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).

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.

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.

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.

Abiotic Reactivity of De Novo Designed Artificial Copper Peptides (ArCuPs): The Case of C-H Activation

2022 ◽  
Author(s):  
Divyansh Prakash ◽  
Suchitra Mitra ◽  
Morgan Murphy ◽  
Saumen Chakraborty
Keyword(s):  
Steric Effect ◽  
De Novo ◽  
Outer Sphere ◽  
The Other ◽  
Easy Access ◽  
Oxygen Species ◽  
Redox Potentials ◽  
Spectroscopic Feature ◽  
Detailed Assessment ◽  
Active Variant

We report a series of de novo designed Artificial Cu Peptides (ArCuPs) that oxidize and peroxygenate C-H bonds of model abiotic substrates via electrochemically generated Cu-oxygen species using H2O2 as the terminal oxidant, akin to native Cu enzymes. Detailed assessment of kinetic parameters established the catalytic nature of the ArCuPs. Selective alteration of outer sphere steric at the d layers above and below the Cu site allows facilitated access of substrates, where a more pronounced effect on catalysis is observed when space is created at the d layer below the Cu site via Ile to Ala mutation producing a kcat of 6.2 s-1, TONmax of 14800 and catalytic proficiency (kcat/KM/kuncat) of 340 M-1 for the oxidation of benzyl alcohol. Independent spectroscopic studied revealed that the rate of formation of the Cu-oxygen species and the spectroscopic feature of the most active variant is distinct compared to the other ArCuPs. Systematic alteration of outer sphere hydrophobicity led to a correlated tuning of the T2 Cu site redox potentials by ~80 mV. The enhanced activity of the ArCuP variant is attributed to a combination of steric effect that allows easy access of substrates, the nature of Cu-oxygen species, and stability of this construct compared to others, where Ile to Ala mutation unexpectedly leads to a higher thermostability which is further augmented by Cu binding.

Aristotle's Unmoved Movers

Traditio ◽  
1946 ◽  
Vol 4 ◽  
pp. 1-30 ◽  
Author(s):  
Philip Merlan
Keyword(s):  
Middle Ages ◽  
The Other ◽  
World Picture ◽  
The Earth ◽  
Concentric Spheres ◽  
The One ◽  
The Universe ◽  
The Middle Ages

According to Aristotle all heavenly movement is ultimately due to the activity of forty-seven (or fifty-five) ‘unmoved movers'. This doctrine is highly remarkable in itself and has exercised an enormous historical influence. It forms part of a world-picture the outlines of which are as follows. The universe consists of concentric spheres, revolving in circles. The outermost of these bears the fixed stars. The other either bear planets or, insofar as they do not, contribute indirectly to the movements of the latter. Each sphere is moved by the one immediately surrounding it, but also possesses a movement of its own, due to its mover, an unmoved, incorporeal being. (It was these beings which the schoolmen designated as theintelligentiae separatae.) The seemingly irregular movements of the planets are thus viewed as resulting from the combination of regular circular revolutions. The earth does not move and occupies the centre of the universe. Such was Aristotle's astronomic system, essential parts of which were almost universally adopted by the Arabic, Jewish, and Christian philosophers of the Middle Ages.

Effects of Viewing Distance on the Responses of Vestibular Neurons to Combined Angular and Linear Vestibular Stimulation

1999 ◽  
Vol 81 (5) ◽  
pp. 2538-2557 ◽  
Author(s):  
Chiju Chen-Huang ◽  
Robert A. McCrea
Keyword(s):  
Eye Movements ◽  
Eye Movement ◽  
Vestibular Stimulation ◽  
The Other ◽  
Viewing Distance ◽  
Vestibuloocular Reflex ◽  
Horizontal Canal ◽  
Head Velocity ◽  
Vestibular Neurons ◽  
Axis Of Rotation

Effects of viewing distance on the responses of vestibular neurons to combined angular and linear vestibular stimulation. The firing behavior of 59 horizontal canal–related secondary vestibular neurons was studied in alert squirrel monkeys during the combined angular and linear vestibuloocular reflex (CVOR). The CVOR was evoked by positioning the animal’s head 20 cm in front of, or behind, the axis of rotation during whole body rotation (0.7, 1.9, and 4.0 Hz). The effect of viewing distance was studied by having the monkeys fixate small targets that were either near (10 cm) or far (1.3–1.7 m) from the eyes. Most units (50/59) were sensitive to eye movements and were monosynaptically activated after electrical stimulation of the vestibular nerve (51/56 tested). The responses of eye movement–related units were significantly affected by viewing distance. The viewing distance–related change in response gain of many eye-head-velocity and burst-position units was comparable with the change in eye movement gain. On the other hand, position-vestibular-pause units were approximately half as sensitive to changes in viewing distance as were eye movements. The sensitivity of units to the linear vestibuloocular reflex (LVOR) was estimated by subtraction of angular vestibuloocular reflex (AVOR)–related responses recorded with the head in the center of the axis of rotation from CVOR responses. During far target viewing, unit sensitivity to linear translation was small, but during near target viewing the firing rate of many units was strongly modulated. The LVOR responses and viewing distance–related LVOR responses of most units were nearly in phase with linear head velocity. The signals generated by secondary vestibular units during voluntary cancellation of the AVOR and CVOR were comparable. However, unit sensitivity to linear translation and angular rotation were not well correlated either during far or near target viewing. Unit LVOR responses were also not well correlated with their sensitivity to smooth pursuit eye movements or their sensitivity to viewing distance during the AVOR. On the other hand there was a significant correlation between static eye position sensitivity and sensitivity to viewing distance. We conclude that secondary horizontal canal–related vestibuloocular pathways are an important part of the premotor neural substrate that produces the LVOR. The otolith sensory signals that appear on these pathways have been spatially and temporally transformed to match the angular eye movement commands required to stabilize images at different distances. We suggest that this transformation may be performed by the circuits related to temporal integration of the LVOR.

Heat Transfer in Rotating Serpentine Passages With Selected Model Orientations for Smooth or Skewed Trip Walls

1994 ◽  
Vol 116 (4) ◽  
pp. 738-744 ◽  
Author(s):  
B. V. Johnson ◽  
J. H. Wagner ◽  
G. D. Steuber ◽  
F. C. Yeh
Keyword(s):  
Heat Transfer ◽  
Turbine Blades ◽  
The Other ◽  
First Passage ◽  
Forward Flow ◽  
Coriolis Forces ◽  
The Third ◽  
Axis Of Rotation ◽  
Trailing Edges ◽  
Serpentine Passages

Experiments were conducted to determine the effects of model orientation as well as buoyancy and Coriolis forces on heat transfer in turbine blade internal coolant passages. Turbine blades have internal coolant passage surfaces at the leading and trailing edges of the airfoil with surfaces at angles that are as large as ±50 to 60 deg to the axis of rotation. Most of the previously presented, multiple-passage, rotating heat transfer experiments have focused on radial passages aligned with the axis of rotation. The present work compares results from serpentine passages with orientations 0 and 45 deg to the axis of rotation, which simulate the coolant passages for the midchord and trailing edge regions of the rotating airfoil. The experiments were conducted with rotation in both directions to simulate serpentine coolant passages with the rearward flow of coolant or with the forward flow of coolant. The experiments were conducted for passages with smooth surfaces and with 45 deg trips adjacent to airfoil surfaces for the radial portion of the serpentine passages. At a typical flow condition, the heat transfer on the leading surfaces for flow outward in the first passage with smooth walls was twice as much for the model at 45 deg compared to the model at 0 deg. However, the differences for the other passages and with trips were less. In addition, the effects of buoyancy and Coriolis forces on heat transfer in the rotating passage were decreased with the model at 45 deg, compared to the results at 0 deg. The heat transfer in the turn regions and immediately downstream of the turns in the second passage with flow inward and in the third passage with flow outward was also a function of model orientation with differences as large as 40 to 50 percent occurring between the model orientations with forward flow and rearward flow of coolant.

Frictional Moment of Flow Between Two Concentric Spheres, One of Which Rotates

1978 ◽  
Vol 100 (1) ◽  
pp. 97-106 ◽  
Author(s):  
K. Nakabayashi
Keyword(s):  
Reynolds Number ◽  
Empirical Formula ◽  
Outer Sphere ◽  
Basic Flow ◽  
Confined Space ◽  
Clearance Ratio ◽  
Spherical Annulus ◽  
Concentric Spheres ◽  
Annulus Flow ◽  
Rotating Bodies

Frictional moment is investigated experimentally for the flow between a rotating inner sphere and a stationary outer one, and for the flow between a rotating outer sphere and a stationary inner one, respectively. The existence of four basic flow regimes for both is verified, and their extent is mapped over a range of Reynolds number-clearance ratio combinations. An empirical formula for the coefficient of frictional moment is obtained for every flow regime. The coefficient of frictional moment obtained for spherical annulus flow is then correlated with that of the flow around a disk rotating in a confined space, and the effect of the surface curvature of rotating bodies is considered.

The transition to Taylor vortices in a closed rapidly rotating cylindrical annulus

1980 ◽  
Vol 372 (1749) ◽  
pp. 201-218 ◽  
Keyword(s):  
Flow Pattern ◽  
Shear Layers ◽  
Vertical Shear ◽  
Taylor Vortices ◽  
Cylindrical Annulus ◽  
Ekman Layers ◽  
Angular Velocities ◽  
The Difference

The effects of the end walls on the flow in a rapidly rotating cylindrical annulus are considered. When the difference in the angular velocities of the cylindrical walls is small the flow is described in terms of Ekman layers and vertical shear layers but for larger values the Ekman layers are non­linear and a flow pattern corresponding to Taylor vortices may be resonated.

Interference Between Two Tandem Cylinders of Different Diameters

2006 ◽  
Author(s):  
Md. Mahbub Alam ◽  
Y. Zhou
Keyword(s):  
Low Frequency ◽  
Shear Layers ◽  
Diameter Ratio ◽  
The Other ◽  
Flow Structures ◽  
Incident Flow ◽  
Tandem Cylinders ◽  
Cylinder Diameter ◽  
Different Diameters ◽  
Dominant Frequencies

This paper presents a detailed investigation of Strouhal frequencies, forces and flow structures resulting from the interference between two tandem cylinders of different diameters. The upstream cylinder diameter (d) was varied from 25 mm to 6 mm, while the downstream cylinder diameter (D) of 25 mm was unchanged, the corresponding diameter ratio, d/D, being 1.0 ∼ 0.24. The spacing between the cylinders was 5.5d. At this spacing, the shear layers separated from the upstream cylinder roll up alternately, forming a vortex street in the gap between and behind the cylinders. Two dominant frequencies are detected behind the downstream cylinder at d/D = 1.0 ∼ 0.4. One is the same as detected between the cylinders, and the other is of relatively low frequency and is probably generated by the downstream cylinder. Whilst the former remains unchanged, the latter increases with decreasing d/D, due to an increase in the incident flow velocity of the downstream cylinder. The time-averaged drag on the downstream cylinder also increases with decreasing d/D, though the fluctuating forces drop because vortices impinging upon the downstream cylinder are impaired by decreasing d/D.

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