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2 APPLICATIONS v Heat and mass transfer rates are enhanced by the oscillation of the surrounding fluid. Useful in combustion, drying and the passage of sound waves through particulate systems. v Particle-laden flows, Brownian motion, suspension rheometry, colloidal suspension, and particle motion in filters.

Rajai3 MODES OF HEAT TRANSFER v RADIATION v CONDUCTION v CONVECTION (forced & free)

Rajai4 GOVERNING EQUATIONS Conservation of momentum where

Rajai5 First law of thermodynamics Conservation of mass

Rajai6 VECTOR RELATIONS

Rajai7 Define:

Rajai8 then,

Rajai9 and,

Rajai10 a g SPHERICAL COORDINATES SCALE FACTORS r

Rajai11 VARIABLES

Rajai12 DIMENSIONLESS NUMBERS kinematic viscosity thermal diffusivity volumetric expansion coef. frequency of oscillation surface temperature far-field temperature

Rajai13 USING THE SPHERICAL COORDINATE SYSTEM, THE EQUATIONS REDUCE TO: (1) (2) (3)

Rajai14 BOUNDARY CONDITIONS

Rajai15 Equations are two dimensional, time-dependent, nonlinear, coupled, and of infinite domain. No explicit boundary conditions for vorticity on the surface. Difficulties with finite- differences such as, indeterminate forms, etc..

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Rajai17 THE METHOD OF SOLUTION

Rajai18 HOW ?!

Rajai19 INTEGRALS NEEDED !! and in general,

Rajai20 where After Mavromatis & Alassar:

Rajai21 where

Rajai22 Saalschutz’s Theorem

Rajai23 3-j SYMBOLS Represent the probability amplitude that three angular momentum j1, j2, and j3 with projections m1, m2, and m3 are coupled to yield zero angular momentum. They are related to Clebsch-Gordan coefficients (C) by:

Rajai24 where Clebsch-Gordan Coefficients

Rajai25 (1) (2) (3)

Rajai26 where,

Rajai27 MODES BOUNDARY CONDITIONS

Rajai28 NUMERICAL METHOD CRANK-NICOLSON F.D. SCHEME

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Rajai30 SPECIALIZED STEP-BY-STEP METHOD

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Rajai33 PHYSICAL PARAMETERS Nusselt Number

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Rajai36 Drag Coefficient

Rajai37 Pressure Coefficient

Rajai38 VALIDATION

Rajai39 SOME RESULTS

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Rajai51 FUTURE RESEARCH Heat transfer from a sphere in a spinning infinite fluid. Heat transfer from other geometries such as oblate and prolate spheroids.

Rajai52 SPHEROIDS

Rajai53 Stream Function Vorticity

Rajai54 Boundary Conditions Energy

Rajai55 A NOTE ON THE PROBLEM OF SPINNING STREAM

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