A power-law fluid flow past a porous sphere

Summary A model has been developed for the flow of a non-Newtonian fluid past a porous sphere. The drag force exerted on a porous sphere moving in a power-law fluid is obtained by an approximate solution of equations of motion in the creeping flow regime. It is predicted that the effect of the pseudoplastic anomaly on the drag force is more pronounced at large porosity parameters.

---

Zusammenfassung Es wird ein Modell für die Strömung einer nichtnewtonschen Flüssigkeit längs einer porösen Kugel entwickelt. Die auf die in einer Ostwald-DeWaele-Flüssigkeit bewegte Kugel ausgeübte Reibungskraft wird durch eine Näherungslösung der Bewegungsgleichungen für schleichende Strömung gewonnen. Man findet, daß der Einfluß der Abweichung vom newtonschen Verhalten um so ausgeprägter wird, je größer die Porosität ist.

---

A, B, C, D a, b, c, d coefficients in eqs. [10] and [18] - F _D drag force - K consistency index in power-law model - k ₁ ,k ₂ coefficients defined by eq. [18] - m porosity parameter - n flow index in power-law model - P pressure - P ^* dimensionless pressure defined by eq. [4] - P pressure difference - R radius of porous sphere - r radial distance from the center of the sphere - U velocity of uniform stream - u _i velocity component - u _i ^* dimensionless velocity component defined by eq. [4] - Y drag force correction factor defined by eq. [27] - _ij rate of deformation tensor - _ij ^* dimensionless rate of deformation tensor defined by eq. [4] - , spherical coordinates - dimensionless radial distance defined by eq. [4] - second invariant of rate of deformation tensor - ^* dimensionless second invariant of rate of deformation tensor defined by eq. [4] - _ij stress tensor - _ij ^* dimensionless stress tensor defined by eq. [4] - stream function - ^* dimensionless stream function defined by eq. [4] - i inside the surface of the sphere - o outside the surface of the sphere With 1 figure and 1 table     

The influence of polyvinylacetate additive in water on turbulent velocity field and drag reduction

The effect of polymer concentration on drag reduction was studied experimentally with diluted water solutions of polyvinylacetate in a 2.4 cm I. D. pipe. The instantaneous local velocities of the velocity fields were measured by a one-channel differential laser-Doppler anemometer DISA Mark II, with forward scattering. Concentrations of water-polyvinylacetate over the range from 10 to 2,000 ppm were used. The drag reduction coefficient is proportional to the concentration and hydrolysis degree of the saponificated polyvinylacetate (PVAC) employed. A mechanical degradation in the turbulent shear flow was not observed.List of Symbols a ₁ coefficient in Eq. (3) - a ₂ coefficient in Eq. (3) - D pipe diameter - k coefficient in modified Blasius equation for friction factor - K consistency parameter given by (1 b) - K _i coefficients in Eq. (5) - m coefficient in Eq. (3) - n flow index Eq. (1a), coefficient in Eq. (3) - n ⁺ dimensionless position parameter defined by Eq. (4) - N ⁺ position parameter defined by Eq. (7) - r radial distance from the pipe center - R pipe radius - Re Reynolds number - Re _g generalized Reynolds number, Eq. (9) - t temperature - u ⁺ dimensionless local velocity, /u ^* - u ^* dynamic friction velocity, w(/8) ^0,5 - U ⁺ dimensionless local mean velocity defined by Eq. (6) - time-averaged local velocity - _m time-averaged local velocity at the pipe center - w average velocity over the cross-section of the pipe - X concentration of polymer in water, w · ppm - y distance from the pipe wall - y ⁺ dimensionless distance from the pipe wall, y u ^* / or as in Eq. (8) - friction factor in drag reduction flow - ₀ friction factor of pure water - degree of drag reduction - viscosity - standard deviation A version of this paper was presented at the 9th National Symposium on the measurement of turbulence with laser Doppler and other anemometers, Bratislava, CSSR, 1986     

Transient temperatures from periodic surface heat flux for slabs,cylinders, and spheres

The transient temperatures resulting from a periodically varying surface heat flux boundary condition have numerous applications. In this work, explicit, analytic solutions are presented for the transient surface and medium temperatures due to periodically varying step changes in surface heat flux for geometries such as a slab, cylinder, and sphere. The nonlinear case allowing for the added effects of radiation from the surface into an external ambient are studied numerically.

---

Temperaturschwankungen aus periodischen Änderungen des Wärmeflusses an der Oberfläche von Platten, Zylindern und Kugeln

Zusammenfassung Temperaturschwankungen herrührend von periodischen Änderungen des Wärmeflusses an der Oberfläche und der Grenzschichtbedingungen haben zahlreiche Anwendungen. In dieser Arbeit wird eine explizite analytische Lösung für die transienten Temperaturen an der Oberfläche und in der Mitte von Platten, Zylindern und Kugeln angegeben, die durch periodische stufenweise Änderungen des Wärmeflusses an der Oberfläche entstehen. Der nichtlineare Fall mit zusätzlichem Einfluß der Wärmestrahlung in die Umgebung wurde numerisch studiert.

---

Nomenclature f ₀ reference heat flux - f() dimensionless applied surface heat flux=q(t)/f₀ - F _i dimensionless stepchange in surface heat flux for linear problem - J _i (z) Bessel function - k thermal conductivity - L half thickness of slab, half radius of cylinder and sphere - N conduction-to-radiation parameter= - P period of on-off surface heat flux - q (t) applied surface heat flux - t time - T(x, t) temperature - T _r reference temperature=(f ₀/)^1/4 - U(z) unit step function - x physical distance Greek symbols thermal diffusivity - _m eigenvalues - ₀ surface emissivity - dimensionless spacial distance=x/2L - (, ) dimensionless temperature=T/T _r - 0 ₀ dimensionless initial temperature - _i dimensionless times at which step changes in surface heat flux occur - dimensionless time=t/2 L² - Stefan-Boltzmann constant - fraction of periodP during which the surface heat flux is non-zero - (, ) dimensionless temperature     

Heat transfer between gas-liquid spray stream flowing perpendicularly to the row of the cylinders

Experimental investigation and analysis of heat transfer process between a gas-liquid spray flow and the row of smooth cylinders placed in the surface perpendicular to the flow has been performed. Among others, there was taken into account in the analysis the phenomenon of droplets bouncing and omitting the cylinder as well as the phenomenon of the evaporation process from the liquid film surface.In the experiments test cylinders were used, which were placed between two other cylinders standing in the row.From the experiments and the analysis the conclusion can be drawn that the heat transfer coefficients values for a row of the cylinders are higher than for a single cylinder placed in the gasliquid spray flow.

---

Wärmeübergang an eine senkrecht anf eine Zylinderreihe auftreffende Gas-Flüssigkeits-Sprüh-Strömung

Zusammenfassung Es wurden Messungen und theoretische Analysen des Wärmeübergangs zwischen einer Gas-FlüssigkeitsSprüh-Strömung und den glatten Oberflächen einer Zylinderreihe durchgeführt, die senkrecht zum Sprühstrahl angeordnet waren. Dabei wurde in der Analyse unter anderem das Phänomen betrachtet, daß die Tropfen die Zylinderwand treffen und verfehlen können und daß sich ein Verdampfungsprozeß aus dem flüssigen Film an der Zylinderoberfläche einstellt.Gemessen wurde an einem zwischen zwei Randzylindern befindlichen Zylinder.Die Experimente und die Analyse gestatten die Schlußfolgerung, daß der Wärmeübergangskoeffizient für eine Zylinderreihe höher ist als für einen einzelnen Zylinder in der Sprühströmung.

---

Nomenclature a distance between axes of cylinders, m - c _l specific heat capacity of liquid, J/kg K - c _g specific heat capacity of gas, J/kg K - D cylinder diameter, m - g _l mass velocity of liquid, kg/m²s - ¯k average volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - k() local volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - L specific latent heat of vaporisation, J/kg - m mass fraction of water in gas-liquid spray flow, dimensionless - M constant in Eq. (9) - p pressure, Pa - p _g statical pressure of gas, Pa - p _w pressure of gas on the cylinder surface, Pa - p external pressure on the liquid film surface, Pa - r cylindrical coordinate, m - R radius of cylinder, m - T temperature, K, °C - T _l, t_l liquid temperature in the gas-liquid spray, K, °C - T _w,t_w temperature of cylinder surface, K, °C - T temperature of gas-liquid film interface, K - U liquid film velocity, m/s - w gas velocity on cylinder surface, m/s - w _g gas velocity in free stream, m/s - W _l liquid vapour mass ratio in free stream, dimensionless - W liquid vapour mass ratio at the edge of a liquid film, dimensionless - x coordinate, m - y coordinate, m - z complex variable, dimensionless - average heat transfer coefficient, W/m²K - local heat transfer coefficient, W/m² K - average heat transfer coefficient between cylinder surface and gas, W/m² K - _g, local heat transfer coefficient between cylinder surface and gas, W/m² K - mass transfer coefficient, kg/m²s - liquid film thickness, m - _lg dynamic diffusion coefficient of liquid vapour in gas, kg/m s - pressure distribution function on a cylinder surface - function defined by Eq. (3) - _l liquid dynamic viscosity, kg/m s - _g gas dynamic viscosity, kg/m s - cylindrical coordinate, rad, deg - _l thermal conductivity of liquid, W/m K - _g thermal conductivity of gas, W/m K - mass transfer driving force, dimensionless - _l density of liquid, kg/m³ - _g density of gas, kg/m³ - _w shear stress on the cylinder surface, N/m² - _w shear stress exerted by gas at the liquid film surface, N/m² - air relative humidity, dimensionless - T -T _w - _w =T _wT_l Dimensionless parameters I= enhancement factor of heat transfer - m ^*=M _l/M_g molar mass of liquid to the molar mass of gas ratio - Nu _g= D/ _g gas Nusselt number - Pr _g=c _{g g}/_g gas Prandtl number - Pr _l=c_lll liquid Prandtl number - ¯r=(r–R)/ dimensionless coordinate - Re _g=w_gD _g/_g gas Reynolds number - Re _g,max=w_g,max D _g/_g gas Reynolds number calculated for the maximal gas velocity between the cylinders - Sc=m ^* _g/_l–g Schmidt number =/R dimensionless film thickness     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Flow of second-order fluids over an enclosed rotating disc

Summary This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters ₁ and ₂. Maximum values ₁ and ₂ of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=₁+₂)]. The velocities at ₁ and ₂ as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.Nomenclature , , z coordinates in a cylindrical polar system - z ₀ distance between rotor and stator (gap length) - =/z ₀, dimensionless radial distance - =z/z ₀, dimensionless axial distance - _s = _s/z₀, dimensionless disc radius - V =(u, v, w), velocity vector - dimensionless velocity components - uniform angular velocity of the rotor - , p fluid density and pressure - P =p/(₂ z ₀₂ ² , dimensionless pressure - ₁, ₂, ₃ kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively - ₁, _{2 2}/z ₀ ² , resp. ₃/z ₀ ² , dimensionless parameters representing the ratio of second-order and inertial effects - m = , mass rate of symmetrical radial outflow - l a number associated with induced circulatory flow - Rm =m/(z ₀₁), Reynolds number of radial outflow - R _l =l/(z ₀₁), Reynolds number of induced circulatory flow - Rz =z ₀ ² /₁, Reynolds number based on the gap - ₁, ₂ maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively - ₁(T), ₂(T) ₁ and ₂ for different T - U _1(T) ⁽⁺⁾ = dimensionless radial velocity, Rm>0 - V _1(T) ⁽⁺⁾ = , dimensionless transverse velocity, Rm>0 - U _2(T) ^(–) = , dimensionless radial velocity, Rm=–Rn<0, m=–n - V _2(T) ^(–) = , dimensionless transverse velocity, Rm<0 - C _m moment coefficient     

Time averaged temperature distribution in a cylindrical resistor with alternating current

The non-uniform heat generation in a cylindrical resistor crossed by an alternating electric current is considered. The time averaged and dimensionless temperature distribution in the resistor is analytically evaluated. Two dimensionless functions are reported in tables which allow one to determine the time averaged temperature field for arbitrarily chosen values of the physical properties and of the radius of the resistor, of the electric current frequency, of the Biot number and of either the power generated per unit length or the effective electric current.

---

Zeitliche Temperaturverteilung in einem zylinderförmigen Wechselstromwiderstand

Zusammenfassung Es wird ungleichförmige Wärmeerzeugung in einem mit Wechselstrom belasteten Widerstand unterstellt, woraus sich die darin einstellende, zeitlich gemittelte, dimensionslose Temperaturverteilung analytisch berechnen läßt. Zwei tabellierte dimensionslose Funktionen gestatten die Bestimmung dieser Temperaturverteilung für beliebige Werte der Stoff- und Feldparameter, des Widerstandhalbmessers, der elektrischen Frequenz, der Biot-Zahl, sowie der erzeugten Leistung pro Längeneinheit oder des effektiven Stroms.

---

Nomenclature A intregration constant introduced in Eq. (15) - Bi Biot numberhr ₀/ - c speed of light in empty space - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E amplitude of the electric field oscillations - electric permittivity - f function ofs and defined in Eq. (22) - function of defined in Eq. (45) - g function ofs and defined in Eq. (34) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=–1 - I electric current - I _eff effective electric currentI _eff=I/2 - Im imaginary part of a complex number - J current density - J _n Bessel function of first kind and ordern - thermal conductivity - magnetic permeability - ₀ magnetic permeability of free space - q _g power generated per unit volume - time average of the power generated per unit volume - Q time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - mass density - s dimensionless radial coordinates=r/r ₀ - s,s integration variables - electric conductivity - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - dimensionless temperature defined in Eq. (27) - x position vector - x arbitrary real variable - x integration variable - Y ₀ Bessel function of second kind and order 0 - z axial coordinate - z unit vector parallel to the axis of the cylinder - angular frequency - dimensionless parameter =r₀ - · modulus of a complex number - equal by definition     

Predictions on momentum,heat and mass transfer in turbulent channel flow with the aid of a boundary layer growth-breakdown model

This paper deals with theoretical aspects of momentum, heat and mass transfer in turbulent channel flow and in particular with phenomena occurring close to the wall. The analysis presented involves the use of a boundary-layer growth-breakdown model. Theoretical expressions have been derived predicting heat and mass transfer at smooth surfaces in the fully developed and entrance region and at surfaces provided with ideal two-dimensional roughness elements. The analysis is restricted to fluids having Prandtl and Schmidt numbers larger than one. Good agreement appears to exist between theoretical predictions and experimental observations.

---

Zusammenfassung Diese Arbeit behandelt die Theorie der Übertragungsvorgänge von Impuls, Wärme und Stoff in turbulenter Kanalströmung unter besonderer Berücksichtigung der Vorgänge in Wandnähe. Das verwendete Modell beruht auf dem Zusammenbruch der anwachsenden Grenzschicht. Für die ausgebildete Strömung und für den Einlaufbereich bei glatter Wand und bei Oberflächen mit idealen zweidimensionalen Rauhigkeitselementen werden theoretische Ausdrücke abgeleitet bei Beschränkung auf Prandtl- und Schmidt-Zahlen über Eins. Zwischen den theoretischen Voraussagen und den Versuchsergebnissen scheint gute Übereinstimmung zu herrschen.

---

Nomenclature a thermal diffusivity [m²/s] - c concentration [kg/m³] - c _p specific heat [J/kg °C] - D molecular diffusivity [m²/s] - G relative increase in friction factor due to surface roughening - d pipe diameter [m] - e height (depth) of roughness element [m] - e ^p+ dimensionless roughness height (depth) - F parameter denoting the ratio - f friction factor for smooth surface and isothermal conditions - f _h friction factor for heating conditions - f _r friction factor for artificially roughened surface - n _av average frequency of fluctuations at the wall [s^–1] - q heat flux [W/m₂] - q _w heat flux at the wall [W/m²] - q _wr heat flux at roughened wall [W/m²] - q _wx wall heat flux to growing laminar boundary layer at positionx [W/m²] - R _ma longitudinal correlation coefficient for mass transfer - R _mo longitudinal correlation coefficient for momentum transfer - T temperature [°C] - T _b bulk temperature of fluid [°C] - T ₀ fluid temperature at edge of viscous boundary layer (edge of viscous region) [°C] - T _w wall temperature [°C] - T _wx wall temperature at positionx for growing laminar boundary layer [°C] - t time [s] - t ₀ characteristic time period associated with boundary layer growth [s] - u local axial fluid velocity, at wall distancey, for turbulent flow also denoting the mean velocity at that distance [m/s] - u _b bulk fluid velocity [m/s] - u ₀ fluid velocity at edge of viscous boundary layer (edge of viscous region) [m/s] - u _0r fluid velocity at edge of viscous region for the case of an artificially roughened wall [m/s] - u axial fluid velocity fluctuation [m/s] - u ⁺ dimengionless fluid velocity,u/(_w/)^1/2 - u _i ⁺ instantaneous value ofu ⁺ - u _min ⁺ minimum value ofu _i ⁺ - u _r ⁺ root mean square value of dimensionless axial velocity - u ₀ ⁺ value ofu ⁺ at edge of viscous region - v fluid velocity normal to flow direction and normal to wall [m/s] - v fluctuation of the velocityv [m/s] - x coordinate in flow direction [m] - x axial distance interval [m] - x ⁺ dimensionless distance interval - x ₀ viscous boundary layer growth length [m] - x ₀ ⁺ dimensionless boundary growth length - x _r axial dixtance between roughness elements [m] - x _r ⁺ dimensionless distance between roughness elements - x _h value of viscous boundary growth length for heating conditions [m] - y distance from wall [m] - y ⁺ dimensionless wall distance - y _v thickness of viscous region [m] - y _v ⁺ dimensionless form ofy _v - z _u unheated (zero mass transfer) part of elementary viscous boundary layer in entrance region [m] - z _h heated (mass transfer) part of elementary viscous boundary layer [m] - z _v lateral extent of elementary viscous boundary layer [m] Greek symbols heat transfer coefficient defined with respect to bulk fluid temperature [W/m² °C] - ₀ viscous region heat transfer coefficient [W/m² °C] - _0h viscous boundary layer heat transfer coefficient averaged over lengthx ₀ for conditions of heating [W/m² °C] - _0hh viscous region heat transfer coefficient averaged over lengthx _h for conditions of heating [W/m² °C] - entrance region heat transfer coefficient at position [W/m² °C - _,t viscous boundary layer heat transfer coefficient at position and timet [W/m² °C] - mass transfer coefficient [m/s] - _av average value of mass transfer coefficient [m/s] - _x mass transfer coefficient for viscous boundary layer at positionx [m/s] - entrance region mass transfer coefficient at position [m/s] - thickness of laminar (viscous) boundary layer evaluated atu=1/2u ₀ [m] - _max maximum value of boundary layer thickness [m] - _i turbulent diffusivity for momentum transfer [m²/s] - _h turbulent diffusivity for heat transfer [m²/s] - _m turbulent diffusivity for mass transfer [m²/s] - turbulent intensity - thermal conductivity [W/m °C] - kinematic viscosity [m²/s] - ₀ value ofv at edge of viscous region [m²/s] - _w value ofv at the wall [m²/s] - density [kg/m³] - shear stress [N/m²] - _tx local value of wall shear stress associated with viscous boundary layer growth [N/m²] - ₀ value of wall shear stress averaged over lengthx ₀ [N/m²] - _0r value of ₀ for the case of an artificially roughened wall [N/m²] - _0h value of ₀ for heating conditions [N/m²] - _h value of wall shear stress for heating conditions, averaged over lengthx _h [N/m²] - _w wall shear stress for conditions of turbulent flow [N/m²] - _wh value of _w for heating conditions [N/m²] - dimensionless axial distancex/x ₀ in extrance region Dimensionless numbers Nu Nusselt number (d/) - Nu _x Entrance region Nusselt number at axial positionx - Nu _h Nusselt number for heating conditions - Nu _r Nusselt number for the case of artificially roughened surface - Pr Prandtl number (v/a) - Re Reynolds number (d u _b/v) - Re _b Boundary layer Reynolds number (1/2 u ₀/v) - Re _ber Critical value ofRe _b - Sh Sherwood number (d/D) - Sh _x entrance region Sherwood number at axial positionx - Sc Schmidt number (v/D)     

Viscous heating correction for thermally developing flows in slit die viscometry

In the thermally developing region, d _yy /dx| _y=h varies along the flow direction x, where _yy denotes the component of stress normal to the y-plane; y = ±h at the die walls. A finite element method for two-dimensional Newtonian flow in a parallel slit was used to obtain an equation relating d _yy /dx/ _y=h and the wall shear stress ₀ at the inlet; isothermal slit walls were used for the calculation and the inlet liquid temperature T₀ was assumed to be equal to the wall temperature. For a temperature-viscosity relation /₀ = [1+(T–T₀]^–1, a simple expression [(hd _yy /dx/ _y=h )/ _w0] = 1–[1-F _c(Na)] [M()+P(Pr) ·Q(Gz ^–1)] was found to hold over the practical range of parameters involved, where Na, Gz, and Pr denote the Nahme-Griffith number, Graetz number, and Prandtl number; is a dimensionless variable which depends on Na and Gz. An order-of-magnitude analysis for momentum and energy equations supports the validity of this expression. The function F _c(Na) was obtained from an analytical solution for thermally developed flow; F _c(Na) = 1 for isothermal flow. M(), P(Pr), and Q(Gz) were obtained by fitting numerical results with simple equations. The wall shear rate at the inlet can be calculated from the flow rate Q using the isothermal equation.Notation x,y Cartesian coordinates (Fig. 2) - , dimensionless spatial variables [Eq. (16)] - dimensionless variable, : = Gz(x)^–1 - dimensionless variable [Eq. (28)] - t,t ^* time, dimensionless time [Eq. (16)] - , velocity vector, dimensionless velocity vector - _x , velocity in x-direction, dimensionless velocity - _y , velocity in y-direction, dimensionless velocity - V average velocity in x-direction - _yy , ^* normal stress on y-planes, dimensionless normal stress - shear stress on y-planes acting in x-direction - _w , _w ^* value of shear stress stress at the wall, dimensionless wall shear stress - _w0, _w0 ^* wall shear stress at the inlet, dimensionless variable - , ^* rate-of-strain tensor, dimensionless tensor - wall shear rate, wall shear rate at the inlet - Q flow rate - T, T ₀, temperature, temperature at the wall and at the inlet, dimensionless temperature - h, w half the die height, width of the die - l,L the distance between the inlet and the slot region, total die length - T ₂, T ₃, T ₄ pressure transducers in the High Shear Rate Viscometer (HSRV) (Fig. 1) - P, P2, P3 pressure, liquid pressures applied to T ₂ and T ₃ - , ₀, ^* viscosity, viscosity at T = T ₀, dimensionless viscosity - viscosity-temperature coefficient [Eq. (8)] - k thermal conductivity - C _p specific heat at constant pressure - Re Reynolds number - Na Nahme-Griffith number - Gz Graetz number - Pr Prandtl number     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

---

Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

---

Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Rotationsviskosimetrie mit kugelförmigen Spindeln

Zusammenfassung Die Rheodynamik der stationären viskometrischen Drehströmung um eine rotierende Kugel wird mit Methoden der Variationsrechnung untersucht. Neben iterativen numerischen Lösungsmethoden, die zu exakten Resultaten führen, wird auch eine approximative Ein-Gradienten-Lösung konstruiert, die durch Quadraturen dargestellt wird. Ausgehend von dieser analytischen Approximation werden einfache Methoden zur Auswertung von Experimentaldaten vorgeschlagen, die mit Hilfe von Eintauch-Rotationsviskosimetern mit kugelförmigen Meßspindeln gewonnen wurden.

---

Summary The rotational viscometric flow around a rotating sphere has been studied by variational methods. The exact numerical, as well as an approximate analytical solutions are given. Employing the analytical approximation, a simple method of evaluating viscometric data from immersional (portable) viscometers with a rotating sphere is proposed.

---

A Achsenschnitt durch den Bereich der Strömung - B - b, c anpaßbare empirische Konstanten - C Kalibrierungsoperator - D Schergeschwindigkeit der viskosimetrischen Strömung - D _ij Komponenten des Deformationsgeschwindigkeitstensors - D _I, _I Stoffkonstanten der VF des Ellis-Modells - g metrischer Koeffizient - H() Funktional der Ein-Gradienten-Approximation, Gl. [27] - J[] energetisches Potential - J _a[] Ein-Gradienten-Approximation fürJ - K Konsistenzkoeffizient, Parameter der VF des Potenzmodells - m Parameter des Ellis-Modells - M Drehmoment - n Parameter des Potenzmodells - n, n Differentialindices der VF, Gl. [20c, d] - n*,n** Differentialindices der RC, Gl. [9], [13] - r, , z polare Zylinderkoordinaten - R Spindelhalbmesser - rheometrischer Operator - S Spindeloberfläche - U(D) energetische Funktion nachBird, Gl. [20e] - v _i physikalische Komponenten der Geschwindigkeit - Z() transformierte VF, Gl. [20f] - (n) durch Gl. [35] definierte Funktion - k Verhältnis der Radien von Spindel und Wand - ( durch Gl. [43] definierte Funktion - natürliche (Radial-)Koordinate - Schubspannung der viskosimetrischen Strömung - _ij Komponenten des Spannungstensors - _S() Spannungsprofil an der Spindeloberfläche - _M Maximalspannung an der Spindeloberfläche - mittlere Spannung an der Spindeloberfläche, Gln. [3], [22] - natürliche (Meridional-) Koordinate - Winkelgeschwindigkeit in der Flüssigkeit - Winkelgeschwindigkeit der Spindelrotation - ( rheometrische Charakteristik Mit 4 Abbildungen und 3 Tabellen     

Evaluating Herschel-Bulkley fluids with the back extrusion (annular pumping) technique

A mathematical model was developed to describe the behavior of Herschel-Bulkley fluids in a back extrusion (annular pumping) device. A technique was also developed to determine the rheological properties (yield stress, flow behavior index, and consistency coefficient) of these fluids. Mathematical terms were expressed in four dimensionless terms, and graphical aids and tables were prepared to facilitate the handling of the expressions.Nomenclature a radius of the plunger, m - dv/dr shear rate, s^–1 - F force applied to the plunger, N - F _b buoyancy force, N - F _cb force corrected for buoyancy, N - F _T recorded force just before the plunger is stopped, N - F _Te recorded force after the plunger is stopped, N - g acceleration due to gravity, m/s² - H(t) momentary height between plunger and container bottom, m - K a/R, dimensionless - L length of annular region, m - L(t) depth of plunger penetration, m - n flow behavior index, dimensionless - p static pressure, Pa - P _L pressure in excess of hydrostatic pressure at the plunger base, Pa - p ₀ pressure at entrance to annulus, Pa - P pressure drop per unit of length, Pa/m - Q total volumetric flow rate through the annulus, m³/s - r radial coordinate, measured from common axis of cylinder forming annulus, m - R radius of outer cylinder of annulus, m - s reciprocal of n, dimensionless - t time, s - T dimensionless shear stress, defined in Eq. (3) - T ₀ dimensionless yield stress, defined in Eq. (4) - T _w dimensionless shear stress at the plunger wall - _p velocity of plunger, m/s - velocity, m/s - mass density of fluid, kg/m³ - Newtonian viscosity, Pa s - P p ₀ –p _L , Pa - consistency coefficient, Pa sⁿ - value of where shear stress is zero - _–, ₊ limits of the plug flow region (Fig. 1) - r/R - shear stress, Pa - _y yield stress, Pa - _w shear stress at the plunger wall, Pa - dimensionless flow rate defined in Eq. (24) - dimensionless velocity defined by Eq. (5) - _–, ₊ dimensionless velocity outside the plug flow region - _max dimensionless maximum velocity in the plug flow region - _p dimensionless velocity at the plunger wall     

Bestimmung des Diffusionskoeffizienten von Wasserstoff in Wasser und wässerigen Polymerlösungen nach der CBS-Methode

Zusammenfassung Die Werte des Diffusionskoeffizienten von Wasserstoff in Wasser und wässerigen Polymerlösungen bei 20 und 30°C und ungefähr l bar Gesamtdruck werden gegeben. Die Bestimmung dieser Werte geschah nach der kürzlich veröffentlichten constant bubble-size-Methode (CBS-Methode).Der Einfluß der freien Konvektion bei der Bestimmung der Diffusionskoeffizienten von mäßig lösbaren Gasen in Flüssigkeiten ist qualitativ untersucht worden. Es wird gezeigt, daß freie Konvektion durch Erhöhung der Viskosität völlig zurückgedrängt wird. Dazu wird die Viskosität durch Zusatz eines Polymerisats erhöht.Weiterhin wurde auch der Zusammenhang zwischen Diffusionskoeffizient und zero-shear-Viskosität quantitativ untersucht. Es wurde die zero-shear-Viskosität dieser wässerigen Polymerlösungen bestimmt. Ferner ergab sich, daß der Zusammenhang zwischen dem Logarithmus des Diffusionskoeffizienten und dem Logarithmus der zero-shear-Viskosität direkt proportional war.Der Diffusionskoeffizient nimmt bei höherem Polymerzusatz leicht ab. Die experimentellen Werte wurden mit Ergebnissen aus dem Schrifttum verglichen.

---

Measurement of the diffusion coefficient of hydrogen in water and aqueous polymer solutions according to the CBS-method

Values of the diffusion coefficient of hydrogen in water and aqueous polymer solutions at 20 and 30°C and about 1 bar total pressure are given. The measurement of these values has been performed according to the recently published constant bubble size method (CBS-method).The influence of free convection on the determination of diffusion coefficients of slightly soluble gases in liquids has been investigated qualitatively. It is shown that by increase of viscosity, free convection is reduced. To this end, the viscosity is increased by addition of a polymer. Furthermore, the relation between diffusion coefficient and zero-shear viscosity has been investigated quantitatively. The zero-shear viscosity of the non-Newtonian polymer solutions has been determined. A directly proportional relation between the logarithm of the diffusion coefficient and the logarithm of the zero-shear viscosity has been found.Increasing values of the polymer concentration result in a small decrease of the diffusion coefficient. The experimental values are compared with other results from literature.

---

Formelzeichen a [m² s^–1] Temperaturleitfähigkeit der Flüssigkeit - A _c [m²] Oberfläche des gesperrten Kugelabschnitts - c _A [mol m^–3] Konzentration des GasesA in der Flüssigkeit - c _p [kg kg^–1] Polymerkonzentration in der Flüssigkeit - c _R [mol m^–3] Konzentration des Gases in der Flüssigkeit an der Oberfläche (r=R) - c _z [mol m^–3] Konzentration der Zusatzmenge - c [mol m^–3] Konzentration des Gases in der Flüssigkeit fürt=0 und zur fürt >0 - d [m] Gasblasendurchmesser - d _c [m] Durchmesser der Spitze des Kegelstumpfs - D _AB [m² s^–1] Diffusionskoeffizient des GasesA in der FlüssigkeitB - D _w [m² s^–1] fusionskoeffizient des Gases in der reinen Flüssigkeit - g [m s^–2] Fallbeschleunigung - L [m] Halbmesser der Innenzelle - m [s^–1] Neigung der Gerade in der Gleichung (10) - n [1] Exponent in Gleichung (12) - N _A ^* [mol] Menge des in der Flüssigkeit absorbierten GasesA - p _R [Pa] Teildruck des Gases zur=R - r [m] Kugelkoordinate - R [m] Halbmesser der Gasblase - R [Jmol^–1K^–1] Gaskonstante (R=8.314 J mol^–1 K^–1) - t [s] Zeit - T [K] Temperatur - T [K] Temperaturdifferenz - v _* ^a [m³] Volumen des in der Flüssigkeit absorbierten GasesA Griechische Formelzeichen [W K^–1 m^–2] Wärmeübergangskoeffizient - [K^–1] Wärmedehnungszahl der Flüssigkeit - [rad] Winkel - [Pa s] Viskosität - _w [Pa s] Viskosität der reinen Flüssigkeit - ₀ [Pa s] Viskosität der Polymerlösung für 0 - [Pa s] Viskosität der Polymerlösung für - [rad] Kugelkoordinate - [W K^–1 m^–1] Wärmeleitfähigkeit der Flüssigkeit - [m² s^–1] kinematische Viskosität der Flüssigkeit - _L [kg m^–3] Dichte der Flüssigkeit - [Pa m] Oberflächenspannung der Flüssigkeit - _D [m² s^–1] Standardabweichung vom Diffusionskoeffizienten - _n [1] Standardabweichung vonn - [Pa] Schubspannung Dimensionslose Kenngrößen Eö [1] Eötvössche Kenngröße (Eö=_L g R²/) - He [1] Henrysche Kenngröße (He=c _RRT/p_R) - Nu [1] Nusseltsche Kenngröße (Nu= L/) - Ra [l] Rayleighsche Kenngröße (Ra=L ³ g T/(a v))     

Helical flow of molten polymers in a cylindrical annulus

Summary The paper is concerned with an analytical investigation of helical flow of a non-Newtonian fluid through an annulus with a rotating inner cylinder. The shear dependence of viscosity is described by a power law and the temperature dependence by an exponential function.Velocity and temperature profiles, energy input and shear along the stream lines, pressure drop, and torque are presented for the range of input parameters encountered in polymer extrusion.The results of the study can be applied to a mixing element in a screw extruder and for a device to control extrudate temperature and output.Nomenclature a thermal diffusivity [m²/s] - b temperature coefficient [K^–1], see eq. [4] - c heat capacity [J/kg K] - h slot width [m] - I ₁,I ₂,I ₃ invariants of the rate of deformation tensor, see eq. [5] - k thermal conductivity [J/m s K] - l, L = 1/h length of the slot - l _T ,l _K thermal and kinematic entrance length - m power law exponent, see eq. [3] - M torque [m N] - p pressure [N/m²] - P dimensionless pressure gradient, see eq. [24] - P _R,P _RZ dimensionless components of the shear stress tensor, see eq. [25] and eq. [26] - r, R = r/r _wa radial coordinate - r _wa, r_wi outer and inner radius of annulus [m] - t time [s]; dwell time in the annulus - T temperature [K] - v , v_r, V_z velocity components [m/s] - v ₀ angular velocity at inner wall [m/s] - average velocity inz-direction [m/s] - V , V_R, V_Z dimensionless velocity components,v /v₀, v_r/v₀, v_z/v₀ - V _z velocity ratio, helical parameter - Y coordinate inr-direction, see eq. [20] - z, Z = z/h Pe axial coordinate - deformation - rate of deformation tensor [s^–1] - apparent viscosity [N s/m²], see eq. [3] - dimensionless temperature,b (T – T ₀) - azimuth coordinate - ratio of radii,r _wi/r_wa - density [kg/m³] - , kl shear stress tensor [N/m²] - fluidity [m²w/Nw s], see eq. [4] - Gf Griffith number, see eq. [12] - Pe Péclet number, see eq. [13] - Re Reynolds number, - 0 initial state, reference state - equilibrium state - e entrance - wi, wa at surface of inner or outer wall - r, R, z, Z, coordinates - i, j radial and axial position of nodal point in the grid - k, l tensor components Presented at Euromech 37, Napoli 6. 20–23. 1972.With 15 figuresDedicated to Prof. Dr.-Ing. G. Schenkel on his 60th birthday     

Injection moulding of thermoplastics: Freezing of variable-viscosity fluids.

The injection moulding of thermoplastics involves, during mould filling, flows of hot polymer melts into mould networks, the walls of which are so cold that frozen layers form on them. An analytical study of such flows is presented here for the case when the Graetz and Nahme numbers are large and the Pearson number is small. Thus the flows are developing and temperature differences due to heat generation by viscous dissipation are sufficiently large to cause significant variations in viscosity (but the difference between the entry temperature of the polymer to a specific part of the mould network and the melting temperature of the polymer is not). Br Brinkman number - Gz Graetz number - h half-height of channel or disc - h ^* half-height of polymer melt region in channel or disc - L length of channel or pipe - m viscosity shear-rate exponent - Na Nahme number - p pressure - P pressure drop - Pe Péclet number - Pn Pearson number - Q volumetric flowrate - r radial coordinate in pipe or disc - R radius of pipe - Re Reynolds number - R _i inner radius of disc - R _o outer radius of disc - R ^* radius of polymer melt region in pipe - T temperature - T _ad adiabatic temperature rise - T _e entry polymer melt temperature - T _m melting temperature of polymer - T _max maximum temperature - T ₀ reference temperature - T _w wall temperature - flow-average temperature rise - u _r radial velocity in pipe or disc - u _x axial velocity in channel - u _y transverse velocity in channel or disc - u _z axial velocity in pipe - w width of channel - x axial coordinate in channel or modified radial coordinate in disc - y transverse coordinate in channel or disc - z axial coordinate in pipe - thermal conductivity of molten polymer - thermal conductivity of frozen polymer - scaled dimensionless axial coordinate in channel or pipe or radial coordinate in disc - ₀ undetermined integration constant - heat capacity of molten polymer - viscosity temperature exponent - dimensionless transverse coordinate in channel or disc - ^* dimensionless half-height of polymer melt region in channel or disc - H ^* scaled dimensionless half-height of polymer melt region in channel or disc or radius of polymer melt region in pipe - dimensionless temperature - ^* dimensionless wall temperature - scaled dimensionless temperature - numerical constant - µ viscosity of molten polymer - µ ₀ consistency of molten polymer - dimensionless pressure gradient - scaled dimensionless pressure gradient - density of molten polymer - dimensionless radial coordinate in pipe or disc - _i dimensionless inner radius of disc - ^* dimensionless radius of polymer melt region in pipe - dimensionless streamfunction - scaled dimensionless streamfunction - dummy variable - streamfunction - similarity variable - similarity variable     

Creeping motion of spheres through shear-thinning elastic fluids described by the Carreau viscosity equation

Summary Previous work on the creeping flow of viscoelastic fluids past a sphere is reviewed. Theoretical analyses available in the literature were obtained for weakly elastic fluids and therefore they predict only a small influence of fluid elasticity on the drag. In this paper, an approximate theoretical analysis is given for the creeping flow past a rigid sphere in an unbounded medium. The analysis uses a variational principle to solve the equations of motion and continuity in conjunction with the Carreau constitutive equation. The theoretical results are presented in terms of a correction factor to the Newtonian drag coefficient. The correction factor is a function of the power law flow behaviour indexn, the ratio of limiting viscosities ( ₀ – )/₀ and a dimensionless time which reflects the elastic nature of the fluids. The results are presented in graphical form covering a realistic range of these dimensionless groups.In order to verify the theoretical predictions, the drag coefficient of a number of spheres was measured in a series of shear thinning elastic test fluids. The flow properties of the test fluids were independently measured with a Weissenberg Rheogoniometer. The power law index of the test fluids varied between 1.0 and 0.4. Particle Reynolds number based on ₀ was in the range of 410^–6 to 410^–2. The difference between theoretically predicted values of drag coefficient and the experimentally measured values is less than ±7.5%. In addition, it is found that the Carreau viscosity equation can be used to predict the elastic parameter of primary normal stress difference with moderate to good accuracy for all the polymer solutions used in this work.

---

Zusammenfassung Einleitend wird ein Überblick über die früheren Untersuchungen betreffend die schleichende Strömung um eine Kugel gegeben. Die in der Literatur vorliegenden theoretischen Analysen sind auf schwach viskoelastische Flüssigkeiten beschränkt und sagen deshalb nur einen geringen Einfluß der Elastizität auf den Widerstand voraus. In dieser Veröffentlichung wird dagegen eine genäherte theoretische Analyse für die schleichende Strömung um eine starre Kugel in einem unendlich ausgedehnten Medium gegeben, bei welcher zur Lösung der Bewegungsgleichungen und der Kontinuitätsgleichung in Verbindung mit den rheologischen Stoffgleichungen vonCarreau ein Variationsprinzip verwendet wird. Die theoretischen Ergebnisse werden mittels eines Korrekturfaktors zum newtonschen Widerstandskoeffizienten beschrieben. Dieser Korrekturfaktor ist eine Funktion des Potenz-Gesetz-Exponentenn, des Verhältnisses der Grenzviskositäten ( ₀ – )/₀ und einer dimensionslosen Zeit, welche das elastische Verhalten kennzeichnet. Die Ergebnisse werden in graphischer Form unter Zugrundelegung eines realistischen Wertebereichs dieser dimensionslosen Gruppen dargestellt.Um diese theoretischen Voraussagen zu verifizieren, wurde der Widerstandskoeffizient für eine Anzahl von Kugeln in einer Reihe von Scherentzähung aufweisenden elastischen Probeflüssigkeiten gemessen. Die Fließeigenschaften dieser Flüssigkeiten wurden zusätzlich mit dem Weissenberg-Rheogoniometer bestimmt. Der Potenz-Gesetz-Exponent variierte dabei zwischen 1,0 und 0,4. Die auf den Kugeldurchmesser und die Nullviskosität bezogenen Reynolds-Zahlen lagen zwischen 410^–6 und 410^–2. Der Unterschied zwischen theoretisch vorausgesagten und experimentell bestimmten Widerstandskoeffizienten war kleiner als ±7,5%. Außerdem wurde noch gefunden, daß die Viskositätsgleichung vonCarreau dazu verwendet werden kann, den elastischen Parameter erste Normalspannungs-Differenz für alle in dieser Untersuchung verwendeten Polymerlösungen mit mäßiger bis guter Genauigkeit vorauszusagen.

---

Notation C _d drag coefficient - d diameter of sphere - f external body forces in equation of motion [2] - F _d drag force - g acceleration due to gravity - J integral defined in eq. [3] - n a parameter in the Carreau viscosity eq. [6] - p isotropic pressure term in equation of motion [2] - r,, spherical coordinates - R radius of sphere - Re ₀, Re₁ Reynolds numbers defined in eq. [16] - t time - u ⁱ ,u ^j velocities in equation of motion [2] - u _r ,u r and components of velocity - V terminal velocity of sphere in unbounded medium - V volume, in eq. [3] - X correction factor to the drag force, eq. [14] - y,z dimensionless spherical coordinates, eq. [9] - ratio of two Reynolds numbers given by eq. [16] - shear rate - apparent viscosity - ₀, zero shear rate and infinite shear rate viscosities respectively - a parameter in the Carreau viscosity eq. [6] - the dimensionless time, defined in eq. [11] - second invariant of the rate of deformation tensor - a parameter in the stream function, eq. [8] - stream function - _p,_f densities of sphere and fluid respectively With 7 figures and 1 table     

Experimental study of mutual interference between two spheres placed on a plane boundary

This paper describes an experimental study of the mutual interference between two spheres placed on a plane boundary. The experiment was carried out in an N. P. L. type wind-tunnel having a working section of 500×500×2000 mm³ in size at a Reynolds number of 4.74×10⁴. The surface-pressure distributions of two spheres were measured for the various relative positions of two spheres and the drag, side-force, and lift coefficients were determined from surface-pressure distributions. The separation of the flow and the formation of vortices were observed by the method of visualization. The distributions of velocities, and turbulent intensities of the flow past two spheres were measured. The experimental results for two spheres were compared with those of a single sphere.List of symbols C _D drag coefficient - C _L lift coefficient - C _p surface-pressure coefficient of sphere=(P-P )(qU ² ) - C _s coefficients of side force - D diameter of sphere [mm] - P static pressure [Pa] - P static pressure in free stream [Pa] - Re Reynolds number= DU/v - S spacing between the centers of two adjoining spheres in plane view [mm] - U time-mean velocity in X-direction [m/s] - [m/s] free stream velocity [m/s] - u, v, w X, Y and Z-components of velocity fluctuation [m/s] - X, Y, Z coordinate axes with origin at the bottom center of test sphere, X, Y, Z axis being taken in the streamwise, lateral and vertical directions respectively [mm] (Fig. 1) - latitude angle [°] - longitude angle [°] - angle between the line connected with the centers of two spheres and wind direction [°] (Fig. 2) - kinematic viscosity of air [m²/s] - density of air [N/m³] This paper was presented at the 10th Symposium on Turbulence, University of Missouri-Rolla, Sept. 22–24, 1986     

Heat conduction in a plane wall subject to periodic temperature changes

Analytical solutions for the heat conduction in a plane wall with periodic temperature variations at the wall surface are presented. Series and asymptotic developments of these solutions are deduced. The results are important for the calculation of the heat transfer in rotary kilns or other rotaring units.

---

Die Wärmeleitung in einer ebenen Wand mit periodischen Temperaturänderungen

Zusammenfassung Es werden analytische Lösungen für die Wärmeleitung in einer ebenen Wand mit periodischen Temperaturänderungen an ihrer Oberfläche mitgeteilt. Reihen- und asymptotische Entwicklungen dieser Lösungen werden abgeleitet. Die Ergebnisse sind wichtig für die Berechnung des Wärmetransportes in Drehrohröfen oder ähnlichen Maschinen.

---

Nomenclature a ² =/ C m²/s thermal diffusivity, Eq. (1) - C J/kg K specific heat - F K initial temperature of the wall, Eq. (4) - F m² surface of the wall - G Green's function, Eq. (10) - G₁ Green's function, Eq. (12) - h m thickness of the wall - H Heaviside function, Eq. (5) - k constant, Eq. (25) - k _x constant, Eq. (25) - Q J total energy, Eq. (17) - Q ^u J total energy from temperatureU, Eq. (18) - Q ^v J total energy from temperatureV, Eq. (19) - s s^–1 Laplace variable - t s time - t ₁ s heating time, Eq. (5) - t ₂ s period, Eq. (5) - T K temperature of the wall - T _i K surface temperature of the wall - T ₁ K surface temperature of the wall during the heating time - T ₂ K surface temperature of the wall during the cooling time - U K temperature of the wall defined in problem 1 - V, K temperature of the wall defined in problem 2 - x m coordinate - ₀ W/m²K overall heat transfer coefficient, Eq. (31) - ₁₀ W/m² K overall heat transfer coefficient, Eq. (32) - ₂₀ W/m² K overall heat transfer coefficient, Eq. (33) - Dirac Delta function - s^–1/2 parameter, Eq. (6) - W/mK thermal conductivity - kg/m³ specific mass - dimensionless time, Eq. (34) - Riemann Zeta function surfix Laplace transformed variable     

The stability of the helical flow of pseudoplastic liquids in a narrow annular gap with a rotating inner cylinder

The stability of a laminar helical flow of pseudoplastic liquids in an annular gap with a rotating inner cylinder is investigated theoretically. The analysis is carried out under the assumption of a torroidal form of the secondary flow (torroidal Taylor vortices) for the narrow gap geometry. The power law model has been applied to describe the pseudoplasticity of liquids. The problem of the stability has been formulated with the aid of the method of small disturbances, and solved using the Galerkin method. In order to describe the stability limit the Reynolds and Taylor numbers defined with the aid of the mean viscosity value have been introduced. It has been found that pseudoplasticity has a considerably destabilizing influence on the Couette motion as well as on the helical flow in the initial range of the Reynolds number values (Re<30). A decrease of the flow index value,n, is accompanied by a decrease of the critical value of the Taylor number. This destabilizing effect of pseudoplasticity vanishes in the range of the larger values of the Reynolds number. In the rangeRe>30, the stability limit of the flow of pseudoplastic liquids can be described by a general dependence of the critical valueTa _c onRe, which is consistent with results obtained for the case of Newtonian fluids. a frequency number (Eq. (27)), 1/s - b wave number (Eq. (27)), 1/m - B = M/N parameter - d = R ₂ –R ₁ gap width, m - f(y, B, k) function of viscosity distribution (Eq. (7)) - f ₀ (x) function of viscosity distribution (narrow gap Eq. (35)) - F(x) = V(x)/V _m dimensionless distribution of axial flow velocity - G(x) = U(x) _i dimensionless distribution of angular flow velocity - K consistency coefficient, N sⁿ/m² - M = (P/L)R ₂ parameter of the stress field (Eq. (1)), N/m² - M ₀ torque per unit length, N - n flow index - N = M ₀/(2R ₂ ² ) parameter of the stress field (Eq. (1)), N/m² - p = 1/2n–1/2 parameter - pressure disturbance amplitude, N/m² - p pressure disturbance, N/m² - (P/L) pressure drop per unit length of the gap, N/m² - r radial coordinate, m - r _m location of the maximum value of the axial velocity, m - R ₁,R ₂ inner, outer radius of the annulus, m - Re = V _m 2d/ _m Reynolds number - S = (P/L · d/N) parameteer of the stress field (narrow gap) - t time, s - Ta = _i d ^3/2 R ₁ ^1/2 / _m Taylor number - U tangential velocity, m/s - U _i tangential velocity at the surface of the inner cylinder, m/s - V axial velocity, m/s - V _m mean axial velocity, m/s - V disturbance vector of velocity field, m/s - amplitude of theV _k -disturbance, m/s - X, Y, Z functions in Eqs. (36–38) - y = r/R ₂ dimensionless radial coordinate - x = (r—(R ₁+R ₂)/2)d radial coordinate (narrow gap) - L ₁ L ₄ linear operators in Eqs. (36–38) - = ad/V _m dimensionless frequency number - = b·d dimensionless wave number - component of the rate of strain tensor, 1/s - component of the rate of strain tensor corresponding to the disturbance, 1/s - = R ₁/R ₂ radius ratio - apparent viscosity, Ns/m² - ₀ apparent viscosity in the main flow, Ns/m² - µ disturbance of the apparent viscosity, Ns/m² - µ _m mean apparent viscosity, Ns/m² - density, kg/m³ - _ij component of the stress tensor, N/m² - angular velocity, rad/s - _i angular velocity of the inner cylinder, rad/s     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

---

Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

---

Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀