Kinetic theory and rheology of dumbbell suspensions with Brownian motion

  • R. B. Bird
  • H. R. WarnerJr.
  • D. C. Evans
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 8)


Shear Rate Kinetic Theory Stress Growth Hydrodynamic Interaction Normal Stress Difference 


Italic and Roman Symbols


dimensionless shear rate [Eq. (6.11)]


cos θ


cos φ




probability density in position-velocity space


force on i th bead through the connector


tension in the connector of the dumbbell


Hooke law constant for a spring


function defined in Eq. (25.4)


\(\sqrt { - 1}\)




normalization constant defined in Eq. (3.14)


equilibrium shear compliance [Eq. (13.3)]


Boltzmann constant


distance between bead centers in a rigid dumbbell


molecular weight


mass of a bead of the dumbbell


Avogadro's number


normal unit vector


number density of dumbbell


spherical harmonics [Eq. (5.5)]


thermodynamic pressure


any function of R


position vector for center of mass of dumbbell


position vector for i th bead (with coordinates x x , y i , z i )


orientation vector with components X, Y, Z (= r 2r 1)


separation between beads of dumbbell (= |R|)


gas constant in Eq. (6.11)


area of surface in Eqs. (4.11) and (4.12)


sin θ


sin φ




absolute temperature




local fluid velocity


perturbation velocity (§ 25)


mole fraction of j th species [Eq. (26.1)]

xi, yi, zi

components of r i

X, Y, Z

components of R

Greek Symbols


secondary normal stress function [Eq. (6.3)]


secondary normal stress function at zero shear rate


gamma function

\(\dot \gamma \)

rate of deformation tensor [after Eq. (3.1)]


ultimate shear recovery [Eq. (12.1)]


infinitesimal strain


unit tensor


Dirac delta function


unit vector in radial direction


unit vector in i th Cartesian direction


friction coefficient of a bead


viscosity function in Eq. (6.1) (non-Newtonian viscosity)


solvent viscosity


zero shear rate viscosity


intrinsic viscosity [Eq. (6.9)]


zero-shear-rate intrinsic viscosity


complex viscosity (η′ − iη″)

\(\bar \eta\)

elongational viscosity


angle down from z-axis in spherical coordinates


primary normal stress function [Eq. (6.2)]


primary normal stress function at zero shear rate


tensor specifying the homogeneous velocity field [Eq. (3.1)]


shear rate in shear flow

\(\bar \kappa\)

elongational rate in elongational flow


operator in Eq. (5.1)


time constant for Hookean dumbbell [Eq. (4.24)]


time constant for rigid dumbbell [Eq. (5.1)]


time constant for rigid dumbbell with hydrodynamic interaction [before Eq. (25.15)]


time constant defined after Eq. (26.2)


probability density in velocity space [Eq. (4.2)]


pressure tensor (total stress tensor)


density of fluid


stress tensor (or extra stress tensor)


solvent contribution to stress tensor [Eq. (4.1)]


polymer contribution to stress tensor [Eq. (4.1)] ( = τ p (b) + τ p (c) )


geometrical ratio in § 21


angle about the z-axis in spherical coordinates


probability density for two beads


probability density for internal configuration of a dumbbell


probability density for orientation of a rigid dumbbell [see Eq. (3.13 a)]


expansion functions in Eq. (6.4)


Oseen tensor in § 25


angular velocity in § 21


operator in Eq. (5.1)


operator in Eq. (15.1)




vorticity tensor [Eq. (20.3)]

Special Symbols

nabla operator


substantial derivative


convected derivative [Eq. (3.17)]



\(\frac{\partial }{{\partial R}}\)

vector operator with components \(\frac{\partial }{{\partial X}},\frac{\partial }{{\partial Y}},\frac{\partial }{{\partial Z}}\)


real part of

〈 〉

expectation value defined in Eq. (3.13)

{A, B, C}

special symbol defined in Eq. (12.13)


transpose of a tensor


complex viscometric functions (see §§7, 8, 9, 22)

Marks above Symbol


time differentiation

expectation value in Eq. (4.25)

complex conjugate in Eq. (7.9)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ballman, R. L.: Extensional flow of polystyrene melt. Rheol. Acta 4, 137–140 (1965).CrossRefGoogle Scholar
  2. 2.
    Benbow, J. J., Howells, E. R.: Normal stress, shear recovery, and viscosity in polydimethyl siloxanes. Polymer 2, 429–436 (1961).CrossRefGoogle Scholar
  3. 3.
    Bird, R. B., Evans, D. C., Warner, H. R., Jr.: Recoil in macromolecular solutions according to rigid dumbbell kinetic theory. Applied Scientific Research 22, 185–192 (1970).Google Scholar
  4. 4.
    — Harris, E. K., Jr.: Analysis of steady state shearing and stress relaxation in the Maxwell orthogonal rheometer. A.I.Ch.E. J. 14, 758–761 (1968); 16, 149 (1970).Google Scholar
  5. 5.
    — Johnson, M. W., Curtiss, C. F.: Potential flows of dilute polymer solutions by Kramers' method. J. Chem. Phys. 51, 3023–3026 (1969).CrossRefGoogle Scholar
  6. 6.
    — — Stevenson, J. F.: Molecular theories of elongational viscosity. Proc. 5th International Congress on Rheology, Vol. 4, pp. 159–168. Tokyo: University of Tokyo Press 1970.Google Scholar
  7. 7.
    — Marsh, B. D.: Viscoelastic hysteresis, Part I. Model predictions. Trans. Soc. Rheol. 12, 479–488 (1968).CrossRefGoogle Scholar
  8. 8.
    Bird, R. B., Stewart, W. E., Lightfoot, E. N.: Transport phenomena. New York: Wiley 1960.Google Scholar
  9. 9.
    — Warner, H. R., Jr.: Creep in macromolecular solutions according to rigid dumbbell kinetic theory. Appl. Sci. Res. 22, 193–196 (1970).Google Scholar
  10. 9a.
    — — Hydrodynamic interaction effects on the behavior of solutions of linear macromolecules. Trans. Soc. Rheol. (1971).Google Scholar
  11. 10.
    — — Ramakka, W. R.: Stress relaxation in solutions of linear macromolecules. J. Chem. Phys. 52, 2001–2002 (1970).CrossRefGoogle Scholar
  12. 11.
    Bogue, D. C., Doughty, J. O.: Comparison of constitutive equations for viscoelastic fluids. Ind. Eng. Chem. Fundamentals 5, 243–252 (1966).CrossRefGoogle Scholar
  13. 12.
    Booij, H. C.: Influence of superposed steady shear flow on the dynamic properties of non-newtonian fluids. Rheol. Acta 5, 215–221; 222–227 (1966).CrossRefGoogle Scholar
  14. 12a.
    — Effect of superimposed steady shear flow on dynamic properties of polymeric fluids. Doctoral Thesis, Leiden, Holland (1970).Google Scholar
  15. 12b.
    Brenner, H.: Hydrodynamic resistance of particles at small Reynolds numbers. Adv. in Chem. Engr., 6, 287–438 (1966).Google Scholar
  16. 12c.
    — Rheology of Two-Phase Systems. Ann. Rev. Fluid Mechanics, 2, 137–176 (1970).CrossRefGoogle Scholar
  17. 13.
    Burgers, J. M.: On the motion of small particles of elongated form suspended in a viscous liquid. Verhandel. Koninkl. Ned. Akad. Wetenschap. 16 (Sect. 1, Chap. 3) 113–184 (1938).Google Scholar
  18. 14.
    Carreau, P. J.: Rheological equations from molecular network theories. Ph. D. Thesis, University of Wisconsin, 1968.Google Scholar
  19. 15.
    — Macdonald, I. F., Bird, R. B.: A nonlinear viscoelastic model for polymer solutions and melts-II. Chem. Eng. Sci. 23, 901–911 (1968).CrossRefGoogle Scholar
  20. 15a.
    Cerf, R.: La dynamique des solutions de macromolécules dans un champ de vitesses. Fortschr. Hochpolym.-Forsch., 1, 382–450 (1959).CrossRefGoogle Scholar
  21. 16.
    Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943).CrossRefGoogle Scholar
  22. 17.
    Cogswell, F. N.: The rheology of polymer melts under tension. Plastic and Polymers 36, 109–111 (1968).Google Scholar
  23. 18.
    — Tensile deformations in molten polymers. Rheol. Acta 8, 187–194 (1969).CrossRefGoogle Scholar
  24. 19.
    Coleman, B. D., Markovitz, H.: Normal stress effects in second-order fluids. J. Appl. Phys. 35, 1–9 (1964).CrossRefGoogle Scholar
  25. 20.
    — — Noll, W.: Viscometric flows of non-newtonian fluids. Berlin-Heidelberg-New York: Springer 1966.Google Scholar
  26. 21.
    — Noll, W.: Foundations of linear viscoelasticity. Rev. Mod. Phys. 33, 239–249 (1961).CrossRefGoogle Scholar
  27. 22.
    Cox, W. P., Merz, E. H.: Correlation of dynamic and steady flow viscosities. J. Polymer Sci. 28, 619–622 (1958).CrossRefGoogle Scholar
  28. 23.
    Dirac, P. A. M.: The principles of quantum mechanics, 3rd Ed., pp. 58–61. Oxford: Oxford University Press 1947.Google Scholar
  29. 24.
    Erdélyi, A., et al.: Tables of integral transforms. New York: McGraw-Hill 1954.Google Scholar
  30. 25.
    Evans, D. C., Warner, H. R., Jr., Ramakka, W. R., Bird, R. B.: Behavior of solutions of linear macromolecules in steady shear flow with superposed oscillations. J. Chem. Phys. 52, 4086–4089 (1970).CrossRefGoogle Scholar
  31. 26.
    Ferry, J. D.: Viscoelastic properties of polymers, 2nd Ed. New York: Wiley 1970.Google Scholar
  32. 27.
    — Holmes, L. A., Lamb, J., Matheson, A. J.: Viscoelastic behavior of dilute polystyrene solutions in an extended frequency range. J. Phys. Chem. 70, 1685–1689 (1966).CrossRefGoogle Scholar
  33. 28.
    Fixman, H.: Dynamics of polymer chains. J. Chem. Phys. 42, 3831–3837 (1965).CrossRefGoogle Scholar
  34. 29.
    Fraenkel, G. K.: Visco-elastic effect in solutions of simple particles. J. Chem. Phys. 20, 642–647 (1952).CrossRefGoogle Scholar
  35. 30.
    Fredrickson, A. G.: Principles and applications of rheology. Englewood Cliffs, N.J.: Prentice-Hall 1964.Google Scholar
  36. 30a.
    Frisch, H. D., Simha, R.: The viscosity of colloidal suspensions and macromolecular solutions. Chapter 14 of Vol. 1 of Rheology (ed. by F. R. Eirich) Academic Press, New York (1956).Google Scholar
  37. 31.
    Giesekus, H.: Das Reibungsgesetz der strukturviskosen Flüssigkeit. (See erratum in fn. 18 of H. Giesekus, Rheol. Acta 1, 404 (1961).) Kolloid-Z. 147–149, 29–45 (1956).CrossRefGoogle Scholar
  38. 32.
    — Die rheologische Zustandsgleichung. Rheol. Acta 1, 2–20 (1958).CrossRefGoogle Scholar
  39. 33.
    — Einige Bemerkungen zum Fließverhalten elasto-viskoser Flüssigkeiten in stationären Schichtströmungen. Rheol. Acta 1, 404–413 (1961). See fn. 18.CrossRefGoogle Scholar
  40. 34.
    — Die Elastizität von Flüssigkeiten. Rheol. Acta 5, 29–35 (1966).CrossRefGoogle Scholar
  41. 35.
    Hirschfelder, J. O., Curtiss, C. F., Bird, R. B.: Molecular theory of gases and liquids. Second Printing with Corrections, pp. 905–912. New York: Wiley 1964.Google Scholar
  42. 36.
    Huppler, J. D., Macdonald, I. F., Ashare, E., Spriggs, T. W., Bird, R. B., Holmes, L. A.: Rheological properties of three solutions. Part II. Relaxation and growth of shear and normal stresses. Trans. Soc. Rheol. 11, 181–204 (1967).CrossRefGoogle Scholar
  43. 36a.
    Janeschitz-Kriegl, H.: Flow birefringence of elastico-viscous polymer systems. Adv. Polymer Sci., 6, 170–318 (1969).CrossRefGoogle Scholar
  44. 36b.
    Jeffery, G. B.: The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc., A 102, 161–179 (1922).CrossRefGoogle Scholar
  45. 37.
    Kirkwood, J. G.: Macromolecules. New York: Gordon and Breach 1967.Google Scholar
  46. 38.
    — The statistical mechanical theory of irreversible processes in solutions of flexible macromolecules. Rec. Trav. Chim. 68, 649–660 (1949).CrossRefGoogle Scholar
  47. 39.
    — Auer, P. L.: The visco-elastic properties of solutions of rod-like macromolecules. J. Chem. Phys. 19, 281–283 (1951).CrossRefGoogle Scholar
  48. 40.
    — Plock, R. J.: Non-Newtonian visco-elastic properties of rod-like macromolecules in solution. J. Chem. Phys. 24, 665–669 (1956).CrossRefGoogle Scholar
  49. 41.
    — Riseman, J.: The intrinsic viscosities and diffusion constants of flexible macromolecules in solution. J. Chem. Phys. 16, 565–573 (1948); 22, 1626–1627 (1954).CrossRefGoogle Scholar
  50. 42.
    Kotaka, T.: Note on the normal stress effect in the solution of rod-like macromolecules. J. Chem. Phys. 30, 1566–1567 (1959).CrossRefGoogle Scholar
  51. 43.
    — Suzuki, H., Inagaki, H.: Shear-rate dependence of the intrinsic viscosity of flexible linear macromolecules. J. Chem. Phys. 45, 2770–2773 (1966).CrossRefGoogle Scholar
  52. 44.
    Kramers, H. A.: Het gedrag van macromoleculen in een stroomende vloeistof. Physica 11, 1–19 (1944).CrossRefGoogle Scholar
  53. 45.
    Kuhn, W., Kuhn, H.: Die Abhängigkeit der Viskosität vom Strömungsgefälle bei hochverdünnten Suspensionen und Lösungen. Helv. Chim. Acta 28, 97–127 (1945).CrossRefGoogle Scholar
  54. 45a.
    — — Buchner, P.: Hydrodynamisches Verhalten von Macromolekülen in Lösung. Ergebnisse der exakten Naturwissenschaften, 25, 1–108 (1951).Google Scholar
  55. 46.
    Lodge, A. S.: Elastic liquids, New York: Academic Press 1964.Google Scholar
  56. 46a.
    — A network theory of flow birefringence and stress in concentrated polymer solutions. Trans. Faraday Soc., 52, 120–130 (1956).CrossRefGoogle Scholar
  57. 47.
    — Constitutive equations from molecular network theories for polymer solutions. Rheol. Acta 7, 379–392 (1968).CrossRefGoogle Scholar
  58. 48.
    — Concentrated polymer solutions. Proc. 5th International Congress on Rheology, Vol. 4. Tokyo: University of Tokyo Press 1969; (also MRC Tech. Summary Report No. 944, Oct. 1968, Mathematics Research Center, University of Wisconsin).Google Scholar
  59. 49.
    Lodge, A. S., Wu, Y.: Constitutive equations for polymer solutions derived from Zimm's bead/spring model. Rheol. Acta (to be submitted).Google Scholar
  60. 50.
    Macdonald, I. F.: Time dependent nonlinear behavior of viscoelastic fluids. Ph. D. Thesis, Univ. of Wisconsin (1968).Google Scholar
  61. 51.
    — Bird, R. B.: Complex modulus of concentrated polymer solutions in steady shear. J. Phys. Chem. 70, 2068–2069 (1966).CrossRefGoogle Scholar
  62. 52.
    — Marsh, B. D., Ashare, E.: Rheological behavior for large amplitude oscillatory motion. Chem. Engr. Sci. 24, 1615–1625 (1969).CrossRefGoogle Scholar
  63. 53.
    Markovitz, H., Coleman, B. D.: Incompressible second-order fluids. Advanc. Appl. Mech. 8, 69–101 (1964).CrossRefGoogle Scholar
  64. 54.
    Marsh, B. D.: Viscoelastic hysteresis. Part II. Numerical and experimental examples. Trans. Soc. Rheol. 12, 489–510 (1968).CrossRefGoogle Scholar
  65. 54a.
    Massa, D. J.: Computerized measurement of the dynamic viscoelastic properties of dilute polymer solutions over an extended range of solvent viscosity. Ph. D. Thesis, University of Wisconsin (1970).Google Scholar
  66. 55.
    Maxwell, B., Chartoff, R. P.: Studies of a polymer melt in an orthogonal rheometer. Trans. Soc. Rheol. 9, 41–52 (1965).CrossRefGoogle Scholar
  67. 56.
    Meissner, J.: Rheometer zur Untersuchung der deformationsmechanischen Eigenschaften von Kunststoff-Schmelzen unter definierter Zugbeanspruchung. Rheol. Acta 8, 78–88 (1969).CrossRefGoogle Scholar
  68. 57.
    — Dehnungsverhalten von Polyäthylen-Schmelzen. Rheol. Acta (submitted for publication).Google Scholar
  69. 58.
    Moore, R. S., McSkimin, H. J., Gieniewski, C., Andreatch, P., Jr.: Dynamic mechanical properties of concentrated polystyrene solutions at 40 MHz. J. Chem. Phys. 47, 3–9 (1967).CrossRefGoogle Scholar
  70. 58a.
    Noda, I., Yamada, Y., Nagasawa, M.: The rate of shear dependence of the intrinsic viscosity of monodisperse polymer. J. Phys. Chem. 72, 2890–2898 (1968).CrossRefGoogle Scholar
  71. 59.
    Oldroyd, J. G.: On the formulation of rheological equations of state. Proc. Roy. Soc. (London) A 200, 523–541 (1950).Google Scholar
  72. 60.
    Osaki, K., Tamura, M., Kurata, M., Kotaka, T.: Complex modulus of concentrated polymer solutions in steady shear. J. Phys. Chem. 69, 4183–4191 (1965).CrossRefGoogle Scholar
  73. 61.
    Oseen, C. W.: Über die Stokes'sche Formel und über eine verwandte Aufgabe in der Hydrodynamik. Ark. f. Mat. Astr. og. Fys. 6, No. 29, 1–20 (1910).Google Scholar
  74. 62.
    Paul, E.: Non-Newtonian viscoelastic properties of rod-like molecules in solution: Comment on a paper by Kirkwood and Plock. J. Chem. Phys. 51, 1271–1272 (1969).CrossRefGoogle Scholar
  75. 63.
    Peterlin, A.: Einfluß der endlichen Moleküllänge auf die Gradientenabhängigkeit des Staudinger-Index. Makromol. Chem. 44–46, 338–346 (1961).CrossRefGoogle Scholar
  76. 63a.
    — Über die Viskosität von verdünnten Lösungen und Suspensionen in Abhängigkeit von der Teilchenform. Zeits. Physik, 111, 232–263 (1938).CrossRefGoogle Scholar
  77. 63b.
    — Non-Newtonian viscosity and the macromolecule. Adv. in Macromolecular Chemistry, 1, 225–281 (1968).Google Scholar
  78. 63c.
    Pipkin, A. C.: Small displacements superposed on viscometric flow. Trans. Soc. Rheol., 12, 397–408 (1968).CrossRefGoogle Scholar
  79. 64.
    Prager, S.: Stress-strain relations in a suspension of dumbbells. Trans. Soc. Rheol. 1, 53–62 (1957).CrossRefGoogle Scholar
  80. 65.
    Radushkevich, B. V., Fikhman, V. D.: Vinogradov, G. V.: Uniaxial uniform-speed elongation of high-elasticity liquids (of low molecular polyisobutylene as an example). Dokl. Akad. Nauk. SSSR 180, 404–407 (1968).Google Scholar
  81. 66.
    — — — A method for investigating the elongation of highly elastic liquids. Mekhan. Polimerov 2, 343–348 (1968).Google Scholar
  82. 67.
    Riseman, J., Kirkwood, J. G.: The intrinsic viscosity, translational and rotatory diffusion constants of rodlike macromolecules in solution. J. Chem. Phys. 18, 512–516 (1950).CrossRefGoogle Scholar
  83. 68.
    Rivlin, R. S.: Further remarks on the stress-deformation relations for isotropic materials. J. Rat. Mech. Analysis 4, 681–702 (1955).Google Scholar
  84. 68a.
    Saitō, N.: Kōbunshi Butsurigaku, Shokabō, Tōkyō, Revised Edition (1968).Google Scholar
  85. 68b.
    — The effect of the Brownian motion on the viscosity of solutions of macromolecules. J. Phys. Soc. Japan, 6–7, 297–301, 302–304 (1951).Google Scholar
  86. 68c.
    Scheraga, H. A.: Non-Newtonian viscosity of solutions of ellipsoidal particles. J. Chem. Phys., 23, 1526–1532 (1955).CrossRefGoogle Scholar
  87. 69.
    Schremp, F. W., Ferry, J. D., Evans, W. W.: Mechanical properties of substances of high molecular weight. IX. Non-Newtonian flow and stress relaxation in concentrated polyisobutylene and polystyrene solutions. J. Appl. Phys. 22, 711–717 (1951).CrossRefGoogle Scholar
  88. 69a.
    Simmons, J. M.: Dynamic modulus of polystyrene solutions in superposed steady shear flow, Rheol. Acta, 7, 184–188 (1968).CrossRefGoogle Scholar
  89. 70.
    Spriggs, T. W.: Constitutive equations for viscoelastic fluids. Ph. D. Thesis, University of Wisconsin (1966).Google Scholar
  90. 71.
    Stevenson, J. F.: Elongational flow of polymer melts. Ph. D. Thesis, University of Wisconsin (1970).Google Scholar
  91. 72.
    Stewart, W. E., Sørensen, J. P.: Trans. Soc. Rheol. (1971).Google Scholar
  92. 73.
    Vinogradov, G. V., Radushkevich, B. V. Fikhman, V. D.: Extension of elastic liquids: Polyisobutylene. J. Polymer Sci. A-2, 8, 1–17 (1970).CrossRefGoogle Scholar
  93. 73a.
    Wada, E.: Effect of rate of shear on viscosity of a dilute linear polymer and of tobacco mosaic virus in solution. J. Polymer Science, 14, 305–307 (1954).CrossRefGoogle Scholar
  94. 74.
    Warner, H. R., Jr.: Rigid structural models of dilute macromolecular solutions. Ph. D. Thesis, University of Wisconsin (1971).Google Scholar
  95. 75.
    — Bird. R. B.: A molecular interpretation of the steady state Maxwell orthogonal rheometer flow. A. I.Ch.E. J. 16, 150 (1970).Google Scholar
  96. 76.
    Williams, M. C.: Normal stresses in polymer solutions with remarks on the Zimm treatment. J. Chem. Phys. 42, 2988–2989 (1965).CrossRefGoogle Scholar
  97. 77.
    — Bird, R. B.: Oscillatory behavior of normal stresses in viscoelastic fluids. Ind. Eng. Chem. Fundamentals 3, 42–49 (1964).CrossRefGoogle Scholar
  98. 78.
    — — Three-constant Oldroy model for viscoelastic fluids. Phys. Fluids. 5, 1126–1128 (1962).CrossRefGoogle Scholar
  99. 78a.
    Yang, J. T.: Non-Newtonian viscosity of poly-γ-benzyl-L-glutamate solutions. J. Amer. Chem. Soc., 80, 1783–1788 (1958).CrossRefGoogle Scholar
  100. 78b.
    — Factors affecting the non-Newtonian viscosity of rigid particles. J. Amer. Chem. Soc., 81, 3902–3907 (1959).CrossRefGoogle Scholar
  101. 79.
    Zimm, B. H.: Dynamics of polymer molecules in dilute solution: Viscoelasticity, flow birefringence and dielectric loss. J. Chem. Phys. 24, 269–278 (1956).CrossRefGoogle Scholar
  102. 80.
    Zwanzig, R.: Langevin theory of polymer dynamics in dilute solution. Adv. Chem. Phys. 15, 325–331 (1969).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • R. B. Bird
    • 1
  • H. R. WarnerJr.
    • 1
  • D. C. Evans
    • 1
  1. 1.Chemical Engineering Department and Rheology Research CenterUniversity of WisconsinMadisonUSA

Personalised recommendations