A B C D E F G H I J K L M N O P Q R S
SAFETY ASSESSMENT SAFFMAN LENGTH SALINE WATER RECLAMATION SALT SALT DILUTION METHOD FOR FILM FLOW RATE MEASUREMENT SALTATION SALTING OUT SAMPLING SAMPLING METHODS, FOR DROPSIZE MEASUREMENT SAND BLASTING SANDIA NATIONAL LABORATORY, SNL Satellite remote sensing SATELLITES SATURATED FLUID PROPERTIES SATURATED SURFACES SATURATED VOLUME SATURATION PRESSURE SATURATION TEMPERATURE SAUTER MEAN DIAMETER SCALE-UP OF PERFORATION PROCESS SCALES OF TURBULENCE SCALING SCATTERING SCATTERING AMPLITUDE SCATTERING EFFICIENCY SCATTERING INDICATRIX SCATTERING OF RADIATION Scattering problem for cylindrical particles SCHEIBEL EQUATION FOR DIFFUSION IN LIQUIDS SCHLIEREN INTERFEROMETRY SCHLIEREN TECHNIQUE SCHMIDT NUMBER SCHMIDT, ERNST (1892-1975) SCHUSTER-HAMAKER MODEL SCHUSTER-SCHWARZCHILD APPROXIMATION, FOR COMBINED RADIATION AND CONDUCTION SCRAPED SURFACE HEAT EXCHANGERS SCREEN SEPARATORS SCREENS SCREW ROTARY COMPRESSOR SCREWS, PLASTICATING SCROLL DISCHARGE CENTRIFUGE SCRUBBERS SEA WATER COMPOSITION SECOND LAW OF THERMODYNAMICS SECOND NORMAL STRESS DIFFERENCE COEFFICIENT Secondary Flows Secondary quantity SECONDARY RECOVERY PROCESSES SEDIMENTATION SEDIMENTING CENTRIFUGES SEEPAGE SEGMENTAL BAFFLES SEGREGATION SEIDER-TATE CORRELATION SELECTIVE FROTH FLOTATION SELF ORGANIZATION SELF-SIMILAR HARDENING BEHAVIOR SELF-SIMILARITY SEMI-CONDUCTOR THERMOMETERS SEMI-SLUG FLOWS Semi-transparent media containing bubbles SEMIANNULAR FLOW SEMICONDUCTOR DIODE LASERS SEMICONDUCTORS SEMITRANSPARENT MEDIA SENSIBLE HEAT STORAGE SEPARATED FLOW MODELS SEPARATED LIQUID FLOWS SEPARATION OF BOUNDARY LAYERS SEPARATION OF EMULSIONS SEPARATION OF FLUID MIXTURES SEPARATION OF GAS AND SOLIDS SEPARATION OF LIQUIDS SEPARATION OF LIQUIDS AND SOLIDS SEPARATION OF PHASES IN GAS-LIQUID FLOWS SEPARATION PROCESSES SEPARATION, LIQUID/LIQUID SEPARATION, PARTICLES/LIQUID SERIES EXPANSIONS SESSILE DROPS AND BUBBLES SETTLING SLURRIES SEVERE ACCIDENTS, IN NUCLEAR REACTORS, CONTAINMENT OF SHADOWGRAPH TECHNIQUE SHAPE MEMORY SHAPE OF VAPOR FORMATIONS IN EXPLOSIVE BOILING SHAPE SELECTIVE CATALYSIS SHEAR FLOW Shear Layer SHEAR MODULUS Shear Stress SHEAR STRESS MEASUREMENT SHEAR STRESS VELOCITY SHEAR THICKENING SHEAR THICKENING FLUIDS SHEAR THINNING FLUIDS SHEAR VISCOSITY SHEARING INTERFEROGRAM SHEATH CHARACTERISTICS SHEET SPLITTING, IN DROP FORMATION SHELL AND TUBE CONDENSERS SHELL AND TUBE HEAT EXCHANGERS SHELL BOILER SHELL PROGRESSIVE MODEL SHELL-SIDE REFRIGERATION CHILLERS SHELLS SHERWOOD NUMBER SHERWOOD, THOMAS KILGORE (1903-1976) SHOCK TUBES SHOCK WAVE PROPAGATION SHOCK WAVES SHOCK WAVES, CONICAL SHORT ROUGHNESS STRIP SHORT TIME LAPSE PHOTOGRAPHY SHORT-TUBE VERTICAL EVAPORATOR SHOT TOWERS SHRINKING CORE MODEL SI UNITS SIDERITES SIEVE, TRAY COLUMN Silica based nanoporous composite materials SILICA GEL SILICON SILICON CARBIDE SILICON SOLAR CELLS SILOS, GRANULAR FLOW FROM SILVER SILVER METHOD SIMILARITY CONDITIONS SIMILARITY, THEORY OF SIMILITUDE Simplest approximations of double spherical harmonics SIMPLEX ATOMIZER SIMPLIFIED BOILING WATER REACTOR, SBWR SIMULATING SUBSURFACE TEMPERATURE SINCLAIR-LA MER AEROSOL GENERATOR Single-phase medium SINGLET STATE SINGLET STATE LIFETIME Singularities SINGULARITIES, HYDRAULIC RESISTANCE IN SINTERING SINUOUS JETS SIPHON CENTRIFUGE SKIMMER PIPE AND KNIFE CENTRIFUGES SKIN EFFECT SKIN FRICTION SLAG FORMATION SLIGHTLY DEFORMED POROUS CIRCULAR CYLINDER SLIGHTLY INCLINED SURFACE-MOUNTED PRISMS Slip ratio SLIT FLOW METERS SLIT FLOWS SLOT-PERFORATED FLAT FINS SLOW MOTION PHOTOGRAPHY Slug flow SLUG FLOW, SOLID SUSPENSIONS SLUG FREQUENCY SLUG LENGTH SLURRIES SMALL ANCLE SCATTERING METHOD, FOR DROPSIZE MEASUREMENT SMELTING SMOKE, AS AN AIR POLLUTANT SMOKES SNELL REFRACTION LAW SNL SOAVE EQUATION SODA ASH SODIUM SODIUM CARBONATE SODIUM CHLORIDE SODIUM COOLED NUCLEAR REACTOR SODIUM HYDROXIDE SOFTENING OF WATER SOFTWARE ENGINEERING SOIL, THERMAL PROPERTIES SOL SOLAR AIR HEATERS SOLAR CELLS SOLAR COOKERS SOLAR DRYING SOLAR ENERGY SOLAR ENERGY THERMAL CONVERSION SOLAR PONDS SOLAR RADIATION SOLAR RADIATION SPECTRUM SOLAR REFRIGERATION SOLAR SELECTIVE SURFACES SOLAR SODIUM EVAPORATOR SOLAR STILLS SOLAR WATER HEATERS SOLENOIDAL FLOW SOLID FUELS SOLID HOLDUP SOLID PROPELLANT SOLID STATE LASERS SOLID-LIQUID-LIQUID FLOWS SOLIDIFICATION SOLIDIFICATION CONSTANT SOLIDOSITY SOLIDS CONCENTRATION SOLIDS IN LIQUIDS, BOILING HEAT TRANSFER SOLIDS SEPARATION SOLIDS, THERMAL CONDUCTIVITY OF SOLITARY WAVE SOLITON SOLUBILITY SOLUBILITY OF GASES IN LIQUIDS SOLUBILITY OF SOLIDS IN LIQUIDS SOLUTE SOLUTE DIFFUSION SOLUTE FUNCTIONALITY Solution algorithm SOLUTIONS Solutions for one-dimensional problems Solutions for One-Dimensional Radiative Transfer Problems SOLVENT SOLVENT EXTRACTION Some applications: electrical arcs and atmospheric re-entry Some applied problems of combined heat transfer Some methods for detailed numerical simulation of radiative transfer Some validity studies SONIC OSCILLATOR SONIC VELOCITY SONOCAPILLARY EFFECT SOOT SORET AND DUFOUR EFFECTS ON FREE CONVECTION SORET EFFECT SORPTION HEAT PUMPS SOUND ABSORPTION SOUND GENERATION SOUND PROPAGATION SOUR GASES SOUTTER-ION PUMP SPACE HEATING SPACERS SPACERS, EFFECT ON CHF SPARK-IGNITION ENGINES SPARSELY PACKED POROUS MEDIUM Spatial discretization schemes SPATIAL-TEMPORAL CORRELATION SPECIFIC HEAT CAPACITY SPECIFIC WORK, IN TURBINES SPECKLE METHOD SPECKLE PHOTOGRAPHY SPECTRA, EMISSION AND ABSORPTION SPECTRAL ANALYSIS SPECTRAL DENSITY FUNCTION SPECTRAL EMISSIVITY SPECTRAL EXTINCTION METHOD Spectral radiative properties of diesel fuel droplets Spectral radiative properties of disperse systems: theoretical modeling and experimental characterization Spectral radiative properties of gases and plasma: theoretical models and experimental data Spectral radiative properties of some important materials: experimental data and theoretical models SPECTROFLUORIMETRY Spectroscopic databases SPECTROSCOPY SPECULAR REFLECTION SPEED OF LIGHT SPEED OF SOUND SPENT FUEL SPHERE, DRAG COEFFICIENT FOR SPHERES, CONVECTIVE HEAT AND MASS TRANSFER SPHERES, DRAG AND LIFT SPHERES, SOLID, DRAG ON Spherical particles SPHERICITY SPIRAL CLASSIFIER SPIRAL HEAT EXCHANGERS SPIRAL TUBES, USE IN BENSON BOILERS SPIROPYRAN SPLATTERING, EFFECT ON JET IMPINGEMENT SPONTANEOUS CONDENSATION SPRAY CHARACTERISTICS SPRAY COLUMNS SPRAY CONDENSERS SPRAY COOLING SPRAY DRYER SPRAY DRYING SPRAY EQUATION SPRAY EVAPORATORS SPRAY FLOWS SPRAY FORMATION SPRAY NOZZLES SPRAY TOWERS SPRAYERS SPRAYING SPRAYS SPREADING OF LIQUIDS ON LIQUIDS SPUTTERING STABILITY STABILITY CRITERIA STABILITY OF EMULSIONS STABILITY OF FLOATING BODIES STACKS, POLLUTION FROM STAGGERED TUBE BANKS STAGNANT FILM MODEL STAGNATION POINT STAGNATION PRESSURE STAGNATION TEMPERATURE STANDARD CONDITIONS STANTON GAUGE STANTON NUMBER STANTON, SIR THOMAS EDWARD (1865-1931) STARK BROADENING STARK NUMBER STARS, FUSION REACTIONS IN Static Head STATIC INSTABILITIES IN TWO-PHASE SYSTEMS STATIC MIXERS STATIC REGENERATORS STATIONARY PHASE, SP, CHROMATOGRAPHY Statistical band models STATISTICAL MECHANICS STATISTICAL THEORY, OF TURBULENT FLOW STATISTICAL THERMODYNAMICS STEAM ENGINES STEAM GAS TURBINE UNITS STEAM GENERATORS, NUCLEAR STEAM JET EJECTORS STEAM JET REFRIGERATION STEAM TABLES STEAM TURBINE STEAM-WATER SEPARATION STEEL AND TUBE CONDENSERS STEELS STEFAN'S LAW STEFAN, JOSEF (1835-1893) STEFAN-BOLTZMANN CONSTANT STEFAN-BOLTZMANN LAW STEFAN-MAXWELL EQUATIONS STEPWISE HEAT RELEASE STEREOSCOPIC IMAGING Stewart number Stewart number STEWARTSON TYPE FLOW STIELTJES' INTEGRAL STIRRED TANK REACTOR STIRRED TANKS STIRRED VESSEL PHASE INVERSION STOCHASTIC DIFFERENTIAL EQUATIONS STOCHASTIC PROCESS STOICHIOMETRIC COMBUSTION STOKES EQUATION STOKES FLOW STOKES LENGTH STOKES PARADOX STOKES PROBLEM STOKES SHIFT STOKES STREAM FUNCTION STOKES' LAW FOR SOLID SPHERES AND SPHERICAL BUBBLES STOKES-EINSTEIN EQUATION STOKES-EINSTEIN EQUATION, FOR DIFFERENTIAL COEFFICIENTS IN LIQUIDS STOMATAL CONTROL OF WATER LOSS FROM PLANTS STOPPING DISTANCE STORE'S FORMULA STRAIN STRAIN GAUGES STRAIN RATE STRANGE ATTRACTORS STRATIFICATION, UNSTABLE AND STABLE Stratified Gas-Liquid Flow STRATIFIED WAVY FLOW STRATOSPHERE STREAM ANALYSIS METHOD STREAM AVAILABILITY Stream Function Streamline Streamline Flow STREAMLINED BODIES, FLOW OVER STREAMLINES STREAMLINES, VISUALIZATION STREAMTUBE STRESS Stress in Fluids STRESS IN SOLID MATERIALS STRESS TENSOR STRESS VECTOR STRESS, NORMAL STRESS, SHEAR STRETCHING SHEET STRETCHING SURFACE STRETCHING/STABILIZING EFFLUX FLUID FILMS STROUHAL NUMBER Structure of plasma spectra STRUCTURED SURFACE STUART NUMBER SUBCHANNEL ANALYSIS SUBCHANNEL MIXING SUBCOOLED TWISTED FLOW SUBCOOLING SUBCOOLING EFFECTS ON POOL BOILING SUBLAYER FENCE SUBLIMATION SUBMERGED COMBUSTION SUBMERGED COMBUSTION EVAPORATORS SUBMERGED JETS SUBROUTINES SUBSTITUTE NATURAL GAS (SNG) SUBSURFACE BARRIER SUBUNDAL FLOW SUCTION SUCTION EFFECTS SULFUR SULFUR DIOXIDE SULFUR HEXAFLUORIDE SULFUR POLLUTION SULFURIC ACID SUN, HEAT TRANSFER IN SUPER-PHENIX SUPERCAVITATION SUPERCOMPUTING SUPERCONDUCTING MAGNETS SUPERCONDUCTORS SUPERCRITICAL HEAT TRANSFER SUPERFICIAL VELOCITY SUPERHEATING SUPERSATURATION SUPERSONIC EXTERNAL FLOW SUPERSONIC FLOW SUPERSONIC FLOW, IN NOZZLES SUPERSONIC HETEROGENEOUS FLOW SUPERSONIC JET SUPERUNDAL FLOW SUPPRESSION OF NUCLEATE BOILING SURFACE ACTIVE SUBSTANCES SURFACE ALLOYING OF METALS Surface and interfacial tension SURFACE CONDENSERS SURFACE DIFFUSION SURFACE DIMPLES SURFACE EFFECTS ON BOILING SURFACE EFFICIENCY SURFACE ENERGY SURFACE EXTENSIONS SURFACE FLOW VISUALIZATION SURFACE ROUGHNESS SURFACE TENSION SURFACE TENSION DEVICES SURFACE TREATMENT SURFACE, CIRCULAR SURFACE, PERMEABLE SURFACTANT COLLECTORS SURFACTANTS SURGE TANKS SUSPENSION OF PARTICLES IN LIQUID SUTHERLAND COEFFICIENT SWEATING SWEETENING OF GASES SWIRL BURNERS SWIRL FLOW DEVICES SWIRLING FLOW SWIRLING TAPES, FOR INCREASING BURNOUT FLUX SYMMETRIC TENSOR SYMMETRY ANALYSIS OF SECOND-GRADE FLOW SYNCHROTON RADIATION SYNOPTIC SCALE CIRCULATION, OF ATMOSPHERE SYNTHETIC ZEOLITES System of units
T U V W X Y Z

STOKES' LAW FOR SOLID SPHERES AND SPHERICAL BUBBLES

Interlinking between Articles
Visual Navigation

Introduction

Stokes' Law is the name given to the formula describing the force F on a stationary sphere of radius a held in a fluid of viscosity η moving with steady velocity V. This is usually expressed in the form

(1)

By translation, this result also applied to a sphere moving with steady velocity V in an otherwise stagnant fluid. Equation (1) can more conveniently be expressed in terms of a drag coefficient and a Reynolds number defined as follows

(2)
(3)

In terms of these variables, Equation (1) takes the form

(4)

Equations (1) and (4) are however applicable for slow flows and should only be used for Re < 1. The reasons for this and the results for higher Reynolds numbers are discussed below.

For a spherical gas bubble the corresponding results are

(5)

and

(6)

These results can be derived as follows.

Derivation

It is more convenient to analyze the situation in which the sphere is held stationary in a fluid that moves with velocity V at great distances from the sphere. Since Stokes Law is restricted to slow steady flow we can begin with the Navier Stokes equation and omit both the time-dependent and inertial terms giving

(7)

The situation is axially symmetrical and therefore there are two components of Eq. (7). Eliminating the pressure between these two components and expressing the resulting equation in terms of the Stokes stream function ψ, gives

(8)

where E2 is the operator,

(9)

For a solid sphere the boundary conditions are that the radial and tangential velocities on the surface are zero by the no-slip condition, i.e.,

(10)
(11)

and that the velocity tends to V at great distances, i.e.,

(12)

The form of these boundary conditions suggests a solution of the form ψ = f(r) sin2 θ and the only possible result is

(13)

Putting in boundary conditions(10)-(12) gives A = 0, B = V/2, C = −3Va/4 and D = Va3/4. Thus,

(14)

Hence we can obtain the velocity components,

(15)
(16)

For a spherical gas bubble, boundary conditions (10) and (12) still apply, but boundary condition (11) has to be replaced by the condition that there is no shear stress on the surface. Hence the shear strain rate is zero where is defined by

(17)

Under these circumstances A = 0, B = V/2, C = - Va/2 and D = 0, giving

(18)
(19)
(20)

The shear stress on the surface can be obtained from

(21)

For a solid sphere,

(22)

and for a bubble, τ is obviously zero.

The skin friction drag Fs is given by

(23)

from which we find that

(24)

for a solid sphere and zero for a bubble.

The pressure distribution can be found by substituting the expression for the velocity components into the Navier Stokes equation, giving

(25)

where p0 is an arbitrary constant.

For a solid sphere,

(26)

For a buble,

(27)

The normal stress on the surface is given by

(28)

where is the normal strain rate ∂vr/∂r.

For a solid sphere,

(29)

and for a bubble,

(30)

The form drag FF is given by

(31)

giving

(32)

for a solid sphere, and

(33)

for a bubble.

These results can be tabulated thus,

Form drag Skin friction drag Total drag
Solid sphere 2πηVa 4πηVa 6πηVa
Bubble 4πηVa 0 4πηVa

giving the results presented above.

Discussion

It must be emphasized that these results are applicable only for low Reynolds numbers. This is because the inertial terms (v. )v have been omitted from the analysis. An extension to the theory, known as Ossen's approximation, can be obtained by replacing these inertial terms by (V. )v, which gives rise to the result

(34)

In fact, this provides an over-correction and the empirical result,

(35)

gives a good correlation of the experimental results up to a Reynolds numbers of about 1000. The reader is referred to Clift et al. (1972) for alternative correlations and extensions to even higher Reynolds numbers.

For bubbles in nonpolar liquids, Equation (6) may be used up to a Reynolds number of about 1.5. However, polar liquids, such as water, are prone to contamination by surface active agents, which collect on the surface of the bubble, effectively immobilizing the surface. Such a surface can support a shear stress and bubbles in polar liquids behave as solid spheres. Indeed circumstances can arise in which bubbles obey the result for solid spheres over a very much larger range of Reynolds numbers than solid spheres themselves. Details of the behavior of bubbles are given by both Clift et al. (1972) and Wallis (1974). Wallis' correlation is probably the most reliable and convenient currently available.

These results can be used for accelerating spheres and bubbles without much loss of accuracy, but care must be exercised if the accelerations are large as then the Basset history term must be included, see Clift et al. (1972).

REFERENCES

Clift, R., Grace, J. R., and Weber, M. E. (1972) Bubbles, Drops and Particles,Academic Press, New York.

Wallis, G. B. (1974) Int. J. Multiphase Flow, 1, 491-511.

References

  1. Clift, R., Grace, J. R., and Weber, M. E. (1972) Bubbles, Drops and Particles,Academic Press, New York.
  2. Wallis, G. B. (1974) Int. J. Multiphase Flow, 1, 491-511.

This article belongs to the following areas:

S in A-Z Index
Number of views: 12131 Article added: 2 February 2011 Article last modified: 9 February 2011 © Copyright 2010-2014 Back to top