Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

An experimental study on effects of solid concentration and ζ-potential on pseudoplastic flow of suspensions

The pseudoplastic flow of suspensions, alumina or styrene-acrylamide copolymer particles in water or an aqueous solution of glycerin has been studied by the step-shear-rate method. The relation between the shear rate,D, and the shear stress,, in the step-shear-rate measurements, where the state of dispersion was considered to be constant, was expressed as = AD ^1/2 +CD. The effective solid volume fraction,ø _F, andA were dependent on the shear rate and expressed byø _F =aD ^b andA = D . Combining the above relations, the steady flow curve was expressed by = D ^{1/2 +} + ₀ (1 – a D ^b/0.74)^–1.85 D, where ₀ is the viscosity of the medium.With an increase in solid volume fraction and a decreases in the absolute value of the-potential, the flow behavior of the suspensions changed from Newtonian ( = = b = 0), slightly pseudoplastic ( = b = 0), pseudoplastic ( = 0) to a Bingham-like behavior.The change in viscosity of the medium had an effect on the change in the effective volume fraction.     

Experimental studies of suspended ferromagnetic fibers in a magnetic field

The analysis of the rotation of a ferromagnetic ellipsoid suspended in a Newtonian fluid and subjected to a uniform magnetic field is extended to include a long, slender cylindrical fiber which is magnetically saturated. Experimental observations of rotating nickel cylinders with aspect ratiosL/D ranging from 5 to 40 agree with the theoretical predictions that: (1) the proper magnetoviscous time constant for the motion is _MV = _s/µ ₀ M _s ² , (2) larger fiber aspect ratios result in considerably longer orientation times; and (3) the strength of the applied external field has only a slight effect on the overall fiber rotation, and has no effect on the maximum angular velocity achieved. Quantitative agreement of theory and experiments is obtained for fibers withL/D 20; for the shorter fibers, the theory tends to overpredict the fiber rotation rate by as much as 30%. D diameter of the cylinder - D ^P (r) position-dependent demagnetization tensor, implicitly defined in eq. (2.5) - D _xx,D _yy,D _zz volume-averaged demagnetizing factors for an ellipsoid equivalent to a uniformly magnetized cylinder, defined in eq. (2.6) - H _i ;H _i magnetic field inside a ferromagnetic body; magnitude ofH _i - H ₀;H ₀ magnetic field applied by external sources; magnitude ofH ₀ - k geometric parameter in the hydrodynamic resistance of a body rotating in a Newtonian fluid, eq. (2.2) - L length of the cylinder - L ^(h);L _z ^(h) hydrodynamic torque exerted on a rotating body; thez-component ofL ^(h) on the cylinder - L ^(m);L _z ^(m) magnetic torque exerted on a magnetic body in a magnetic field, eq. (2.4); thez-component ofL ^(m) on the cylinder - M the magnetization of a magnetic material - M _s the saturation magnitude ofM, approached by all ferromagnetic materials asH _i becomes large - r position vector of a point within a ferromagnetic body - V volume of a magnetic particle - x, y, z rectangular coordinate axes fixed in the cylinder according to figure 1 - angle of inclination of the axis of the cylinder with respect toH ₀ - shear rate - small parameter of slender body theory,=1/ln (2L/D) - _s constant viscosity of the suspending fluid - µ ₀ the magnetic permeability of free space,µ ₀=4 · 10^–7 H/m - _MV the magnetoviscous time constant, a characteristic time for a process involving a competition of viscous and magnetic stresses - ₁ the first normal-stress coefficient - ; _z angular velocity of a rotating body; angular velocity of a cylinder about thez-axis, _z =– d/dt     

Flow in porous media I: A theoretical derivation of Darcy's law

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to aclosure problem, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem ofabrupt changes in the structure of a porous medium is not considered.Roman Letters A interfacial area of the - interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the - interface contained within the averaging volume, m² - A ^* interfacial area of the - interface contained within a unit cell, m² - Ae area of entrances and exits for the -phase contained within a unit cell, m² - B second order tensor used to represent the velocity deviation (see Equation (3.30)) - b vector used to represent the pressure deviation (see Equation (3.31)), m^–1 - d distance between two points at which the pressure is measured, m - g gravity vector, m/s² - K Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the -phase (see Figure 2), m - characteristic length scale for the -phase (see Figure 2), m - n unit normal vector pointing from the -phase toward the -phase (n =–n ) - n _e unit normal vector for the entrances and exits of the -phase contained within a unit cell - p pressure in the -phase, N/m² - p intrinsic phase average pressure for the -phase, N/m² - p –p , spatial deviation of the pressure in the -phase, N/m² - r ₀ radius of the averaging volume and radius of a capillary tube, m - v velocity vector for the -phase, m/s - v phase average velocity vector for the -phase, m/s - v intrinsic phase average velocity vector for the -phase, m/s - v –v , spatial deviation of the velocity vector for the -phase, m/s - V averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ Greek Letters V/V, volume fraction of the -phase - mass density of the -phase, kg/m³ - viscosity of the -phase, Nt/m² - arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s - arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s^–1     

Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C ₁, C ₂ apparatus constants - D damping of the pendulum - e _x , e _y , e _z Cartesian basis - e _r , e , e _z orthonormal cylindrical basis - E geometrical constant - E _t , ⁰ E _t , _t relative strain tensor - f function of shear rate - F _t relative deformation tensor - G (t) memory function - G ^* complex shear modulus - G Re(G ^* ) - G Im(G ^* ) - h distance between plates - H ^* transfer function - , functional - i imaginary unit: i ²= – 1 - I moment of inertia - J _exc excitation current - J ₀ amplitude of J _exc - k ^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M _dr driving torque - M _liq torque exerted by the liquid - ⁰ M _liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R ₀, R _c Re(Z ^*, Z ₀ ^* , Z _c ^* ) - s time - t, t time - T temperature - T, ⁰ T, stress tensor - u velocity - U lock-in output - ₀ velocity - V _det detector output voltage - V _sig, V _cr signal and cross-talk part of V _det - x Cartesian coordinate - X , X ₀, X _c Im(Z ^*, Z ₀ ^* , Z _c ^* ) - y Cartesian coordinate - z Cartesian coordinate, axial position     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - a _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

The rotation of a suspended axisymmetric ellipsoid in a magnetic field

Under the influence of a uniform and parallel magnetic field, a ferromagnetic fiber suspended in a Newtonian fluid rotates to align with the field direction. This study examines the field-induced rotation process for an individual non-Brownian axisymmetric ellipsoid suspended in a stagnant Newtonian fluid. Theoretical predictions are derived by a perturbation analysis for the limiting case where the strength of the applied magnetic field far exceeds the saturation magnetization of the ellipsoid. Numerical calculations are performed for the more general problem of an ellipsoid with known isotropic, non-hysteretic magnetic properties, using nickel and a stainless steel as examples. The analysis encompasses materials with field-induced, nonlinear magnetic properties, distinguishing these results from the simpler cases where the particle magnetization is either independent of, or linearly dependent on, the strength of the applied external field. In this study, predictions indicate that when the ellipsoid is magnetically saturated, the particle rotation is governed by the magnetoviscous time constant, _MV = _s/₀ M _s ² . It is found that the rotation rate depends strongly on the aspect ratio,a/b, of the ellipsoid, but only weakly on the dimensionless magnetization,M _s/H ₀. A geometric parameter for an ellipsoid, defined in eq. (2.5) - a, b major, minor semi-axes of an axisymmetric ellipsoid - D demagnetization tensor for an ellipsoid - D ^M magnetometric demagnetization tensor, the volume-average ofD ^P (r) - D ^P (r) position dependent demagnetization tensor, implicitly defined in eq. (2.12) - D _xx,D _yy,D _zz demagnetization factors, the diagonal elements ofD. Values for ellipsoids are defined in eq. (2.15) - F ^(m) magnetic force exerted on a body in a magnetic field - H _i ;H _i magnetic field inside a ferromagnetic body; magnitude ofH _i - H ₀;H ₀ magnetic field applied by external sources; magnitude ofH ₀ - h _i ;h _ix,h _iy Cartesian components of dimensionless internal magnetic field,h _i =H _i /H ₀ - I moment of inertia tensor - k geometric parameter for hydrodynamic resistance of a body rotating in a Newtonian fluid given in eq. (2.3) - L ^(h);L _z ^(h) hydrodynamic torque exerted on a rotating body; thez-component of the hydrodynamic torque - L ^(m);L _z ^(m) magnetic torque exerted on a magnetic body in a magnetic field, eq. (2.10); thez-component of the magnetic torque - M;M the magnetization, or dipole moment density, of a magnetic material; the magnitude ofM - M _s the saturation value ofM, approached by all ferromagnetic materials asH _i becomes large (figure 3) - m _s the dimensionless saturation magnetization,M _s/H ₀ - r position vector of a point within a ferromagnetic body - s dummy integration variable in eq. (2.5) - t time - U magnetoquasistatic potential energy of a magnetic body in a magnetic field, given in eq. (2.8) - u curve-fitting variable in eq. (4.1);u = logH _i - V volume of a magnetic particle; for an axisymmetric ellipsoid,V = (4/3) ab ² - x, y, z rectangular coordinate axes fixed in the ellipsoid (figure 1) - angle of inclination of the major axis of the ellipsoid with respect toH ₀ - _s viscosity of the Newtonian suspending medium - µ ₀ the magnetic permeability of free space,µ ₀ =4 · 10^–7H/m - _MV the magnetoviscous time constant, a characteristic time for a process involving a competition of viscous and magnetic stresses - the magnetic susceptibility of a magnetic material, = M/H _i - ; _z angular velocity of a rotating body; angular velocity about thez-axis of an ellipsoid, _z=–d/dt     

The very slow flow of a Powell-Eyring fluid around a sphere

Summary The very slow flow of a Powell-Eyring type non-Newtonian fluid around a sphere is investigated by a variational technique. The result, a correction factor that is applied to the Stokes' equation, is given as a plot and as an equation which is empirically fit to the plot. Also, a comparison of the very slow flows of a simplified viscoelastic Oldroyd fluid and the Powell-Eyring fluid is made which indicates that in a certain restricted region of the very slow flows, both models give essentially the same results. The Oldroyd and Powell-Eyring model parameters are interrelated by forcing both models to fit the same tube flow viscosity data.Nomenclature B dimensionless quantity, v /R - C dimensionless second invariant - c ₁ constant determined by variational method - D dimensionless variational integral - D _2j , D _j+k position-independent variables used in specification of trial functions - E _2j , E _j+k position-independent variables used in specification of trial functions - f friction factor - f _corr friction factor correction - F _drag drag force on sphere - g, g ₀, g ₁ general trial function; first and second terms in the general trial function - G, H terms in the expression for C - j index - J variational integral - k index - K term in the expression for C - p, q integers - r integer, radial coordinate - R radius of sphere - Re Reynolds number - Re ₀ Reynolds number at point of zero shear rate - Re Reynolds number at infinite distance from sphere - Re _NN Reynolds number based on variable part of viscosity - u, v dimensionless position coordinates - V volume considered - v _i ith velocity component - v _r , v , v _z velocity components in the r, , and z-directions - v approach velocity of the fluid - x/ parameter in Powell-Eyring model - x _i i-position coordinate - parameter in Powell-Eyring model - rate of deformation - , _c , _N , ₀ coefficient of viscosity; cross viscosity; parameter in Powell-Eyring model; viscosity in limit of zero shear rate - spherical coordinate - , _ij rate of deformation tensor; ij-component of rate of deformation tensor - ₁, ₂ parameters in Oldroyd model - Newtonian viscosity - ₁, ₂ parameters in Oldroyd model - dimensionless radial coordinate, r/R - second invariant - fluid density - spherical coordinate - stream function     

Slit die viscometry at shear rates up to 5 × 106s−1: An analytical correction for small viscous heating errors

If the viscosity can be expressed in the form = (T)f(), the walls are at a constant temperatureT ₀, and the extra stress, velocity and temperature fields are fully developed, then the wall shear rate can be calculated by applying the Weissenberg-Rabinowitsch operator toF _c Q instead of to the flow rateQ, whereF _c is a correction factor which differs from 1 when the temperature field is non-uniform; the isothermal equation relating the wall shear stress and pressure gradient is still valid. For the case in whcih = ₀|| ⁿ /(1 +(T–T ₀)), wheren, ₀, and are independent of shear stress and temperatureT, an exact analytical expression forF _c in terms of the Nahme-Griffith numberNa andn is obtained. Use of this expression gives agreement with data obtained for degassed decalin ( = 2.5 mPa s) from a new miniature slit-die viscometer at shear rates up to 5 × 10⁶s^–1; here, the correction is only 7%,Na is 1.3, andGz, the Graetz number at the die exit, is 119. For a Cannon standard liquidS6 ( = 9 mPa s), agreement extends up to 5 × 10⁵s^–1; at 2×10⁶s^–1 (whereNa = 7.2 andGz = 231), the corrections are 11% (measured) and 36% (calculated).Notation x, y Cartesian coordinates - v _x ,v velocity inx-direction, dimensionless velocity - p _xx ,p _yy normal stress onx- andy-planes - N ₁ first normal stress difference - shear stress ony-planes acting inx-direction - _w value of shear stress at the wall - shear rate, shear rate at the wall - Q, Q flow rate (Eqs. (2.13), (2.15)) - T, T ₀ temperature, temperature at the wall - ø, dimensionless temperature (Eqs. (2.24), (2.25)) - h, w half of die height, width of die - R diameter of a tube - , ₀ viscosity, viscosity atT = T ₀ - viscosity-temperature coefficient - k thermal conductivity - c _p specific heat at constant pressure - n, m dimensionless parameters characterizing shear stress dependence of viscosity - Na Nahme Griffith number (Eq. (2.21)) - Gz Graetz number (Eq. (5.1)) - F _c viscous heating correction factor (Eq. (2.18)) - ( ) a function characterizing temperature dependence of viscosity (Eq. (2.8)) - J _k ( ) Bessel function of the first kind This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

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     

Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

Fragile networks and rheology of concentrated suspensions

Suspensions consisting of particles of colloidal dimensions have been reported to form connected structures. When attractive forces act between particles in suspension they may flocculate and, depending on particle concentration, shear history and other parameters, flocs may build-up in a three-dimensional network which spans the suspension sample. In this paper a floc network model is introduced to interpret the elastic behavior of flocculated suspensions at small deformations. Elastic percolation concepts are used to explain the variation of the elastic modulus with concentration. Data taken from the suspension rheology literature, and new results with suspensions of magnetic -Fe₂O₃ and non-magnetic -Fe₂O₃ particles in mineral oil are interpreted with the model proposed.Non-zero elastic modulus appeared at threshold particle concentrations of about 0.7 vol.% and 0.4 vol.% of the magnetic and non-magnetic suspensions, respectively. The difference is attributed to the denser flocs formed by magnetic suspensions. The volume fraction of particles in the flocs was estimated from the threshold particle concentration by transforming this concentration into a critical volume concentration of flocs, and identifying this critical concentration with the theoretical percolation threshold of three-dimensional networks of different coordination numbers. The results obtained indicate that the flocs are low-density structures, in agreement with cryo-scanning electron micrographs. Above the critical concentration the dynamic elastic modulus G was found to follow a scaling law of the type G ( _f - _f ^c ) ^f , where _f is the volume fraction of flocs in suspension, and _f ^c is its threshold value. For magnetic suspensions the exponent f was found to rise from a low value of about 1.0 to a value of 2.26 as particle concentration was increased. For the non-magnetic a similar change in f was observed; f changed from 0.95 to 3.6. Two other flocculated suspension systems taken from the literature showed a similar change in exponent. This suggests the possibility of a change in the mechanism of stress transport in the suspension as concentration increases, i.e., from a floc-floc bond-bending force mechanism to a rigidity percolation mechanism.     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate