Skip to main content
Log in

Hydrodynamic stability of non-newtonian media

  • Published:
Soviet Applied Mechanics Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Literature Cited

  1. V. B. Amfilokhiev, A. Ya. Voitkunskaya, and I. V. Ivanov, “Stability of laminar flow of a power non-Newtonian fluid in the boundary layer of a flat plate,” Republication Interdepartmental Collection “Bionics,” [in Russian], No. 6, Khimiya, Leningrad (1972).

    Google Scholar 

  2. V. B. Amfilokhiev and I. V. Ivanov, “Stability of laminar flow of a power non-Newtonian fluid in a plane channel,” Trudy Leningr. Karablestorit. In-ta, No. 80 (1972).

  3. G. I. Barenblatt, I. G. Bulina, V. P. Myasnikov, and G. I. Sholomovich, “Effect of small additions of soluble high-molecular compounds on the regime of motion of a fluid,” Zh. Prikl. Mekhan. i Tekh. Fiz., No. 4 (1965).

  4. V. A. Gorodtsov and A. I. Leonov, “Nonlinear instability of plane-parallel Couette flow of a viscoelastic fluid,” Prikl. Matem. i Mekhan.31, No. 2 (1967).

  5. V. A. Gorodtsov and A. I. Leonov, “Kinematics, nonequilibrium thermodynamics, and rheological relations in nonlinear viscoelasticity theory,” ibid. Prikl. Matem. i Mekhan.,31, No. 1 (1968).

  6. V. A. Gorodtsov and A. I. Leonov, “Role of scalar structural parameter in description of rheological behavior of viscoelastic fluids,” Mekhan. Polim., No. 6 (1969).

  7. S. H. Davis, “Principle of variation of stability,” in: Mechanics [Russian translation], No. 4 (1970).

  8. Yu. P. Ivanilov, “Stability of plane-parallel flow of a viscous fluid over an inclined floor,” Prikl. Matem. i Mekhan.,14, No. 2 (1960).

  9. E. N. Korzhov and V. I. Rubezhanskii, “Squire's theorem in gravitational convection of microstructural fluids,” in: Proc. Voronezh University [in Russian], No. 2 (1971).

  10. S. S. Kutateladze, V. I. Popov, and E. M. Khabakhapasheva, “Hydrodynamics of fluids having variable viscosity,” Zh. Prikl. Mekhan. i Tekh. Fiz., No. 1 (1966).

  11. V. B. Lemberskii, “Stability of flow of a structurally viscous fluid in the boundary layer on a plate,” Inzh.-Fiz. Zh.,12, No. 3 (1967).

  12. A. I. Leonov, “Description of rheological behavior of viscoelastic media for large elastic deformations,” Trudy In-ta Problem Mekhaniki Akad. Nauk SSSR [in Russian], Moscow (1973).

  13. C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge (1955).

  14. A. T. Listrov, “Stability of parallel flows of non-Newtonian media,” Dokl. Akad. Nauk SSSR,164, No. 5 (1965).

  15. A. T. Listrov, “Stability of flow of a viscoelastic fluid flowing along an inclined plane,” Zh. Prikl. Mekhan. i Tekh. Fiz., No. 5 (1965).

  16. A. T. Listrov, “Stability of flow of a layer of fluid of the Grad model along an inclined plane,” Izv. Akad. Nauk SSSR, Mekhan. Zhidk. i Gaza, No. 6 (1966).

  17. A. T. Listrov and S. P. Levitskii, “Squire's theorem for a layer of a nematic liquid crystal flowing along an inclined plane,” Trudy Voronezhsk. Un-ta, No. 6 (1972).

  18. A. M. Makarov, A. K. Martinson, and K. B. Pavlov, “Stability of plane flow of a non-Newtonian fluid with a rheological power law,” Inzh.-Fiz. Zh.,16, No. 5 (1969).

  19. L. I. Sedov, Mechanics of a Continuous Medium [in Russian], Vol. 1, Nauka, Moscow (1970).

    Google Scholar 

  20. V. V. Skripachev, “Stability of the boundary layer of a power non-Newtonian fluid,” Izv. Akad. Nauk SSSR, Mekhan. Zhidk. i Gaza, No. 2 (1971).

  21. H. A. Barnes and K. Walters, “On the flow of viscous and elastico-viscous liquids through straight and curved pipes,” Proc. Roy. Soc. Ser. A,314, No. 1516 (1969).

  22. D. W. Beard, M. H. Davies, and K. Walters, “The stability of elastico-viscous flow between rotating cylinders,” Part 3, Overstability in viscous and Maxwell fluids,” J. Fluid Mech.,24, No. 2 (1966).

  23. R. Betchov, “Stability of parallel flows with frequency-dependent viscosity,” Phys. Fluids,8, No. 10 (1965).

  24. R. K. Bhatnagar and H. Giesekus, “On the stability of viscoelastic fluids flow, II, Plane channel flow,” Rheologica Acta,9, No. 1 (1970).

  25. R. K. Bhatnagar and H. Giesekus, “On the stability of viscoelastic fluid flow, III, Flow in a cylindrical tube and an annulus,” ibid. Rheologica Acta,9, No. 3 (1970).

  26. B. Bird and P. J. Carraeu, “A nonlinear viscoelastic model for polymer solutions and melts,” J. Chem. Engng. Sci.,28, No. 5 (1968).

  27. C. F. Chan Man Fong, “Stability of plane Poiseuille flow of a slightly viscoelastic fluid,” Rheologica Acta,7, No. 4 (1968).

  28. C. F. Chan Man Fong, “Stability of flow of viscoelastic fluids between arbitrary spaced cylinders,” Z. Angew. Math. und Phys.,21, No. 6 (1970).

  29. C. F. Chan Man Fong, “Nonlinear Taylor stability of viscoelastic fluids,” Appl. Sci. Res.,23, Nos. 1–2 (1970).

  30. C. F. Chan Man Fong and K. Walters, “The solution of flow problems in the case of materials with memory, Part II, The stability of plane Poiseuille flow of slightly viscoelastic liquids,” J. de Mechanique,4, No. 4 (1965).

  31. S. Chandrasekhar, “The stability of viscous flow between rotating cylinders,” Mathematika,1 (1954).

  32. S. Chandrasekhar, “Hydrodynamic and hydromagnetic stability,” Clarendon Press, Oxford (1961).

    Google Scholar 

  33. S. Chandrasekhar and W. H. Reid, “On the expansion of functions with four boundary conditions,” Proc. Nat. Acad. Sci., Washington,43 (1957).

  34. D. H. Chun and W. H. Schwarz, “Stability of a plane Poiseuille flow of a second-order fluid,” Phys. Fluids,11, No. 1 (1968).

  35. B. D. Coleman and H. Markovitz, “Normal stress effects in second-order fluids,” J. Appl. Phys.,35, No. 1 (1964).

  36. B. D. Coleman, R. J. Duffin, and V. J. Mizel, “Instability, uniqueness, and nonexistence theorems for the equation ut=uxx−uxtx on a strip,” Arch. Rat. Mech. Anal.,19, No. 2 (1965).

  37. B. D. Coleman and W. Noll, “On certain steady flows of general fluids,” ibid. Arch. Rat. Mech. Anal.,3, No. 4 (1959).

  38. B. D. Coleman and W. Noll, “An approximation theorem for functionals with applications in continuum mechanics,” ibid. Arch. Rat. Mech. Anal.,6, No. 5 (1960).

  39. B. D. Coleman and W. Noll, “Foundations of linear viscoelasticity,” Rev. Mod. Phys.,33, No. 2 (1961).

  40. B. D. Coleman and W. Noll, “Recent results in the continuum theory of viscoelastic fluids,” Ann. N.Y. Acad. Sci.,89, No. 4 (1961).

  41. R. R. Cousins, “The effect of stress relaxation parameters on the stability of plane Poiseuille flow,” Int. J. Engng. Sci.,10, No. 3 (1972).

  42. R. R. Cousins, “Stability of plane Poiseuille flow to finite-amplitude disturbance in viscoelastic fluids,” Int. J. Engng. Sci.,10, No. 6 (1972).

  43. A. D. D. Craik, “A note on the static stability of an elastico-viscous fluid,” J. Fluid Mech.,33, No. 1 (1968).

  44. S. D. Datta, “Note on the stability of an elastico-viscous liquid in Couette flow,” Phys. Fluids,7, No. 3 (1964).

  45. W. R. Dean, “Fluid motion in a curved channel,” Proc. Roy. Soc., Ser. A,121 (1928).

  46. M. M. Denn and J. J. Roisman, “Rotational stability and measurement of normal stress functions in dilute polymer solutions,” AIChE J.,15, No. 3 (1969).

  47. J. L. Ericksen, “Theory of anisotropic fluids,” Trans. Soc. Rheol.,4, No. 1 (1960).

  48. J. L. Ericksen, “Anisotropic fluids,” Arch. Rat. Mech. Anal.,4, No. 4 (1960).

  49. G. E. Gadd, “Turbulence damping and drag reduction produced by certain additives in water,” Nature,206, No. 4983 (1965).

  50. S. M. Genesky, “A general theorem concerning the stability of a particular non-Newtonian fluid,” Quart. Appl. Math.,18, No. 3 (1960).

  51. H. Giesekus, “Zur stabilität von strömungen viscoelastischer flüssigkeiten, I, Ebene und kreisformige Couette-strämung,” Rheologica Acta,5, No. 3 (1966).

  52. H. Giesekus and R. K. Bhatnagar, “On the stability of viscoelastic fluid flow, IV, Overstability in plane Couette flow,” ibid. Rheologica Acta,10, No. 2 (1971).

  53. R. F. Ginn and M. M. Denn, “Rotational stability in viscoelastic liquids theory,” AIChE J.,15, No. 3 (1969).

  54. H. Grad, “Statistical mechanics, thermodynamics, and fluid dynamics of systems with arbitrary number of integrals,” Comm. Pure Appl. Math.,5, No. 4 (1952).

  55. A. E. Green and R. S. Rivlin, “The mechanics of nonlinear materials with memory, Part I,” Arch. Rat. Mech. Anal.,1, No. 1 (1957).

  56. A. E. Green, R. S. Rivlin, and A. J. M. Spenser, “The mechanics of nonlinear materials with memory, Part II,” ibid. Arch. Rat. Mech. Anal.,3, No. 1 (1959).

  57. A. E. Green and R. S. Rivlin, “The mechanics of nonlinear materials with memory, Part III,” ibid. Arch. Rat. Mech. Anal.,4, No. 5 (1960).

  58. A. S. Gupta, “Stability of a viscoelastic liquid film flowing down an inclined plane,” J. Fluid Mech.,28, No. 1 (1967).

  59. A. S. Gupta and L. Rai, “Stability of an elastico-viscouc liquid film flowing down an inclined plane,” Proc. Camb. Phil. Soc.,63, No. 2 (1967).

  60. D. M. Herbert, “On the stability of viscoelastic liquids in heated plane Couette flow,” J. Fluid Mech.,17, No. 3 (1963).

  61. W. M. Johnes and D. E. Marshall, “Relaxation effects in Couette flow between rotating cylinders,” Brit. J. Appl. Phys.,2, No. 6 (1969).

  62. D. T. Johnes and K. Walters, “Some remarks on the stability of parallel flows of non-Newtonian fluids,” AIChE J.,14, No. 4 (1968).

  63. J. N. Kapur and S. Goel, “A stability theorem for general non-Newtonian fluid,” Appl. Sci. Res., Ser. A,11, No. 3 (1963).

  64. P. K. Kundu, “Small disturbance stability of plane Poiseuille flow of Oldroyd fluid,” Phys. Fluids,15, No. 7 (1972).

  65. W. Lai, “Stability of an elastico-viscous liquid film flowing down an inclined plane,” ibid. Phys. Fluids,10, No. 4 (1967).

  66. F. M. Leslie, “The stability of Couette flow of certain anisotropic fluids,” Proc. Camb. Phys. Soc.,60, No. 4 (1964).

  67. C. H. Li, “Stability of two superposed elastico-viscous liquids in plane Couette flow,” Phys. Fluids,12, No. 3 (1969).

  68. C. C. Lin, “On the stability of two-dimensional parallel flows, Part 2, Stability in an inviscid fluid,” Quart. Appl. Math.,3, No. 3 (1946).

  69. F. J. Lockett, “On Squire's theorem for viscoelastic fluids,” Int. J. Engng. Sci.,7, No. 3 (1969).

  70. F. J. Lockett and R. S. Rivlin, “Stability in Couette flow of a Rivlin-Ericksen fluid,” Nat. Phys. Lab. Rept., No. 67 (1967).

  71. F. J. Lockett and R. S. Rivlin, “Stability in Couette flow of a viscoelastic fluid, Part I,” J. de Mecanique,7, No. 4 (1968).

  72. H. Markovitz and B. D. Coleman, “Incompressible second-order fluids,” in: Advances in Applied Mechanics, Vol. 8, Academic Press, New York (1964).

    Google Scholar 

  73. L. V. McIntire, “Use of retarded motion expansion for simple fluids in hydrodynamic stability analysis,” Phys. Fluids,14, No. 6 (1971).

  74. L. V. McIntire and C. H. Lin, “Finite amplitude instability of second-order fluids in plane Poiseuille flow,” J. Fluid Mech.,52, No. 2 (1972).

  75. L. V. McIntire and W. R. Schowalter, “Stability of viscoelastic fluids: Plane Couette flow with superposed temperature gradient,” Trans. Soc. Rheol.,14, No. 2 (1970).

  76. L. V. McIntire and W. R. Schowalter, “Hydrodynamic stability of viscoelastic fluids: Importance of fluid model, overstability, and form of disturbance,” AIChE J.,18, No. 1 (1972).

  77. A. B. Metzner and M. G. Park, “Turbulent flow characteristics of viscoelastic fluids,” J. Fluid Mech.,20, No. 2 (1964).

  78. D. H. Michael, “The stability of plane Poiseuille flow of a dusty gas,” ibid. J. Fluid Mech.,18, No. 1 (1964).

  79. M. J. Miller and E. B. Christiansen, “The stress state of elastic fluids in viscometric flow,” AICh E. J.,18, No. 3 (1972).

  80. D. T. Mook, “Stability equation for viscoelastic liquids of arbitrary memory,” Phys. Fluids,14, No. 12 (1971).

  81. D. T. Mook, “Stability of parallel flows of second-order liquids,” ibid. Phys. Fluids,15, No. 2 (1972).

  82. D. T. Mook and W. P. Graebel, “Stability of plane Poiseuille flows of viscoelastic liquids: an asymptotic solution,” J. Hydronaut.,5, No. 1 (1971).

  83. U. Müller, “Das stabilitätsverhalten einer ebenen Couette-strömung in einer leheitzen nichtlinear viscosen flüssigkeit,” Acta Mech.,6, No. 1 (1968).

  84. W. A. Noll, “A mathematical theory of the mechanical behavior of continuous media,” Arch. Rat. Mech. Anal.,2, No. 3 (1958).

  85. J. G. Oldroyd, “On the formulation of rheological equations of state,” Proc. Roy. Soc., Ser. A,200, No. 1063 (1950).

  86. J. G. Oldroyd, “Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids,” ibid. Proc. Roy. Soc., Ser. A,245, No. 1241 (1958).

  87. R. W. Paterson and F. H. Abernathy, “Turbulent flow drag reduction and degradation with dilute polymer solutions,” J. Fluid Mech.,43, No. 4 (1970).

  88. J. R. A. Pearson and C. J. S. Petrie, “On the melt flow instability of extruded polymers” Proc. 4th Intern. Congr. Rheol., Providence, 1963, Part 3, Interscience, New York (1965).

    Google Scholar 

  89. A. Pellew and R. V. Southwell, “On maintained convective motion in a fluid heated from below,” Proc. Roy. Soc., Ser. A,176, No. 966 (1940).

  90. J. Platten and R. S. Schechter, “Stability of the flow of a slightly viscoelastic fluid,” Phys. Fluids,13, No. 3 (1970).

  91. K. C. Porteous and M. M. Denn, “Linear stability of plane Poiseuille flow of viscoelastic liquids,” Trans. Soc. Rheol.,16, No. 2 (1972).

  92. K. C. Porteous and M. M. Denn. “Nonlinear stability of plane Poiseuille flow of viscoelastic liquids,” ibid. Trans. Soc. Rheol.,16, No. 2 (1972).

  93. W. H. Reid, “On the stability of viscous flow in a curved channel,” Proc. Roy. Soc., Ser. A,244, No. 1237 (1958).

  94. P. L. Rimmer, “The stability of the plane Poiseuille flow of a non-Newtonian fluid,” Rheologica Acta,10, No. 4 (1971).

  95. R. S. Rivlin, “Further remarks on the stress-deformation relations for isotropic materials,” J. Rat. Mech. Anal.,4, No. 5 (1955).

  96. R. S. Rivlin, Research Frontiers in Fluid Dynamics, Interscience, New York (1965).

    Google Scholar 

  97. R. S. Rivlin and J. L. Ericksen, “Stress-deformation relations for isotropic materials,” J. Rat. Mech. Anal.,4, No. 2 (1955).

  98. R. L. Sani, “Remarks on the Bénard and related problems,” Z. Angew. Math. und Phys.,17, No. 5 (1966).

  99. F. A. Seyer, “Friction reduction in turbulent flow of polymer solution,” J. Fluid Mech.,40, No. 2 (1970).

  100. M. M. Smith and R. S. Rivlin, “Stability in Couette flow of a viscoelastic fluid, Part II,” J. de Mecanique,11, No. 1 (1972).

  101. J. T. Stuart, “On the stability of viscous flow between parallel planes in the presence of a coplanar magnetic field,” Proc. Roy. Soc.,A 221, No. 1145 (1954).

  102. J. T. Stuart, “On the nonstability mechanics of hydrodynamic stability,” J. Fluid Mech.,4, No. 1 (1958).

  103. R. H. Thomas and K. Walters, “The stability of the flow of an elastico-viscous liquid in a curved channel,” Proc. Roy. Soc., Ser. A,274, No. 1352 (1963).

  104. R. H. Thomas and K. Walters, “The stability of elastico-viscous flow between rotating cylinders, Part I,” J. Fluid Mech.,18, No. 1 (1964).

  105. R. H. Thomas and K. Walters, “The stability of elastico-viscous flow between rotating cylinders, Part 2,” ibid. J. Fluid Mech.,19, No. 4 (1964).

  106. G. Tlapa and B. Bernstein, “Stability of a relaxation-type viscoelastic fluid with slight elasticity,” Phys. Fluids,13, No. 3 (1970).

  107. B. A. Thoms, “Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers,” Proc. First Int. Congr. Rheol.,2 (1948).

  108. C. Truesdell and W. Noll, “The nonlinear field theories of mechanics,” Handbuch der Physik, Vol. III/3, Springer-Verlag, Berlin (1965).

    Google Scholar 

  109. P. D. S. Verma, “Couette flow of certain anisotropic fluids,” Arch. Rat. Mech. Anal.,10, No. 2 (1962).

  110. K. Walters, “The solution of flow problems in the case of materials with memory, Part I,” J. de Mechanique,1, No. 4 (1962).

  111. K. Walters, The Solution of Flow Problems in Elasticity, Plasticity, and Fluid Dynamics, Pergamon Press, London (1964).

    Google Scholar 

  112. K. Walters, “Relation between Coleman-Noll, Rivlin-Ericksen, Green-Rivlin, and Oldroyd fluids,” Z. Angew. Math. und Phys.,21, No. 4 (1970).

  113. J. Watson, “On the nonlinear mechanics of wave disturbances in stable and unstable flows, Part 2,” J. Fluid Mech.,9, No. 2 (1960).

  114. D. A. White, “Correlation of pressure drop data in pipe of dilute polymer solutions,” Chem. Eng. Sci.,25, No. 7 (1970).

  115. J. L. White and A. B. Metzner, “Constantive equations for viscoelastic fluids with application to rapid external flows,” AIChE J.,11, No. 2 (1965).

  116. J. Yerushalmi, S. Katz, and R. Shinnar, “The stability of steady shear flows of some viscoelastic fluids,” Chem. Eng. Sci.,25, No. 12 (1970).

  117. C.-S. Yih, “Stability of liquid flow down an inclined plane,” Phys. Fluids,6, No. 3 (1963).

  118. C.-S. Yih, “Stability of a non-Newtonian liquid film flowing down an inclined plane,” ibid. Phys. Fluids,8, No. 7 (1965).

Download references

Authors

Additional information

Kazan' Chemicotechnical Institute. Translated from Prikladnaya Mekhanika, Vol. 10, No. 8, pp. 3–25, August, 1974.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garifullin, F.A., Galimov, K.Z. Hydrodynamic stability of non-newtonian media. Soviet Applied Mechanics 10, 807–824 (1974). https://doi.org/10.1007/BF00882508

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00882508

Keywords

Navigation