scholarly journals Slow Rotation of Concentric Spheres with Source at Their Centre in a Viscous Fluid

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Deepak Kumar Srivastava
Keyword(s):  
Angular Velocity ◽  
Outer Sphere ◽  
Present Situation ◽  
Dissipated Energy ◽  
Slow Rotation ◽  
Azimuthal Component ◽  
Concentric Spheres ◽  
Special Cases ◽  
Inner Sphere ◽  
Angular Velocities

The problem of concentric pervious spheres carrying a fluid source at their centre and rotating slowly with different uniform angular velocities , about a diameter has been studied. The analysis reveals that only azimuthal component of velocity exists, and the couple, rate of dissipated energy is found analytically in the present situation. The expression of couple on inner sphere rotating slowly with uniform angular velocity , while outer sphere also rotates slowly with uniform angular velocity , is evaluated. The special cases, like (i) inner sphere is fixed (i.e., ), while outer sphere rotates with uniform angular velocity , (ii) outer sphere is fixed (i.e., ), while inner sphere rotates with uniform angular velocity , and (iii) inner sphere rotates with uniform angular velocity , while outer sphere rotates at infinity with angular velocity , have been deduced.

Stokes Flow Around Slowly Rotating Concentric Pervious Spheres

2013 ◽  
Vol 60 (2) ◽  
pp. 165-219 ◽  
Author(s):  
Deepak Kumar Srivastava ◽  
Raja Ram Yadav ◽  
Supriya Yadav
Keyword(s):  
Angular Velocity ◽  
Stokes Flow ◽  
Outer Sphere ◽  
Present Situation ◽  
Subject Classification ◽  
Dissipated Energy ◽  
Azimuthal Component ◽  
Special Cases ◽  
Inner Sphere ◽  
Angular Velocities

In this paper, the problem of concentric pervious spheres carrying a fluid sink at their centre and rotating slowly with different uniform angular velocities 1, 2 about a diameter has been studied. The analysis reveals that only azimuthal component of velocity exists and the torque, rate of dissipated energy is found analytically in the present situation. The expression of torque on inner sphere rotating slowly with uniform angular velocity 1, while outer sphere also rotates slowly with uniform angular velocity Ω2, is evaluated. The special cases like, (i) inner sphere is fixed (i.e. Ω1 = 0), while outer sphere rotates with uniform angular velocity Ω2, (ii) outer sphere is fixed (i.e. Ω2 = 0), while inner sphere rotates with uniform angular velocity Ω1, (iii.) inner sphere rotates with uniform angular velocity 1, while outer rotates at infinity with angular velocity 2; have been deduced. The corresponding variation of torque with respect to sink parameter has been shown via figures. AMS subject classification - 76 D07

Finite difference code for velocity and surface traction of a Fluid between Two Eccentric Spheres

2015 ◽  
Vol 11 (1) ◽  
pp. 2960-2971
Author(s):  
M.Abdel Wahab
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Finite Difference ◽  
Numerical Study ◽  
Outer Sphere ◽  
Equation Of Motion ◽  
Annular Region ◽  
Surface Traction ◽  
Angular Velocities ◽  
Axis Passing

The Numerical study of the flow of a fluid in the annular region between two eccentric sphere susing PHP Code isinvestigated. This flow is created by considering the inner sphere to rotate with angular velocity 1  and the outer sphererotate with angular velocity 2  about the axis passing through their centers, the z-axis, using the three dimensionalBispherical coordinates (, ,) .The velocity field of fluid is determined by solving equation of motion using PHP Codeat different cases of angular velocities of inner and outer sphere. Also Finite difference code is used to calculate surfacetractions at outer sphere.

On the flow properties of a fluid between concentric spheres

1971 ◽  
Vol 47 (4) ◽  
pp. 799-809 ◽  
Author(s):  
S. G. H. Philander
Keyword(s):  
Outer Sphere ◽  
Azimuthal Velocity ◽  
Ekman Layer ◽  
Shear Layers ◽  
The Other ◽  
Axial Motion ◽  
Flow Properties ◽  
Axis Of Rotation ◽  
Concentric Spheres ◽  
Angular Velocities

Proudman (1956) and Stewartson (1966) analyzed the dynamical properties of a fluid occupying the space between two concentric rotating spheres when the angular velocities of the spheres are slightly different, in other words, when the motion relative to a reference frame rotating with one of the spheres is due to an imposed azimuthal velocity which is symmetric about the equator. The consequences of forcing motion across the equator are explored here. Whereas the flow inside the cylinder [Cscr ] circumscribing the inner sphere and having generators parallel to the axis of rotation is similar to that which results when the driving is symmetric, the flow outside [Cscr ] is quite different. The Ekman layer on the outer sphere persists outside [Cscr ] - its dynamics is modified in the vicinity of the equator - and is instrumental in transferring fluid from one hemisphere to the other. The divergence of this Ekman layer causes slow, axial motion in the inviscid region outside [Cscr ]. On [Cscr ], two shear layers of thicknessO(R−2/7) andO(R−1/3) (whereRis the Reynolds number, assumed large) remove discontinuities in the flow field and return fluid from one hemisphere to the other (rather than one Ekman layer to the other as is the case when the driving is azimuthal).

Forced Convection in Concentric-Sphere Heat Exchangers

1968 ◽  
Vol 90 (1) ◽  
pp. 125-129 ◽  
Author(s):  
H. A. Rundell ◽  
E. G. Ward ◽  
J. E. Cox
Keyword(s):  
Heat Transfer ◽  
Forced Convection ◽  
Empirical Relation ◽  
Outer Sphere ◽  
Coolant Fluid ◽  
Transfer Rates ◽  
Concentric Spheres ◽  
Temperature Heat ◽  
Inner Sphere ◽  
Visualization Techniques

The first experimental investigation of forced convection of a fluid through the annulus formed by two concentric spheres was performed. The fluid enters and leaves the annulus through diametrically opposed openings in the outer sphere. The inner sphere serves as a constant-temperature heat source. Four sphere-size combinations were tested. The flow patterns in the annulus were established by flow visualization techniques; characteristic photographic results are presented. Energy considerations include heat transfer rates, inner-sphere surface temperatures, and temperature traverses of the coolant fluid. An empirical relation correlates the heat transfer data.

scholarly journals Ekman-Hartmann Boundary Layers and the Length of Day Variations

1993 ◽  
Vol 157 ◽  
pp. 453-455
Author(s):  
N. Kleeorin ◽  
I. Rogachevskii ◽  
A. Ruzmaikin
Keyword(s):  
Angular Velocity ◽  
Cylindrical Surface ◽  
Outer Sphere ◽  
Solid Core ◽  
Length Of Day ◽  
Slip Boundary ◽  
Velocity Gradients ◽  
Earth Core ◽  
Liquid Part ◽  
Angular Velocities

In the first issue of Journal of Fluid Mechanics Ian Proudman published a paper on the dynamical properties of a fluid between two concentric rotating spheres (Proudman, 1956). The angular velocities of the spheres were assumed only slightly different and the Reynolds number of the flow was large. It was found under non-slip boundary conditions that the cylindrical surface that touches the inner sphere and parallel to the axis of rotation is a singular surface in which velocity gradients are very large. Outside the cylinder the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder the fluid rotates with an angular velocity intermediate to the angular velocities of the spheres and there is also a meridional circulation. Later Stewartson (1966) presented a detailed investigation of structure of the shear layer near the cylindrical surface. One of the present authors (Ruzmaikin, 1989) pointed out a possible geophysical importance of these solutions. The liquid part of the Earth core occupying a shell between the inner solid core and the rock mantle can be considered as the fluid between two rotating spheres.

scholarly journals Stewartson-layer instability in a wide-gap spherical Couette experiment: Rossby number dependence

2019 ◽  
Vol 878 ◽  
pp. 522-543
Author(s):  
Michael Hoff ◽  
Uwe Harlander
Keyword(s):  
Spherical Shell ◽  
Experimental Studies ◽  
Outer Sphere ◽  
Outer Radius ◽  
Rossby Number ◽  
Mean Flow ◽  
Concentric Spheres ◽  
Angular Velocities ◽  
The Difference ◽  
Spherical Couette

Instabilities of a viscous fluid between two fast but differentially rotating concentric spheres, the so-called spherical Couette flow, with a fixed radius ratio of $\unicode[STIX]{x1D702}=r_{i}/r_{o}=1/3$ are studied, where $r_{i}$ is the inner and $r_{o}$ the outer radius of the spherical shell. Of particular interest is the difference between cases where the Rossby number $Ro=(\unicode[STIX]{x1D6FA}_{i}-\unicode[STIX]{x1D6FA}_{o})/\unicode[STIX]{x1D6FA}_{o}>0$ and cases with $Ro<0$, where $\unicode[STIX]{x1D6FA}_{i}$ and $\unicode[STIX]{x1D6FA}_{o}$ are the inner- and outer-sphere angular velocities. The basic state in both situations is an axisymmetric shear flow with a Stewartson layer situated on the tangent cylinder. The tangent cylinder is given by a cylinder that touches the equator of the inner sphere with an axis parallel to the axis of rotation. The experimental results presented fully confirm earlier numerical results obtained by Hollerbach (J. Fluid Mech., vol. 492, 2003, pp. 289–302) showing that for $Ro>0$ a progression to higher azimuthal wavenumbers $m$ can be seen as the rotation rate $\unicode[STIX]{x1D6FA}_{0}$ increases, but $Ro<0$ gives $m=1$ over a large range of rotation rates. It is further found that in the former case the modes have spiral structures radiating away from Stewartson layer towards the outer shell whereas for $Ro<0$ the modes are trapped in the vicinity of the Stewartson layer. Further, the mean flow excited by inertial mode self-interaction and its correlation with the mode’s amplitudes are investigated. The scaling of the critical $Ro$ with Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{o}\,d^{2})$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $d$ the gap width, is well within the bounds that have been established in a number of experimental studies using cylindrical geometries and numerical studies using spherical cavities. However, the present work is the first that experimentally examines Stewartson-layer instabilities as a function of the sign of $Ro$ for the true spherical-shell geometry.

Convection in a spherical capacitor

2002 ◽  
Vol 450 ◽  
pp. 297-316 ◽  
Author(s):  
KAMEL AMARA ◽  
JOHN HEGSETH
Keyword(s):  
Heat Transfer ◽  
Constant Temperature ◽  
Large Scale ◽  
Experimental System ◽  
Outer Sphere ◽  
Parameter Range ◽  
Polarization Force ◽  
Voltage Difference ◽  
Concentric Spheres ◽  
Inner Sphere

Real-time holographic interferometry and shadowgraph visualization are used to study convection in the fluid between two concentric spheres when two distinct buoyancy forces are applied to the fluid. The heated inner sphere has a constant temperature that is greater than the constant temperature of the outer sphere by ΔT. In addition to the usual gravitational buoyancy from temperature induced density differences, another radial buoyancy is imposed by applying an a.c. voltage difference, ΔV between the inner and outer spheres. The resulting electric field gradient in this spherical capacitor produces a central polarization force. The temperature dependence of the dielectric constant results in the second buoyancy force that is especially large near the inner sphere. The normal buoyancy is always present and, within the parameter range explored in our experiment, always results in a large-scale cell that is axisymmetric about the vertical. We have found that this flow becomes unstable to toroidal or spiral rolls that form near the inner sphere and travel vertically upward when ΔT and ΔV are suffciently high. These rolls start near the centre sphere's equator and travel upward toward its top. The onset of this instability depends on both the temperature difference at onset ΔTc and the voltage difference at onset ΔVc and these two quantities appear to be related, within the parameter range accessible to our experimental system, by a power law ΔVc ∝ ΔT1/3c. Measurements of the heat transfer show that these travelling rolls increase the heat transfer at onset. Far above onset, the heat transfer may actually decrease with increasing ΔT. The travelling roll's frequency increases with increasing ΔT near onset and with increasing ΔV far above onset. These results have been interpreted in terms of a flow structure that includes a thermal boundary-layer-like behaviour. This layer has a radial width that increases from the bottom pole to an unstable ‘latitude’ near the equator where the rolls appear.

Heat Transfer by Natural Convection between Vertically Eccentric Spheres

1973 ◽  
Vol 95 (1) ◽  
pp. 47-52 ◽  
Author(s):  
N. Weber ◽  
R. E. Powe ◽  
E. H. Bishop ◽  
J. A. Scanlan
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Convective Motion ◽  
Outer Sphere ◽  
Correlation Equation ◽  
Mapping Technique ◽  
Transfer Rates ◽  
Concentric Spheres ◽  
Wide Range ◽  
Inner Sphere

Natural convection to a cooled sphere from an enclosed, vertically eccentric, heated sphere is described in this paper. Water and two silicone oils were utilized in conjunction with four different combinations of sphere sizes and six eccentricities for each of these combinations. Both heat-transfer rates and temperature profiles are presented. The effect of a negative eccentricity (inner sphere below center of outer sphere) on the temperature distribution was an enhancement of the convective motion, while a positive eccentricity tended to stabilize the flow field and promote conduction rather than convection. As for concentric spheres, a multicellular flow pattern was postulated to explain the thermal field observed for the largest inner sphere utilized. In all cases the heat-transfer rates were increased by moving the inner sphere to an eccentric position, and the utilization of a conformal-mapping technique to transform the eccentric spheres to concentric spheres enabled the application of existing empirical correlations for concentric spheres to the eccentric-sphere data. It is significant to note that this technique yields a single correlation equation, in terms of only keff/k and a modified Rayleigh number, which is valid for an extremely wide range of diameter ratios, eccentricities, Rayleigh numbers, and Prandtl numbers.

The almost-rigid rotation of viscous fluid between concentric spheres

1956 ◽  
Vol 1 (5) ◽  
pp. 505-516 ◽  
Author(s):  
Ian Proudman
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Distribution ◽  
Outer Sphere ◽  
Reynolds Numbers ◽  
Rotating Systems ◽  
Velocity Gradients ◽  
Concentric Spheres ◽  
Viscous Stresses

Two concentric spheres are supposed to rotate about the same axis with almost the same angular velocity, so that the viscous stresses over the surfaces of the spheres induce a flow which may be represented by a small perturbation superimposed upon a rigid body rotation of the fluid as a whole. The governing equations are therefore linearized in the magnitude of the perturbation, and it appears that the validity of this linearization is independent of the Reynolds number of the primary rotation. Attention is then restricted to the case in which the Reynolds number is large, the principal object of the note being to exemplify some of the properties of rotating systems at large Reynolds numbers in terms of a particularly simple mathematical model.It is found that the cylindrical surface that touches the inner sphere (the axis being the axis of rotation) is a singular surface in which velocity gradients are very large. Everywhere outside this cylinder, the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder, the velocity distribution in the central (inviscid) core of the motion is shown to be determined by the velocity distribution in the boundary layers over the spheres, and explicit solutions are obtained for all these velocity distributions. The mechanics of the cylindrical shear layer itself is also discussed, though no explicit solution is obtained in this case.

The Analysis of Proprioception Alteration during First Five Months after Anterior Cruciate Ligament Reconstruction

2018 ◽  
Vol 1 (84) ◽  
Author(s):  
Vilma Jurevičienė ◽  
Albertas Skurvydas ◽  
Juozas Belickas ◽  
Giedra Bušmanienė ◽  
Dovilė Kielė ◽  
...  
Keyword(s):  
Anterior Cruciate Ligament ◽  
Angular Velocity ◽  
Knee Joint ◽  
Cruciate Ligament ◽  
Position Sense ◽  
Joint Position Sense ◽  
Joint Position ◽  
Starting Position ◽  
Angular Velocities ◽  
Anterior Cruciate

Research  background  and  hypothesis.  Proprioception  is  important  in  the  prevention  of  injuries  as  reduced proprioception  is  one  of  the  factors  contributing  to  injury  in  the  knee  joint,  particularly  the  ACL.  Therefore, proprioception appears not only important for the prevention of ACL injuries, but also for regaining full function after ACL reconstruction.Research aim. The aim of this study was to understand how proprioception is recovered four and five months after anterior cruciate ligament (ACL) reconstruction.Research methods. The study included 15 male subjects (age – 33.7 ± 2.49 years) who had undergone unilateral ACL reconstruction with a semitendinosus/gracilis (STG) graft in Kaunas Clinical Hospital. For proprioceptive assessment, joint position sense (JPS) was measured on both legs using an isokinetic dynamometer (Biodex), at knee flexion of 60° and 70°, and at different knee angular velocities of 2°/s and 10°/s. The patients were assessed preoperatively and after 4 and 5 months, postoperatively.Research results. Our study has shown that the JPS’s (joint position sense) error scores  to a controlled active movement is significantly higher in injured ACL-deficient knee than in the contralateral knee (normal knee) before surgery and after four and five months of rehabilitation.  After 4 and 5 months of rehabilitation we found significantly lower values in injured knees compared to the preoperative data. Our study has shown that in injured knee active angle reproduction errors after 4 and 5 months of rehabilitation were higher compared with the ones of the uninjured knee. Proprioceptive ability on the both legs was  independent of all differences angles for target and starting position for movement. The knee joint position sense on both legs depends upon the rate of two different angular velocities and the mean active angle reproduction errors at the test of angular velocity slow speed was the highest compared with the fast angular velocity. Discussion and conclusions. In conclusion, our study shows that there was improvement in mean JPS 4 and 5 months after ACL reconstruction, but it did not return to normal indices.Keywords: knee joint, joint position sense, angular velocity, starting position for movement.

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