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Theory of the vibrational hydrodynamic top
Acta Astronautica 114 (2015) 123–129
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Theory of the vibrational hydrodynamic top$ Nikolai Kozlov Laboratory of Vibrational Hydromechanics, Perm State Humanitarian Pedagogical University, 24, Sibirskaya av., 614990 Perm, Russia
a r t i c l e i n f o
abstract
Article history: Received 27 November 2014 Received in revised form 14 February 2015 Accepted 15 April 2015 Available online 23 April 2015
Dynamics of a viscous fluid is investigated theoretically in an annulus with the free inner cylinder under conditions of rotation in an external inertial or gravitational field. The twodimensional formulation is used, corresponding to two long coaxial cylinders. The inner cylinder is free and occupies a steady position on the rotation axis under the action of the centrifugal force due to the fact that it is lighter than surrounding fluid. The action of external force, oriented perpendicular to the rotation axis, induces inertial circular, of the tidal-like type, oscillations of the inner cylinder (core). As a result of the oscillations, the core is brought into rotation relative to the cavity (the outer cylinder) on the background of a steady streaming in the annulus. The mechanism of this differential rotation consists in the generation of an average mass force, of the azimuthal direction, in the oscillating viscous boundary layers on the walls of the core and the cavity. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Rotation Vibration Core oscillation Differential rotation Circular oscillation
1. Introduction Rotating systems are widely spread in nature and technology. Here, important role belongs to the inertia forces: centrifugal and Coriolis forces. Due to their action the rotating fluid acquires non-trivial properties, not found in the absence of rotation. Speaking about momentum and energy transfer in a rotating fluid, it is worth to distinguish viscous interaction in boundary layers on the walls of a container or a body moving in fluid, and inertial waves [1]. Examples of rotating hydrodynamic systems are celestial bodies: stars and planets. There are planets with a solid inner core surrounded by a fluid, e.g. a liquid core in case of the Earth. In the field of an external massive satellite, inside the planet tidal oscillations may be excited, influencing significantly the dynamics of the core. A simple model of such situation, e.g. influence of a satellite on a planet's solid core and liquid core motion, is the ☆ The research was supported by the Russian Scientific Foundation (Project 14-11-00476). E-mail address: [email protected]
http://dx.doi.org/10.1016/j.actaastro.2015.04.010 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
rotation of a two-phase system – comprising a solid core and a liquid shell – in an external inertial field. Vibrations are an actual topic both for fundamental science and technology [2]. Circular vibrations of the inner cylinder in an immobile outer cylinder generate fluid oscillations in the annulus. In the presence of the temperature difference between the surfaces of the cylinders, in the fluid a non-viscous average “vibrational” force is generated, directed radially [3]. This leads to the generation of a steady azimuthal flow. In [4] circular vibrations of quasi-concentric cylinders are reported to induce intensive vortical motion of a viscous fluid filling the annulus. It is known that fluid oscillations near a solid surface or an interface lead to the generation of mean flows in Stokes layers, known as steady streaming [5,6] or acoustic streaming [7]. In the study of dynamics of a fluid layer with the free surface in a partially filled rotating cylinder, the radial gas column displacement under the gravity action is found, stationary in the laboratory frame [8]. Meanwhile, the surface retains the circular cross section, and its motion in the cavity frame represents circular oscillations. In consideration
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N. Kozlov / Acta Astronautica 114 (2015) 123–129
Ωrot
Ωs
2
1
R1
bvibΩ2vib
R2
g
l
Fig. 1. The coaxial layer: 1 – the body (radius R1), 2 – the cavity (radius R2).
of this problem from the positions of vibrational hydromechanics [9] it is shown that such inertial oscillations of a fluid layer lead to the generation of a steady streaming in a Stokes boundary layer. Effect of vibrations, perpendicular to the rotation axis, on the described system leads to a resonant excitation of surface oscillations resulting in the steady streaming [10]. In [11] the rotating cylinder containing a “weightless” free cylinder and entirely fluid-filled is studied experimentally and theoretically. Differential rotation of the inner cylinder is reported and an analytical solution is made, which overestimates rotation velocity measured in the experiment. The action of an external force, perpendicular to the rotation axis, on a free inner cylinder in a rotating outer cylinder with liquid excites circular oscillations of the first. Due to generation of a steady streaming in the viscous boundary layers, the inner cylinder spins [12]. We name a body rotating fast relative to a rotating fluid under the action of a vibrational force “vibrational hydrodynamic top”. In the presented work, the role of viscous exchange between the liquid and solid is considered in conditions of inertial tidal oscillations of the solid core, as well as the influence of oscillating viscous boundary layers on the flow in a coaxial layer. The problem is solved in a twodimensional formulation, however it will be shown that a significant part of parameters does not depend on geometry and may be applied for description of dynamics of both cylindrical and spherical bodies. The problem of fluid motion in a coaxial layer formed by two cylinders and subject to transverse vibrations was studied experimentally and theoretically in [12]. The boundary conditions used in the cited work are valid for the case of the relative radius R ¼ R1 =R2 close to unity, e.g. R ¼0.9, and underestimate the intensity of the core differential rotation for R 0:6, used in experiments, by approximately 2. In the present work a more precise boundary condition is introduced for finding the velocity of fluid oscillations beyond the boundary layer. This allows us to extend the theory validity to an arbitrary value of R. 2. Problem formulation The inner cylinder 1 (the body) is free and lighter than the liquid. The outer cylinder 2 is at constant rotation with the angular velocity Ωrot . Under rotation the body is positioned on the rotation axis under the action of centrifugal force (Fig. 1). The layer is assumed infinitely long, so that the derivative with respect to the axial coordinate is zero, and the effect of the ends is neglected. To achieve this in the experiment, in most cases the aspect ratio of the layer 0 l l=ðR2 R1 Þ is of the order of 10. Here, l is the length of
Fig. 2. The annulus in the cross section.
the body. Thus, the solution reduces to finding the velocity field in a two-dimensional layer of isothermal and homogeneous in density fluid (Fig. 2). The described system is subject to the action of an external inertial field, static or vibrating in the laboratory frame with the frequency Ωvib . The vector of the external force is perpendicular to the rotation axis. In the cavity frame the external force rotates with the frequency Ωosc ¼ Ωvib Ωrot and excites body circular oscillations. In the case of a static external force, Ωvib ¼ 0 and Ωosc ¼ Ωrot . The body oscillation amplitude is supposed to be small compared to the annulus gap and to the body size: bs 5 R2 R1 ;
bs 5R1 :
ð1Þ
The problem is solved in the approximation of high oscillation frequency: sffiffiffiffiffiffiffiffiffiffiffi 2ν R2 R1 b : ð2Þ
Ωosc
This means that the viscous forces act only near the solid walls of the body and the cavity. Here, Ωosc is the frequency, in the cavity frame, of body oscillations induced by an external oscillating force F. The fluid is split into two domains: viscous, inside the Stokes boundary layers localized on the solid walls, and non-viscous, in the annulus beyond the boundary layers. The fluid dynamics in rotating cavity is described by Navier–Stokes and continuity equations: ∂v þ ðv∇Þv ¼ ∇P þ ν∇2 v þ Ωrot r Ωrot þ 2v Ωrot ; ∂t div v ¼ 0: ð3Þ
2.1. Experimental technique and results The body 1 and the cuvette 2 (Fig. 1) are made of organicglass tubes sealed from the ends. The volume between the
N. Kozlov / Acta Astronautica 114 (2015) 123–129
cylinders is filled by a fluid with no air bubbles. The assembled experimental model is installed horizontally in bearing supports on a vibrator platform. The cuvette is brought into rotation via a coaxial flexible transmission by a stepper motor Electroprivod FL86STH80 with rotation stability 0.01 rps. The vertical vibrations are produced by an electrodynamic vibrator VEB GRW Teltow 3.12.39, the amplitude of a spurious harmonic does not exceed 10%. A stroboscopic illumination is used for observation and for measurement of rotation velocity of the cylinders. At rotation in the gravity field, the light body makes inertial circular oscillations in the cavity frame with the frequency of rotation, this results in its lagging rotation. At vibrations, similar inertial oscillations occur with the vibration frequency when it coincides with one of the system eigenfrequencies. Depending on the frequency n Ωvib =Ωrot , the vibrations induce either overrunning (at n 41) or lagging (at n o1) body rotation. In both cases the body differential rotation intensity increases with the amplitude of external forcing. 3. Solution The problem is considered in non-inertial frame. The role of the Coriolis force manifests itself in production of circular body oscillations in response to the external forcing. The body makes inertial oscillations with the frequency Ωosc and the amplitude bs . Its velocity in the cavity frame is given by the expression
We find that A1 ¼
1 2
1 R
; A2 ¼
Φ ¼ bs Ωosc
ð4Þ
In the considered formulation, applying the operator rot to Eq. (4), one can demonstrate that the flow should be potential. Thus, to find the fluid velocity in the nonviscous domain it is sufficient to solve Laplace's equation for the velocity potential: ∇2 Φ ¼ 0:
ð5Þ
We are seeking the solution in the form bs Ωosc f ðrÞ sin ðφ Ωosc tÞ, with f(r) being a function of the radial coordinate. The potential of the velocity of this flow is A Φ ¼ bs Ωosc A1 r þ 2 sin φ Ωosc t : r The constants A1 and A2 are determined by the following boundary conditions: ∂Φ ¼ 0; ∂r r ¼ R2 : U r ¼ bs Ωosc sin ðφ Ωosc tÞ:
r ¼ R1 : U r ¼
rþ
2
1R
R21 r
!
sin Ωosc t φ ;
where R R1 =R2 . The fluid velocity is given by ∂Φ 1 ∂Φ 1 er þ U¼ eφ ¼ bs Ωosc r ∂φ ∂r 1 R2 " ! ! # 2 R R2 1 21 sin Ωosc t φ er 1 þ 21 cos Ωosc t φ eφ : r r ð6Þ The amplitude of the oscillatory velocity in the coaxial layer near the body wall is found from (6): 1 U φ jðr ¼ R1 Þ ¼ 2bs Ωosc : ð7Þ 1 R2 The pulsation velocity near the cavity wall should be sought analogically, but applying the boundary condition written in the cavity reference frame: r ¼ R1 : U r ¼ bs Ωosc sin ðφ Ωosc tÞ; r ¼ R2 : U r ¼ 0: We obtain R2 2
1 R
; A2 ¼ R2 2
1R
3.1. Non-viscous domain
∂U ¼ ∇P þ 2U Ωrot : ∂t
1 R2
1
Φ ¼ bs Ωosc
First, the momentum equations are solved in the annulus (non-viscous domain). As the body oscillation amplitude is small, the second term in the left part of Eq. (3) can be neglected. The velocity in the non-viscous domain is noted as U, the equation is written as follows:
R21
and
A1 ¼
bs Ωosc ð sin ðΩosc tÞi þ cos ðΩosc tÞjÞ:
125
U ¼ bs Ωosc "
1
R22 r2
R2
R21 2 1 ! R
;
R2 rþ 2 r
!
sin Ωosc t φ ;
!
1 R2
!
R2 sin Ωosc t φ er 1 þ 22 r
!
# cos Ωosc t φ eφ :
ð8Þ The amplitude of the velocity oscillations in the annulus, near the cavity wall, is ! R2 U φ jðr ¼ R2 Þ ¼ 2bs Ωosc : ð9Þ 1 R2 Expression (7) is the key adjustment to the theory presented in [12]. It allows calculation of the velocity of the body differential rotation for a vibrational hydrodynamic top with arbitrary values of R, and not only those close to unity. From the expression (6) it follows that due to the circular body oscillations, an azimuthal wave propagates in fluid in the direction coinciding with the body oscillation direction. 3.2. Viscous domain Next, the momentum equations are solved in the viscous boundary layer. Either the body surface or the cavity wall surface is chosen as a reference frame.
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Respectively, the amplitude of pulsation velocity beyond the boundary layer is set equal to (7), or (9). In view of condition (2), the wall curvature may be neglected in the scale of the boundary layer. The problem is formulated in the Cartesian frame, the x-axis is oriented in the direction of propagation of fluid oscillations, the yaxis is normal to the wall. The velocity of tangential flow is much greater than the velocity of radial flow: ∂ ∂ b : ∂y ∂x
ux b uy ;
It is convenient to apply a method, which is usually for vibrational mechanics [2,3]. The motion in the boundary layer may be written as a sum of the pulsating and the mean terms: v ¼ u þ V;
P ¼ p þP av :
ð10Þ
First, pulsation velocity will be found and then averaging over the fast time 2π =Ωosc will be used to subtract the pulsational motion from consideration. Equations for the pulsation velocity read as follows: ∂ux 1 ∂p ∂2 ux ∂ux ∂uy þν 2 ; ¼ þ ¼ 0; ∂t ρL ∂x ∂x ∂y ∂y
ð11Þ
where ρL stands for fluid density. The boundary conditions are the following: y ¼ 0: y
δ
ux ¼ uy ¼ 0; ux ¼ U φ jðr ¼ R1 ;R2 Þ cos ðΩosc t kxÞ:
b 1:
ð12Þ
Here, the choice between R1 and R2 is made taking into account whether the boundary layer is situated on the body wall or the cavity one, kx ¼ φ, k ¼ ð1=rÞjðr ¼ R1 ;R2 Þ , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ ¼ 2ν=Ωosc is the Stokes boundary layer thickness. The pressure in the boundary layer is determined by the type of oscillations in the non-viscous domain. The pulsation velocity is obtained from (11): ux ¼ U φ jðr ¼ R1 ;R2 Þ ðe ζ cos ðθ ζ Þ cos θÞ;
ð13Þ
uy ¼
δU φ
2r ðr ¼ R1 ;R2 Þ
fe ζ ½ sin ðθ ζ Þ cos ðθ ζ Þ þ ð1 þ2ζ Þ cos θ sin θg;
equation of steady streaming is written as follows: ∂2 V x ¼ ρL 〈u∇ux 〉; ∂y2
η
where angular brackets stand for time averaging. As follows from (15), due to viscous momentum exchange in the oscillating boundary layer, a mean mass force is generated, which after substituting (13) and (14) is written as follows: F vib ¼ ρL 〈u∇ux 〉 2 ρ U φ ¼ L 2 r
where θ Ωosc t φ; ζ y=δ.
ðr ¼ R1 ;R2 Þ
R2 R1 b 10δ; which means
ω b100:
ð16Þ
Thus, the tangential fluid oscillations in the viscous boundary layers, that form on the walls of the body and the cavity, lead to the generation of an average mass force which is directed in the sense of propagation of an azimuthal wave excited by the body vibrations. As the body is free, it is entrained by this force and comes into rotation relative to the cavity. The torque of the force applied to the body unit length is found by integration across the boundary layer and along the circumference: Z 1 1 jM vib j ¼ 2π R21 ρL 〈u∇ux 〉∂ζ ¼ πρL δR1 U 2φ jðr ¼ R1 Þ : ð17Þ 2 0 Substituting (7) into (17), we obtain 2 1 2 2 : jM vib j ¼ 2πρL δR1 bs Ωosc 2 1R
ð18Þ
The vibrational torque is balanced by the torque of viscous forces according to the case of rotation of coaxial cylinders [13]. In the experiment, it is calculated through the velocity of the differential rotation of the body ΔΩ Ωs Ωrot : M vib ¼ 4πη
R22 R21
:
ð19Þ
Now let us express the angular velocity of the body rotation in the cavity frame out of the equality of the torques (17) and (19):
3.3. Steady streaming
ΔΩ ¼ The mean flow is considered stationary (or slowly varying), hence the time derivative for its terms is set equal to zero. As the boundary layer is closed, the mean pressure gradient in projection to the tangential direction (x-axis) is zero. For a thin boundary layer, transfer acceleration may be expressed as radial pressure. As the tangential flow is dominant, the convective term V∇V is omitted. After the substitution of (10) into (3) and time averaging, taking into account the abovementioned, the
e ζ e ζ ζ sin ζ þ ζ 1 cos ζ :
As follows from the expression above, the average force is localized inside 10 thicknesses of the Stokes layer. It reduces by an order of magnitude at the distance about 5 boundary layer thicknesses. Hence, the condition (2) could be strengthened to the following:
ΔΩR21 R22
ð14Þ
ð15Þ
U 2φ jðr ¼ R1 Þ 1 R2 : 4Ωosc R1 δ
ð20Þ
4. Rotation in the gravity field Considering the rotation of the studied system in the gravity field in the absence of external vibrations, we can talk about the vibrational mechanism of excitation of the differential rotation of the body, as the gravity vector
N. Kozlov / Acta Astronautica 114 (2015) 123–129
127
rotates in the cavity frame and this leads to forced inertial oscillations of the body. In [8] it was theoretically shown that the effect of gravity on a rotating cylindrical fluid layer with a free surface leads to a redistribution of the mass of the liquid. As a result, the free surface retains the shape of a circular cylinder and undergoes a stationary in the laboratory frame displacement downward with respect to the axis of the cavity. The amplitude of the radial displacement is found in [8]. Mathematically, this problem corresponds to the dynamics of an infinitely light body. In order to take into account the body density, we have to change the boundary condition for the pressure on the body surface (i.e. interface). The total force applied by the fluid to the surface of a unit length of the body, can be written as I F¼ p dS;
Substituting (25) in (17), we obtain the expression for the case without vibrations:
where p is the fluid pressure, dS ¼ R 1 dθ, θ is the angular coordinate in the laboratory frame with zero value in the lowest point. Since we consider the stationary solution, and the axis of the body in the laboratory system is radially shifted downwards and motionless, the pressure force applied to the body is balanced by the body weight in liquid. It ensues from the form of disturbance that the pressure on the surface of the body varies as
The dimensioless frequency ω h =δ . High ω values correspond to thin Stokes layers. From (17) it follows that the vibrational torque is generated in the viscous boundary layers. According to (27), with increase of ω, as δ becomes infinitesimal, jMj should vanish, which is easy to understand for the boundary layers that are torque generators. The vibrational torque could not be measured in the experiment. However, an estimation was done, allowing qualitative comparison. The torque M vib was calculated using the formula (19) for experimental parameters and then substituted into (27). The comparison revealed the same tendency of non-dimensionalized torque to decrease with the dimensionless frequency (Fig. 3). For proper quantification of vibrational effect in experiment, a dimensionless velocity of differential rotation, which is measured directly, should be used. An expression for it can be found by substituting (25) into (20). Considering that for the gravitational excitation of vibrations
pjξ ¼ R þ δ1 ¼ A cos θ: Here, A is some constant. The pressure force projection on the vertical axis is written as F γ ¼ R1
Z 2π 0
A cos 2 θ dθ ¼ π R1 A:
ð21Þ
It is balanced by the body weight
ρs gπ R21 :
ð22Þ
From the equality of the expressions (21) and (22) we obtain the boundary condition for the pressure on the surface of the cylindrical body: pjξ ¼ R þ δ1 ¼ ρs gR1 cos θ:
ð23Þ
π
2
bs ¼
1 Γ R1 1 ρ 2
1 R2 ;
jMj
jM vib j
ρL ghR21
π
¼ pffiffiffi Γ ð1 ρÞ2 ω 1=2 : 2
jMj
Γ ð1 ρÞ2
π
¼ pffiffiffi ω 1=2 : 2
ð25Þ
ð27Þ 2
2
/M/ Γ(1−ρ)2 10-1
1 2 3 4 5
2
U φ jðr ¼ R1 Þ ¼ Γ R1 Ωosc ð1 ρÞ:
ð26Þ
To consider the role of viscosity at a given forcing, one can bring (26) to the following expression:
ð24Þ
where Γ ¼ g=Ωrot R1 is the dimensionless acceleration of gravity and ρ ¼ ρs =ρL is the relative density of the body in fluid. In the limiting case ρ ¼ 0 this result converges to the one obtained in [8]. Substituting (24) into (7) we obtain
ν ; Ωosc h2
where, h ¼ R2 R1 . 2 Expressing the dimension of the torque as ρL ghR1 and 2 putting ω Ωosc h =ν, we can write the dimensionless torque applied to the unit length of the body, given that Ωosc ¼ Ωrot :
Here, δ1 ¼ ðbs =R2 Þ cos θ is a small deviation of the body surface from the coaxial position in the laboratory frame. By solving the equations from [8] and using the boundary condition (23), the oscillation amplitude of the finite density body is obtained:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω2
jM vib j ¼ pffiffiffiρL ghR1 Γ ð1 ρÞ2 osc 2 Ω2rot
10-2 2 10
6
3
10
10
4
ω
Fig. 3. Comparison of the experimental torque estimation for ν ¼ 1.0 (1), 2.7 (2), 5.8 (3), 27.7 (4), and 47.8 cSt (5) with the theoretical dependence 0 (27) (6). R ¼0.60 and l ¼ 9:3.
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N. Kozlov / Acta Astronautica 114 (2015) 123–129
of the body Ωosc ¼ Ωrot , we have ΔΩ 1 R1 2 Γ ð1 ρÞ2 1 R2 : ¼ 4 δ Ωrot
ð28Þ
Comparing this result to [12], one can constatate that the solution obtained in the present work gives higher differential rotation intensity. For example at R ¼0.6 and equal other parameters, the value of (28) is greater by approximately 2 than that in [12] where this value was underestimated. Expression (28) allows a direct comparison of the experimental and theoretical results (Figs. 4, 5), where R Γ 2R 1 Γ 2 ð1 ρÞ2 1 R2 :
δ
On the plane Γ R ; jΔΩj=Ωrot experimental results for different parameters are presented. In the case of water (Fig. 4, points 1) experimental points agree with theory (28) at small values of dimensionless amplitude but show 2 a discrepancy at Γ R 10 2 . However, a slight increase in viscosity reduces the scattering and discrepancy, points 2 on Fig. 4 are in good agreement with (28). Experimental results obtained at different values of ρ and R (Fig. 4, points 3 and 4, respectively) are consistent with other results and the theory. The discrepancy between the theory and experiments with water may be associated with some spurious effects. For example, an instability of the body position on the axis is observed in the experiment under intensive forcing (e.g. low Ωrot values, which correspond to high Γ; or vibrations of high amplitude and frequency). For the cylindrical body this instability manifests itself in the angular three-dimensional body oscillations and circular high-amplitude auto-oscillations. These phenomena are beyond the scope of the problem due to their substantially non-linear properties. Coming back to points 1, which are not considered as obtained under intensive forcing, our suggestion is that there is an additional source of differential rotation. This other mechanism might be responsible for the difference between points 1 and 2, and it may be 2
10
-1
|ΔΩ| Ωrot
1 2 3 4
10-2
10
-3
5
10-2
10-1
ΓR
2
10 10 0
Fig. 4. Comparison of the experimental results for R ¼0.60 and l ¼ 9:3 (1–3), 0.43 and 3.4 (4) with the theoretical dependence (28) (5). The working fluids are water (1, 3, 4) and the aqueous solution of glycerol, ν ¼2.7 cSt (2); ρ ¼0.54, (1, 4), 0.50 (2), and 0.084 (3).
considered secondary as it is easily damped by a slight viscosity increase. From this we conclude that the increase in the system stability with fluid viscosity allows the body dynamics to remain two-dimensional and linear and be well described by the developed theory.
5. Discussion From (20) it follows that the action of an external field perpendicular to the rotation axis leads to the excitation of inertial oscillations of a solid core and to the generation in the viscous boundary layers of a mean differential rotation. The quadratic dependence of ΔΩ=Ωrot on U φ points to the nonlinear nature of the steady streaming. Nevertheless, in the obtained solution for the rotation in the gravity field, the amplitude of body oscillations (24) is in a linear dependence on the amplitude of external forcing Γ, this corresponds to the approximation (1) made in the beginning. The analytical solution found in [11] for the limiting case R 5 1 and ρ 5 0 can be written as ΔΩ=Ωrot ðR1 =δÞΓ 2 . It is consistent with the present theory. From (28) it follows that the intensity of the core differential rotation is determined by the dimensionless frequency, R1 =δ ω1=2 , characterizing the viscous momentum transfer; the amplitude of external forcing: Γ2; and the inertial properties of the core with respect to the fluid: ð1 ρÞ2 . All three parameters do not depend on the problem geometry, the core radius R1 may be used to describe the body of both cylindrical and spherical shapes. The only parameter belonging exclusively to a cylindrical system is ð1 R2 Þ. That is why it would be natural to expect 2 2 that the dimensionless complex Γ ρ ¼ ðR1 =δÞΓ ð1 ρÞ2 be the governing parameter for the description of rotation of a spherical core in an external static field. This idea is supported by experimental investigations [14,15], their results agreeing with the present theory: ΔΩ=Ωrot 2 ðR1 =δÞΓ ð1 ρÞ2 . Thus, the mechanism described in the present work – generation of a mean vibrational force in the oscillating boundary layers – determines the differential rotation intensity under tidal-like oscillations for a core of different geometry. The study of the flow structure in a rotating cylinder with a low-viscosity fluid showed that the inertial circular oscillations of a spherical core may trigger a shear flow instability [15]. In [14] it is demonstrated that the devia2 tion from the dependence ΔΩ=Ωrot Γ takes place in the domain of the Reynolds number values, calculated through ΔΩ, lying above the threshold of the instability onset. This confirms the validity of the developed theory in the condition of the small core oscillation amplitude. The developed theory cannot predict the dynamics of a real planet but describes a mechanism of excitation of core differential rotation, this mechanism having a general character. Although the quantitative results are valid for the high frequency asymptotic case (ω b100), the qualitative result – viscous generation of a vibrational force due to core oscillations – belongs to a wider range of frequencies (and fluid viscosities). In this scope the obtained results could be interesting for the physics of planets and stars.
N. Kozlov / Acta Astronautica 114 (2015) 123–129
6. Conclusion Oscillations of a free cylinder in a rotating cylindrical cavity with liquid are studied under the action of an external force perpendicular to the rotation axis. The body makes inertial circular oscillations relative to the external cylinder. In the annulus the azimuthal steady streaming is formed, the mechanism of which is in generation of the average mass force in oscillating boundary layers on a solid surface. As a result, the inner cylinder rotates with a differential velocity relative to the outer one. A special case, which allows a direct comparison between the theory and experiment, is the rotation in an external static field (e.g. gravity field). The experimental results of the study of the dynamics of a light cylindrical body in a rotating cylindrical container with a fluid agree with the developed theoretical model. The study of a spherical body dynamics in a rotating cylinder [15] and rotating spherical cavity [14] with liquid reveals that the intensity of the differential rotation of the spherical core is determined by the parameter Γ ρ , which is obtained in the present theory. Thus, the described mechanism of the steady streaming generation in oscillating boundary layers is valid in the formulation of a problem with various geometry of the core and the cavity. References [1] H.P. Greenspan, The Theory of Rotating Fluids, University Press, Cambridge, 1968. [2] I.I. Blechman, Vibrational Mechanics, Allied Publishers, New Delhi, 2003, isbn 9788177644579. [3] G.Z. Gershuni, D.V. Lubimov, Thermal Vibrational Convection, Wiley, New York, 1998.
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Theory of the vibrational hydrodynamic top$ Nikolai Kozlov Laboratory of Vibrational Hydromechanics, Perm State Humanitarian Pedagogical University, 24, Sibirskaya av., 614990 Perm, Russia
a r t i c l e i n f o
abstract
Article history: Received 27 November 2014 Received in revised form 14 February 2015 Accepted 15 April 2015 Available online 23 April 2015
Dynamics of a viscous fluid is investigated theoretically in an annulus with the free inner cylinder under conditions of rotation in an external inertial or gravitational field. The twodimensional formulation is used, corresponding to two long coaxial cylinders. The inner cylinder is free and occupies a steady position on the rotation axis under the action of the centrifugal force due to the fact that it is lighter than surrounding fluid. The action of external force, oriented perpendicular to the rotation axis, induces inertial circular, of the tidal-like type, oscillations of the inner cylinder (core). As a result of the oscillations, the core is brought into rotation relative to the cavity (the outer cylinder) on the background of a steady streaming in the annulus. The mechanism of this differential rotation consists in the generation of an average mass force, of the azimuthal direction, in the oscillating viscous boundary layers on the walls of the core and the cavity. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Rotation Vibration Core oscillation Differential rotation Circular oscillation
1. Introduction Rotating systems are widely spread in nature and technology. Here, important role belongs to the inertia forces: centrifugal and Coriolis forces. Due to their action the rotating fluid acquires non-trivial properties, not found in the absence of rotation. Speaking about momentum and energy transfer in a rotating fluid, it is worth to distinguish viscous interaction in boundary layers on the walls of a container or a body moving in fluid, and inertial waves [1]. Examples of rotating hydrodynamic systems are celestial bodies: stars and planets. There are planets with a solid inner core surrounded by a fluid, e.g. a liquid core in case of the Earth. In the field of an external massive satellite, inside the planet tidal oscillations may be excited, influencing significantly the dynamics of the core. A simple model of such situation, e.g. influence of a satellite on a planet's solid core and liquid core motion, is the ☆ The research was supported by the Russian Scientific Foundation (Project 14-11-00476). E-mail address: [email protected]
http://dx.doi.org/10.1016/j.actaastro.2015.04.010 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
rotation of a two-phase system – comprising a solid core and a liquid shell – in an external inertial field. Vibrations are an actual topic both for fundamental science and technology [2]. Circular vibrations of the inner cylinder in an immobile outer cylinder generate fluid oscillations in the annulus. In the presence of the temperature difference between the surfaces of the cylinders, in the fluid a non-viscous average “vibrational” force is generated, directed radially [3]. This leads to the generation of a steady azimuthal flow. In [4] circular vibrations of quasi-concentric cylinders are reported to induce intensive vortical motion of a viscous fluid filling the annulus. It is known that fluid oscillations near a solid surface or an interface lead to the generation of mean flows in Stokes layers, known as steady streaming [5,6] or acoustic streaming [7]. In the study of dynamics of a fluid layer with the free surface in a partially filled rotating cylinder, the radial gas column displacement under the gravity action is found, stationary in the laboratory frame [8]. Meanwhile, the surface retains the circular cross section, and its motion in the cavity frame represents circular oscillations. In consideration
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Ωrot
Ωs
2
1
R1
bvibΩ2vib
R2
g
l
Fig. 1. The coaxial layer: 1 – the body (radius R1), 2 – the cavity (radius R2).
of this problem from the positions of vibrational hydromechanics [9] it is shown that such inertial oscillations of a fluid layer lead to the generation of a steady streaming in a Stokes boundary layer. Effect of vibrations, perpendicular to the rotation axis, on the described system leads to a resonant excitation of surface oscillations resulting in the steady streaming [10]. In [11] the rotating cylinder containing a “weightless” free cylinder and entirely fluid-filled is studied experimentally and theoretically. Differential rotation of the inner cylinder is reported and an analytical solution is made, which overestimates rotation velocity measured in the experiment. The action of an external force, perpendicular to the rotation axis, on a free inner cylinder in a rotating outer cylinder with liquid excites circular oscillations of the first. Due to generation of a steady streaming in the viscous boundary layers, the inner cylinder spins [12]. We name a body rotating fast relative to a rotating fluid under the action of a vibrational force “vibrational hydrodynamic top”. In the presented work, the role of viscous exchange between the liquid and solid is considered in conditions of inertial tidal oscillations of the solid core, as well as the influence of oscillating viscous boundary layers on the flow in a coaxial layer. The problem is solved in a twodimensional formulation, however it will be shown that a significant part of parameters does not depend on geometry and may be applied for description of dynamics of both cylindrical and spherical bodies. The problem of fluid motion in a coaxial layer formed by two cylinders and subject to transverse vibrations was studied experimentally and theoretically in [12]. The boundary conditions used in the cited work are valid for the case of the relative radius R ¼ R1 =R2 close to unity, e.g. R ¼0.9, and underestimate the intensity of the core differential rotation for R 0:6, used in experiments, by approximately 2. In the present work a more precise boundary condition is introduced for finding the velocity of fluid oscillations beyond the boundary layer. This allows us to extend the theory validity to an arbitrary value of R. 2. Problem formulation The inner cylinder 1 (the body) is free and lighter than the liquid. The outer cylinder 2 is at constant rotation with the angular velocity Ωrot . Under rotation the body is positioned on the rotation axis under the action of centrifugal force (Fig. 1). The layer is assumed infinitely long, so that the derivative with respect to the axial coordinate is zero, and the effect of the ends is neglected. To achieve this in the experiment, in most cases the aspect ratio of the layer 0 l l=ðR2 R1 Þ is of the order of 10. Here, l is the length of
Fig. 2. The annulus in the cross section.
the body. Thus, the solution reduces to finding the velocity field in a two-dimensional layer of isothermal and homogeneous in density fluid (Fig. 2). The described system is subject to the action of an external inertial field, static or vibrating in the laboratory frame with the frequency Ωvib . The vector of the external force is perpendicular to the rotation axis. In the cavity frame the external force rotates with the frequency Ωosc ¼ Ωvib Ωrot and excites body circular oscillations. In the case of a static external force, Ωvib ¼ 0 and Ωosc ¼ Ωrot . The body oscillation amplitude is supposed to be small compared to the annulus gap and to the body size: bs 5 R2 R1 ;
bs 5R1 :
ð1Þ
The problem is solved in the approximation of high oscillation frequency: sffiffiffiffiffiffiffiffiffiffiffi 2ν R2 R1 b : ð2Þ
Ωosc
This means that the viscous forces act only near the solid walls of the body and the cavity. Here, Ωosc is the frequency, in the cavity frame, of body oscillations induced by an external oscillating force F. The fluid is split into two domains: viscous, inside the Stokes boundary layers localized on the solid walls, and non-viscous, in the annulus beyond the boundary layers. The fluid dynamics in rotating cavity is described by Navier–Stokes and continuity equations: ∂v þ ðv∇Þv ¼ ∇P þ ν∇2 v þ Ωrot r Ωrot þ 2v Ωrot ; ∂t div v ¼ 0: ð3Þ
2.1. Experimental technique and results The body 1 and the cuvette 2 (Fig. 1) are made of organicglass tubes sealed from the ends. The volume between the
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cylinders is filled by a fluid with no air bubbles. The assembled experimental model is installed horizontally in bearing supports on a vibrator platform. The cuvette is brought into rotation via a coaxial flexible transmission by a stepper motor Electroprivod FL86STH80 with rotation stability 0.01 rps. The vertical vibrations are produced by an electrodynamic vibrator VEB GRW Teltow 3.12.39, the amplitude of a spurious harmonic does not exceed 10%. A stroboscopic illumination is used for observation and for measurement of rotation velocity of the cylinders. At rotation in the gravity field, the light body makes inertial circular oscillations in the cavity frame with the frequency of rotation, this results in its lagging rotation. At vibrations, similar inertial oscillations occur with the vibration frequency when it coincides with one of the system eigenfrequencies. Depending on the frequency n Ωvib =Ωrot , the vibrations induce either overrunning (at n 41) or lagging (at n o1) body rotation. In both cases the body differential rotation intensity increases with the amplitude of external forcing. 3. Solution The problem is considered in non-inertial frame. The role of the Coriolis force manifests itself in production of circular body oscillations in response to the external forcing. The body makes inertial oscillations with the frequency Ωosc and the amplitude bs . Its velocity in the cavity frame is given by the expression
We find that A1 ¼
1 2
1 R
; A2 ¼
Φ ¼ bs Ωosc
ð4Þ
In the considered formulation, applying the operator rot to Eq. (4), one can demonstrate that the flow should be potential. Thus, to find the fluid velocity in the nonviscous domain it is sufficient to solve Laplace's equation for the velocity potential: ∇2 Φ ¼ 0:
ð5Þ
We are seeking the solution in the form bs Ωosc f ðrÞ sin ðφ Ωosc tÞ, with f(r) being a function of the radial coordinate. The potential of the velocity of this flow is A Φ ¼ bs Ωosc A1 r þ 2 sin φ Ωosc t : r The constants A1 and A2 are determined by the following boundary conditions: ∂Φ ¼ 0; ∂r r ¼ R2 : U r ¼ bs Ωosc sin ðφ Ωosc tÞ:
r ¼ R1 : U r ¼
rþ
2
1R
R21 r
!
sin Ωosc t φ ;
where R R1 =R2 . The fluid velocity is given by ∂Φ 1 ∂Φ 1 er þ U¼ eφ ¼ bs Ωosc r ∂φ ∂r 1 R2 " ! ! # 2 R R2 1 21 sin Ωosc t φ er 1 þ 21 cos Ωosc t φ eφ : r r ð6Þ The amplitude of the oscillatory velocity in the coaxial layer near the body wall is found from (6): 1 U φ jðr ¼ R1 Þ ¼ 2bs Ωosc : ð7Þ 1 R2 The pulsation velocity near the cavity wall should be sought analogically, but applying the boundary condition written in the cavity reference frame: r ¼ R1 : U r ¼ bs Ωosc sin ðφ Ωosc tÞ; r ¼ R2 : U r ¼ 0: We obtain R2 2
1 R
; A2 ¼ R2 2
1R
3.1. Non-viscous domain
∂U ¼ ∇P þ 2U Ωrot : ∂t
1 R2
1
Φ ¼ bs Ωosc
First, the momentum equations are solved in the annulus (non-viscous domain). As the body oscillation amplitude is small, the second term in the left part of Eq. (3) can be neglected. The velocity in the non-viscous domain is noted as U, the equation is written as follows:
R21
and
A1 ¼
bs Ωosc ð sin ðΩosc tÞi þ cos ðΩosc tÞjÞ:
125
U ¼ bs Ωosc "
1
R22 r2
R2
R21 2 1 ! R
;
R2 rþ 2 r
!
sin Ωosc t φ ;
!
1 R2
!
R2 sin Ωosc t φ er 1 þ 22 r
!
# cos Ωosc t φ eφ :
ð8Þ The amplitude of the velocity oscillations in the annulus, near the cavity wall, is ! R2 U φ jðr ¼ R2 Þ ¼ 2bs Ωosc : ð9Þ 1 R2 Expression (7) is the key adjustment to the theory presented in [12]. It allows calculation of the velocity of the body differential rotation for a vibrational hydrodynamic top with arbitrary values of R, and not only those close to unity. From the expression (6) it follows that due to the circular body oscillations, an azimuthal wave propagates in fluid in the direction coinciding with the body oscillation direction. 3.2. Viscous domain Next, the momentum equations are solved in the viscous boundary layer. Either the body surface or the cavity wall surface is chosen as a reference frame.
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Respectively, the amplitude of pulsation velocity beyond the boundary layer is set equal to (7), or (9). In view of condition (2), the wall curvature may be neglected in the scale of the boundary layer. The problem is formulated in the Cartesian frame, the x-axis is oriented in the direction of propagation of fluid oscillations, the yaxis is normal to the wall. The velocity of tangential flow is much greater than the velocity of radial flow: ∂ ∂ b : ∂y ∂x
ux b uy ;
It is convenient to apply a method, which is usually for vibrational mechanics [2,3]. The motion in the boundary layer may be written as a sum of the pulsating and the mean terms: v ¼ u þ V;
P ¼ p þP av :
ð10Þ
First, pulsation velocity will be found and then averaging over the fast time 2π =Ωosc will be used to subtract the pulsational motion from consideration. Equations for the pulsation velocity read as follows: ∂ux 1 ∂p ∂2 ux ∂ux ∂uy þν 2 ; ¼ þ ¼ 0; ∂t ρL ∂x ∂x ∂y ∂y
ð11Þ
where ρL stands for fluid density. The boundary conditions are the following: y ¼ 0: y
δ
ux ¼ uy ¼ 0; ux ¼ U φ jðr ¼ R1 ;R2 Þ cos ðΩosc t kxÞ:
b 1:
ð12Þ
Here, the choice between R1 and R2 is made taking into account whether the boundary layer is situated on the body wall or the cavity one, kx ¼ φ, k ¼ ð1=rÞjðr ¼ R1 ;R2 Þ , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ ¼ 2ν=Ωosc is the Stokes boundary layer thickness. The pressure in the boundary layer is determined by the type of oscillations in the non-viscous domain. The pulsation velocity is obtained from (11): ux ¼ U φ jðr ¼ R1 ;R2 Þ ðe ζ cos ðθ ζ Þ cos θÞ;
ð13Þ
uy ¼
δU φ
2r ðr ¼ R1 ;R2 Þ
fe ζ ½ sin ðθ ζ Þ cos ðθ ζ Þ þ ð1 þ2ζ Þ cos θ sin θg;
equation of steady streaming is written as follows: ∂2 V x ¼ ρL 〈u∇ux 〉; ∂y2
η
where angular brackets stand for time averaging. As follows from (15), due to viscous momentum exchange in the oscillating boundary layer, a mean mass force is generated, which after substituting (13) and (14) is written as follows: F vib ¼ ρL 〈u∇ux 〉 2 ρ U φ ¼ L 2 r
where θ Ωosc t φ; ζ y=δ.
ðr ¼ R1 ;R2 Þ
R2 R1 b 10δ; which means
ω b100:
ð16Þ
Thus, the tangential fluid oscillations in the viscous boundary layers, that form on the walls of the body and the cavity, lead to the generation of an average mass force which is directed in the sense of propagation of an azimuthal wave excited by the body vibrations. As the body is free, it is entrained by this force and comes into rotation relative to the cavity. The torque of the force applied to the body unit length is found by integration across the boundary layer and along the circumference: Z 1 1 jM vib j ¼ 2π R21 ρL 〈u∇ux 〉∂ζ ¼ πρL δR1 U 2φ jðr ¼ R1 Þ : ð17Þ 2 0 Substituting (7) into (17), we obtain 2 1 2 2 : jM vib j ¼ 2πρL δR1 bs Ωosc 2 1R
ð18Þ
The vibrational torque is balanced by the torque of viscous forces according to the case of rotation of coaxial cylinders [13]. In the experiment, it is calculated through the velocity of the differential rotation of the body ΔΩ Ωs Ωrot : M vib ¼ 4πη
R22 R21
:
ð19Þ
Now let us express the angular velocity of the body rotation in the cavity frame out of the equality of the torques (17) and (19):
3.3. Steady streaming
ΔΩ ¼ The mean flow is considered stationary (or slowly varying), hence the time derivative for its terms is set equal to zero. As the boundary layer is closed, the mean pressure gradient in projection to the tangential direction (x-axis) is zero. For a thin boundary layer, transfer acceleration may be expressed as radial pressure. As the tangential flow is dominant, the convective term V∇V is omitted. After the substitution of (10) into (3) and time averaging, taking into account the abovementioned, the
e ζ e ζ ζ sin ζ þ ζ 1 cos ζ :
As follows from the expression above, the average force is localized inside 10 thicknesses of the Stokes layer. It reduces by an order of magnitude at the distance about 5 boundary layer thicknesses. Hence, the condition (2) could be strengthened to the following:
ΔΩR21 R22
ð14Þ
ð15Þ
U 2φ jðr ¼ R1 Þ 1 R2 : 4Ωosc R1 δ
ð20Þ
4. Rotation in the gravity field Considering the rotation of the studied system in the gravity field in the absence of external vibrations, we can talk about the vibrational mechanism of excitation of the differential rotation of the body, as the gravity vector
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rotates in the cavity frame and this leads to forced inertial oscillations of the body. In [8] it was theoretically shown that the effect of gravity on a rotating cylindrical fluid layer with a free surface leads to a redistribution of the mass of the liquid. As a result, the free surface retains the shape of a circular cylinder and undergoes a stationary in the laboratory frame displacement downward with respect to the axis of the cavity. The amplitude of the radial displacement is found in [8]. Mathematically, this problem corresponds to the dynamics of an infinitely light body. In order to take into account the body density, we have to change the boundary condition for the pressure on the body surface (i.e. interface). The total force applied by the fluid to the surface of a unit length of the body, can be written as I F¼ p dS;
Substituting (25) in (17), we obtain the expression for the case without vibrations:
where p is the fluid pressure, dS ¼ R 1 dθ, θ is the angular coordinate in the laboratory frame with zero value in the lowest point. Since we consider the stationary solution, and the axis of the body in the laboratory system is radially shifted downwards and motionless, the pressure force applied to the body is balanced by the body weight in liquid. It ensues from the form of disturbance that the pressure on the surface of the body varies as
The dimensioless frequency ω h =δ . High ω values correspond to thin Stokes layers. From (17) it follows that the vibrational torque is generated in the viscous boundary layers. According to (27), with increase of ω, as δ becomes infinitesimal, jMj should vanish, which is easy to understand for the boundary layers that are torque generators. The vibrational torque could not be measured in the experiment. However, an estimation was done, allowing qualitative comparison. The torque M vib was calculated using the formula (19) for experimental parameters and then substituted into (27). The comparison revealed the same tendency of non-dimensionalized torque to decrease with the dimensionless frequency (Fig. 3). For proper quantification of vibrational effect in experiment, a dimensionless velocity of differential rotation, which is measured directly, should be used. An expression for it can be found by substituting (25) into (20). Considering that for the gravitational excitation of vibrations
pjξ ¼ R þ δ1 ¼ A cos θ: Here, A is some constant. The pressure force projection on the vertical axis is written as F γ ¼ R1
Z 2π 0
A cos 2 θ dθ ¼ π R1 A:
ð21Þ
It is balanced by the body weight
ρs gπ R21 :
ð22Þ
From the equality of the expressions (21) and (22) we obtain the boundary condition for the pressure on the surface of the cylindrical body: pjξ ¼ R þ δ1 ¼ ρs gR1 cos θ:
ð23Þ
π
2
bs ¼
1 Γ R1 1 ρ 2
1 R2 ;
jMj
jM vib j
ρL ghR21
π
¼ pffiffiffi Γ ð1 ρÞ2 ω 1=2 : 2
jMj
Γ ð1 ρÞ2
π
¼ pffiffiffi ω 1=2 : 2
ð25Þ
ð27Þ 2
2
/M/ Γ(1−ρ)2 10-1
1 2 3 4 5
2
U φ jðr ¼ R1 Þ ¼ Γ R1 Ωosc ð1 ρÞ:
ð26Þ
To consider the role of viscosity at a given forcing, one can bring (26) to the following expression:
ð24Þ
where Γ ¼ g=Ωrot R1 is the dimensionless acceleration of gravity and ρ ¼ ρs =ρL is the relative density of the body in fluid. In the limiting case ρ ¼ 0 this result converges to the one obtained in [8]. Substituting (24) into (7) we obtain
ν ; Ωosc h2
where, h ¼ R2 R1 . 2 Expressing the dimension of the torque as ρL ghR1 and 2 putting ω Ωosc h =ν, we can write the dimensionless torque applied to the unit length of the body, given that Ωosc ¼ Ωrot :
Here, δ1 ¼ ðbs =R2 Þ cos θ is a small deviation of the body surface from the coaxial position in the laboratory frame. By solving the equations from [8] and using the boundary condition (23), the oscillation amplitude of the finite density body is obtained:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω2
jM vib j ¼ pffiffiffiρL ghR1 Γ ð1 ρÞ2 osc 2 Ω2rot
10-2 2 10
6
3
10
10
4
ω
Fig. 3. Comparison of the experimental torque estimation for ν ¼ 1.0 (1), 2.7 (2), 5.8 (3), 27.7 (4), and 47.8 cSt (5) with the theoretical dependence 0 (27) (6). R ¼0.60 and l ¼ 9:3.
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N. Kozlov / Acta Astronautica 114 (2015) 123–129
of the body Ωosc ¼ Ωrot , we have ΔΩ 1 R1 2 Γ ð1 ρÞ2 1 R2 : ¼ 4 δ Ωrot
ð28Þ
Comparing this result to [12], one can constatate that the solution obtained in the present work gives higher differential rotation intensity. For example at R ¼0.6 and equal other parameters, the value of (28) is greater by approximately 2 than that in [12] where this value was underestimated. Expression (28) allows a direct comparison of the experimental and theoretical results (Figs. 4, 5), where R Γ 2R 1 Γ 2 ð1 ρÞ2 1 R2 :
δ
On the plane Γ R ; jΔΩj=Ωrot experimental results for different parameters are presented. In the case of water (Fig. 4, points 1) experimental points agree with theory (28) at small values of dimensionless amplitude but show 2 a discrepancy at Γ R 10 2 . However, a slight increase in viscosity reduces the scattering and discrepancy, points 2 on Fig. 4 are in good agreement with (28). Experimental results obtained at different values of ρ and R (Fig. 4, points 3 and 4, respectively) are consistent with other results and the theory. The discrepancy between the theory and experiments with water may be associated with some spurious effects. For example, an instability of the body position on the axis is observed in the experiment under intensive forcing (e.g. low Ωrot values, which correspond to high Γ; or vibrations of high amplitude and frequency). For the cylindrical body this instability manifests itself in the angular three-dimensional body oscillations and circular high-amplitude auto-oscillations. These phenomena are beyond the scope of the problem due to their substantially non-linear properties. Coming back to points 1, which are not considered as obtained under intensive forcing, our suggestion is that there is an additional source of differential rotation. This other mechanism might be responsible for the difference between points 1 and 2, and it may be 2
10
-1
|ΔΩ| Ωrot
1 2 3 4
10-2
10
-3
5
10-2
10-1
ΓR
2
10 10 0
Fig. 4. Comparison of the experimental results for R ¼0.60 and l ¼ 9:3 (1–3), 0.43 and 3.4 (4) with the theoretical dependence (28) (5). The working fluids are water (1, 3, 4) and the aqueous solution of glycerol, ν ¼2.7 cSt (2); ρ ¼0.54, (1, 4), 0.50 (2), and 0.084 (3).
considered secondary as it is easily damped by a slight viscosity increase. From this we conclude that the increase in the system stability with fluid viscosity allows the body dynamics to remain two-dimensional and linear and be well described by the developed theory.
5. Discussion From (20) it follows that the action of an external field perpendicular to the rotation axis leads to the excitation of inertial oscillations of a solid core and to the generation in the viscous boundary layers of a mean differential rotation. The quadratic dependence of ΔΩ=Ωrot on U φ points to the nonlinear nature of the steady streaming. Nevertheless, in the obtained solution for the rotation in the gravity field, the amplitude of body oscillations (24) is in a linear dependence on the amplitude of external forcing Γ, this corresponds to the approximation (1) made in the beginning. The analytical solution found in [11] for the limiting case R 5 1 and ρ 5 0 can be written as ΔΩ=Ωrot ðR1 =δÞΓ 2 . It is consistent with the present theory. From (28) it follows that the intensity of the core differential rotation is determined by the dimensionless frequency, R1 =δ ω1=2 , characterizing the viscous momentum transfer; the amplitude of external forcing: Γ2; and the inertial properties of the core with respect to the fluid: ð1 ρÞ2 . All three parameters do not depend on the problem geometry, the core radius R1 may be used to describe the body of both cylindrical and spherical shapes. The only parameter belonging exclusively to a cylindrical system is ð1 R2 Þ. That is why it would be natural to expect 2 2 that the dimensionless complex Γ ρ ¼ ðR1 =δÞΓ ð1 ρÞ2 be the governing parameter for the description of rotation of a spherical core in an external static field. This idea is supported by experimental investigations [14,15], their results agreeing with the present theory: ΔΩ=Ωrot 2 ðR1 =δÞΓ ð1 ρÞ2 . Thus, the mechanism described in the present work – generation of a mean vibrational force in the oscillating boundary layers – determines the differential rotation intensity under tidal-like oscillations for a core of different geometry. The study of the flow structure in a rotating cylinder with a low-viscosity fluid showed that the inertial circular oscillations of a spherical core may trigger a shear flow instability [15]. In [14] it is demonstrated that the devia2 tion from the dependence ΔΩ=Ωrot Γ takes place in the domain of the Reynolds number values, calculated through ΔΩ, lying above the threshold of the instability onset. This confirms the validity of the developed theory in the condition of the small core oscillation amplitude. The developed theory cannot predict the dynamics of a real planet but describes a mechanism of excitation of core differential rotation, this mechanism having a general character. Although the quantitative results are valid for the high frequency asymptotic case (ω b100), the qualitative result – viscous generation of a vibrational force due to core oscillations – belongs to a wider range of frequencies (and fluid viscosities). In this scope the obtained results could be interesting for the physics of planets and stars.
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6. Conclusion Oscillations of a free cylinder in a rotating cylindrical cavity with liquid are studied under the action of an external force perpendicular to the rotation axis. The body makes inertial circular oscillations relative to the external cylinder. In the annulus the azimuthal steady streaming is formed, the mechanism of which is in generation of the average mass force in oscillating boundary layers on a solid surface. As a result, the inner cylinder rotates with a differential velocity relative to the outer one. A special case, which allows a direct comparison between the theory and experiment, is the rotation in an external static field (e.g. gravity field). The experimental results of the study of the dynamics of a light cylindrical body in a rotating cylindrical container with a fluid agree with the developed theoretical model. The study of a spherical body dynamics in a rotating cylinder [15] and rotating spherical cavity [14] with liquid reveals that the intensity of the differential rotation of the spherical core is determined by the parameter Γ ρ , which is obtained in the present theory. Thus, the described mechanism of the steady streaming generation in oscillating boundary layers is valid in the formulation of a problem with various geometry of the core and the cavity. References [1] H.P. Greenspan, The Theory of Rotating Fluids, University Press, Cambridge, 1968. [2] I.I. Blechman, Vibrational Mechanics, Allied Publishers, New Delhi, 2003, isbn 9788177644579. [3] G.Z. Gershuni, D.V. Lubimov, Thermal Vibrational Convection, Wiley, New York, 1998.
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