the initial disturbance). The authors of [4] term this phenomenon an autorotation or a vortical dynamo.The proposed hypothesis is extremely interesting, may have a significant influence on the approach to the explanation of many natural phenomena, and has a fundamental nature. Therefore it is very important to obtain convincing proof of the validity or incorrectness of this proposal.We shall present arguments that yield a basis for serious doubts concerning the validity of the proposed hypothesis and the existence of the spontaneous swirling phenomenon.If the fluid viscosity is constant (independent of the coordinates) and the fluid motion is laminar, stationary, and rotationally symmetric, then the equation for F = rv~ in a cylindrical coordinate system with the axis of symmetry z has the form (in the conventional notations)The two-sided maximum principle is valid for this equation: the maximum and minimum of r are reached on the boundary. This circumstance was noted in [3,4]. This implies that spontaneous swirling is not possible for stationary axisymmetric laminar flows with the condition r = 0 on the boundary, which is a surface of revolution.However, it is stated in [4] that if the conditions of the absence of rotational flow are specified on part of the boundary, then spontaneous swirling is possible even in a homogeneous fluid. An example of such flow is presented in [3], where the problem is examined of the flow between a porous rotating disc and a plane surface.We can show that this statement is erroneous. Spontaneous swirling does not arise even if on part of the boundary condition I" = 0 is specified, and on the boundary segment which is a free surface there are given the conditions ~v = %,n, + %,n, =. 0, u n, + %n, = O,where z~, is the azimuthal component of the tangential stress vector on the free surface; n = (n r, 0, n z) is the outward normal to this surface; r ~ O'~z ----" r o37,Novosibirsk.
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