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Electroosmotic flow of a rheological fluid in non-uniform micro-vessels

Abstract

The paper deals with a theoretical study of electrokinetic flow of a rheological Herschel–Bulkley fluid through a cylindrical tube of variable cross section. The aim of this study is to analyze combined pressure-driven and electroosmotic flow of Herschel–Bulkley fluid. The wall potential is considered to vary slowly and periodically along the axis of the tube. With reference to flow in the micro-vessels, the problem has been solved using the lubrication theory. The Helmholtz–Smoluchowski (HS) slip boundary condition has been employed in this study. Volumetric flow rate Q is found to be significantly affected by the yield stress parameter \(\nu \) only if an applied pressure force is active. The linear superposition of flow components separately due to the hydrodynamic and electric force occurs only for a strictly uniform tube. This linear relationship fails if non-uniformity appears in either tube radius or in distribution of the electrokinetic slip boundary condition. Moreover, converging/diverging nature of the mean tube radius plays a crucial role on the fluid transport. For the benefit of readers, along with the original contribution, some applications of external electrical stimulation (ES) in the human body and HS slip velocity, studied in the past by previous researchers, have been discussed in the paper.

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References

  1. Li DQ (2004) Electrokinetics in microfluidics. Elsevier, Amsterdam

    Google Scholar 

  2. Jacobson SC, Hergenröder R, Koutny LB, Warmack RJ, Ramsey JM (1994) Effects of injection schemes and column geometry on the performance of microchip electrophoresis devices. Anal Chem 66:1107–1113

    Article  Google Scholar 

  3. Ermakov SV, Jacobson SC, Ramsey JM (2000) Computer simulations of electrokinetic injection techniques in microfluidic devices. Anal Chem 72:3512–3517

    Article  Google Scholar 

  4. Hjerten S (1967) Free zone electrophoresis. Chromatogr Rev 9:122

    Article  Google Scholar 

  5. Chen HS, Chang HT (1999) Electrophoretic separation of small DNA fragments in the presence of electroosmotic flow using poly (ethylene oxide) solutions. Anal Chem 71:2033–2036

    Article  Google Scholar 

  6. Dutta P, Beskok A (2001) Analytical solution of combined electroosmotic/pressure driven flow in two dimensional straight channels: finite Debye layer effects. Anal Chem 73:1979–1986

    Article  Google Scholar 

  7. Reuss FF (1809) Charge induced flow. Proc Imp Soc Nat Moscou 3:327–344

    Google Scholar 

  8. Hunter RJ (1981) Zeta potential in colloid science: principle and applications. Academic Press, London

    Google Scholar 

  9. Zhao CL, Yang C (2013) Electrokinetics of non-Newtonian fluids: a review. Adv Colloid Interfaces Sci 201:94–108

    Article  Google Scholar 

  10. Wiedemann G (1852) First quantitative study of electrical endomose. Pogg Ann 87:321

    Google Scholar 

  11. Helmholtz H (1879) Studien über electrische Grenzschichten. Ann Phys 243:337–382

    MATH  Article  Google Scholar 

  12. Smoluchowski M (1903) Contribution à la théorie l’endosmose électrique et de quelques phénomènes corrélatifs. Bull Int Acad Sci Cracovie 8:182–199

    MATH  Google Scholar 

  13. Probstein RF (1994) Physicochemical hydrodynamics, 2nd edn. Wiley, New York

    Book  Google Scholar 

  14. Park HM, Lee WM (2008) Helmholtz–Smoluchowski velocity for viscoelastic electroosmotic flows. J Colloid Interfaces Sci 317:631–636

    Article  Google Scholar 

  15. Patankar NA, Hu HH (1998) Numerical simulation of electroosmotic flow. Anal Chem 70:1870

    Article  Google Scholar 

  16. Mitchell MJ, Qiao R, Aluru NR (2000) Meshless analysis of steady state electro-osmotic transport. J Microelectromech Syst 9:435

    Article  Google Scholar 

  17. MacInnes JM, Du X, Allen RWK (2003) Prediction of electrokinetic and pressure flow in microchannel T-junction. Phys Fluids 15:1992–2005

    MATH  Article  Google Scholar 

  18. Craven TJ, Rees JM, Zimmerman WB (2008) On slip velocity boundary conditions for electroosmotic flow near sharp corners. Phys Fluids 20:43603

    MATH  Article  Google Scholar 

  19. Xuan X, Li D (2005) Electroosmotic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge. J Colloid Interfaces Sci 289:291–303

    Article  Google Scholar 

  20. Ghosal S (2002) Lubrication theory for electro-osmotic flow in a microfluidic channel of slowly varying cross-section and wall charge. J Fluid Mech 459:103–128

    MATH  Article  Google Scholar 

  21. Sinton D, Xuan C, Li D (2004) Thermally induced velocity gradients in electroosmotic microchannel flows: the cooling influence of optical infrastructure. Exp Fluid 37:872–882

    Article  Google Scholar 

  22. Xuan X, Li D (2005) Band broadening in capillary zone electrophoresis with axial temperature gradients. Electrophoresis 26:166–175

    Article  Google Scholar 

  23. Bianchi F, Ferrigno R, Girault HH (2000) Finite element simulation of an electroosmotic driven flow division at a t-junction of microscale dimensions. Anal Chem 72:1987–1993

    Article  Google Scholar 

  24. Erickson D, Li D (2002) Influence of surface heterogeneity on electrokinetically driven microfluidic mixing. Langmuir 18:1883–1892

    Article  Google Scholar 

  25. Ajdari A (2000) Pumping liquids using asymmetric electrode arrays. Phys Rev E 61:R45–R48

    Article  Google Scholar 

  26. Oddy MH, Santiago JG, Mikkelsen JC (2001) Electrokinetic instability micromixing. Anal Chem 73:5822–5832

    Article  Google Scholar 

  27. Dutta P, Beskok A (2001) Analytical solution of time periodic electroosmotic flows: analogies to Stokes’ second problem. Anal Chem 73:5097–5102

    Article  Google Scholar 

  28. Ermakov SV, Jacobson SC, Ramsey JM (1998) Computer simulations of electrokinetic transport in microfabricated channel structures. Anal Chem 70(21):4494–4504

    Article  Google Scholar 

  29. Hunter RJ (1992) Foundations of colloid science, vol I. Oxford University Press, Oxford

    Google Scholar 

  30. Hunter RJ (1992) Foundations of colloid science, vol II. Oxford University Press, Oxford

    Google Scholar 

  31. MacInnes JM (2002) Computation of reacting electrokinetic flow in micro-channel geometries. Chem Eng Sci 57:4539

    Article  Google Scholar 

  32. Zimmerman WB, Rees JM, Craven TJ (2006) Rheometry of non-Newtonian electrokinetic flow in a microchannel T-junction. Microfluid Nanofluid 2:481–492

    Article  Google Scholar 

  33. Molho JM, Herr AE, Desphande M, Gilbert JR, Garguilo MG, Paul PH, John PM, Woudenberg TM, Connel C (1998) Fluid transport mechanisms in microfluidic devices. Proc ASME (MEMS) 66:69–76

    Google Scholar 

  34. Paul PH, Garguilo MG, Rakestraw DJ (1998) Imaging of pressure and electrokinetically driven flows through open capillaries. Anal Chem 70(13):2459–2467

    Article  Google Scholar 

  35. Herr AE, Molho JI, Santiago JG, Mungal MG, Kenny TW, Garguilo MG (2000) Electroosmotic capillary flow with nonuniform zeta potential. Anal Chem 72:1053–1057

    Article  Google Scholar 

  36. Rubin S, Tulchinsky A, Gat AD, Bercovici M (2017) Elastic deformations driven by non-uniform lubrication flows. J Fluid Mech 812:841–865

    MathSciNet  MATH  Article  Google Scholar 

  37. Yariv E (2004) Electro-osmotic flow near a surface charge discontinuity. J Fluid Mech 521:181–189

    MATH  Article  Google Scholar 

  38. Khair AS, Squires TM (2008) Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity. J Fluid Mech 615:323–334

    MATH  Article  Google Scholar 

  39. Vinogradova OI (1999) Slippage of water over hydrophobic surfaces. Int J Miner Process 56(1):31–60

    Article  Google Scholar 

  40. Baudry J, Charlaix E, Tonck A, Mazuyer D (2001) Experimental evidence for a large slip effect at a nonwetting fluid–solid interface. Langmuir 17(17):5232–5236

    Article  Google Scholar 

  41. Iadecola C, Yang G, Ebner T, Chen G (1997) Local and propagated vascular responses evoked by focal synaptic activity in cerebellar cortex. Neurophysiology 78:651–659

    Article  Google Scholar 

  42. Pulgar VM (2015) Direct electric stimulation to increase cerebrovascular function. Front Syst Neurosci 9:54

    Article  Google Scholar 

  43. Lang N, Siebner HR, Ward NS, Lee L, Nitsche MA, Paulus W, Rothwell JC, Lemon RN, Frackowiak RS (2005) How does transcranial DC stimulation of the primary motor cortex alter regional neuronal activity in the human brain? Eur J Neurosci 22:495–504

    Article  Google Scholar 

  44. Nonnekes J, Arrogi A, Munneke MAM, van Asseldonk EHF, Oude Nijhuis LB, Geurts AC, Weerdesteyn V (2014) Subcortical structures in humans can be facilitated by transcranial direct current stimulation. PLoS ONE 9:e107731

    Article  Google Scholar 

  45. Thakral G, Lafontaine J, Najafi B, Talal TK, Kim P, Lavery LA (2013) Electrical stimulation to accelerate wound healing. Diabet Foot Ankle 4:22081

    Article  Google Scholar 

  46. Velmahos GC, Petrone P, Chan LS, Hanks SE, Brown CV, Demetriades D (2005) Electrostimulation for the prevention of deep venous thrombosis in patients with major trauma: a prospective randomized study. Surgery 137:493–498

    Article  Google Scholar 

  47. Doran FS, Drury M, Sivyer A (1964) A simple way to combat the venous stasis which occurs in the lower limbs during surgical operations. Br J Surg 51:486–492

    Article  Google Scholar 

  48. Doran FS, White HM (1967) A demonstration that the risk of postoperative deep venous thrombosis is reduced by stimulating the calf muscles electrically during the operation. Br J Surg 54:686–689

    Article  Google Scholar 

  49. Doran FS, White M, Drury M (1970) A clinical trial designed to test the relative value of two simple methods of reducing the risk of venous stasis in the lower limbs during surgical operations, the danger of thrombosis, and a subsequent pulmonary embolus, with a survey of the problem. Br J Surg 57:20–30

    Article  Google Scholar 

  50. Rajendran SB, Challen K, Wright KL, Hardy JG (2021) Electrical stimulation to enhance wound healing. J Funct Biomater 12:40

    Article  Google Scholar 

  51. Ud-Din S, Bayat A (2014) Electrical stimulation and cutaneous wound healing: a review of clinical evidence. Healthcare 2:445–467

    Article  Google Scholar 

  52. Hampton S, Collins F (2006) Treating a pressure ulcer with bio-electric stimulation therapy. Br J Nurs 15:S14–S18

    Article  Google Scholar 

  53. Arora M, Harvey LA, Glinsky JV, Nier L, Lavrencic L, Kifley A, Cameron ID (2020) Electrical stimulation for treating pressure ulcers. Cochrane Database Syst Rev 1:Cd012196

    Google Scholar 

  54. Kloth LC (2014) Electrical stimulation technologies for wound healing. Adv Wound Care 3:81–90

    Article  Google Scholar 

  55. Watanabe H, Takahashi H, Nakao M, Walton K, Llins RR (2009) Intravascular neural interface with nanowire electrode. Electron Commun Jpn 92:29–37

    Article  Google Scholar 

  56. Clover AJ, McCarthy MJ, Hodgkinson K, Bell PR, Brindle NP (2003) Noninvasive augmentation of microvessel number in patients with peripheral vascular disease. J Vasc Surg 38:1309–1312

    Article  Google Scholar 

  57. Allen NJ, Barres BA (2009) Neuroscience: Glia—more than just brain glue. Nature 457(7230):675–677

    Article  Google Scholar 

  58. Tsytsarev V, Hu S, Yao J, Maslov K, Barbour DL, Wang LV (2011) Photoacoustic microscopy of microvascular responses to cortical electrical stimulation. J Biomed Opt 16:076002

    Article  Google Scholar 

  59. Tsytsarev V, Premachandra K, Takeshita D, Bahar S (2008) Imaging cortical electrical stimulation in vivo: fast intrinsic optical signal versus voltage-sensitive dyes. Opt Lett 33(9):1032–1034

    Article  Google Scholar 

  60. Zonta M, Angulo MC, Gobbo S, Rosengarten B, Hossmann KA, Pozzan T, Carmignoto G (2003) Neuron-to-astrocyte signaling is central to the dynamic control of brain microcirculation. Nat Neurosci 6(1):43–50

    Article  Google Scholar 

  61. Choi YS, Hsueh YY, Koo J et al (2020) Stretchable, dynamic covalent polymers for soft, long-lived bioresorbable electronic stimulators designed to facilitate neuromuscular regeneration. Nat Commun 11:5990

    Article  Google Scholar 

  62. Jones S, Man WD, Gao W, Higginson IJ, Wilcock A, Maddocks M (2016) Neuromuscular electrical stimulation for muscle weakness in adults with advanced disease. Cochrane Database Syst Rev 10:CD009419

  63. Hong Z, Sui M, Zhuang Z, Liu H, Zheng X, Cai C, Jin D (2018) Effectiveness of neuromuscular electrical stimulation on lower limbs of patients with hemiplegia after chronic stroke: a systematic review. Arch Phys Med Rehabil 99:1011.e1-1022.e1

    Google Scholar 

  64. Willand MP, Rosa E, Michalski B, Zhang JJ, Gordon T, Fahnestock M, Borschel GH (2016) Electrical muscle stimulation elevates intramuscular BDNF and GDNF mRNA following peripheral nerve injury and repair in rats. Neuroscience 334:93–104

    Article  Google Scholar 

  65. Loaiza LA, Yamaguchi S, Ito M, Ohshima N (2002) Vasodilatation of muscle microvessels induced by somatic afferent stimulation is mediated by calcitonin gene-related peptide release in the rat. Neurosci Lett 333(2):136–40

    Article  Google Scholar 

  66. Tan JS, Lin CC, Chen GS (2020) Vasomodulation of peripheral blood flow by focused ultrasound potentiates improvement of diabetic neuropathy. BMJ Open Diabetes Res Care 8:e001004

    Article  Google Scholar 

  67. Berli CLA, Olivares ML (2008) Electrokinetic flow of non-Newtonian fluids in microchannels. J Colloid Interfaces Sci 320:582–589

    Article  Google Scholar 

  68. Berli CLA (2010) Output pressure and efficiency of electrokinetic pumping of non-Newtonian fluids. Microfluid Nanofluid 8:197–207

    Article  Google Scholar 

  69. Bandopadhyay A, Chakraborty S (2011) Steric-effect induced alterations in streaming potential and energy transfer efficiency of non-Newtonian fluids in narrow confinements. Langmuir 27:12243–12252

    Article  Google Scholar 

  70. Misra JC, Maiti S (2012) Peristaltic transport of rheological fluid: model for movement of food bolus through esophagus. Appl Math Mech 33(3):15–32

    MathSciNet  Article  Google Scholar 

  71. Misra JC, Maiti S (2012) Peristaltic pumping of blood through small vessels of varying cross-section. J Appl Mech ASME 22(8):061003

    Article  Google Scholar 

  72. Maiti S, Misra JC (2013) Non-Newtonian characteristics of peristaltic flow of blood in micro-vessels. Commun Nonlinear Sci Numer Simul 18:1970–1988

    MathSciNet  MATH  Article  Google Scholar 

  73. Maiti S, Misra JC (2012) Peristaltic transport of a couple stress fluid: some applications to hemodynamics. J Mech Med Biol 12(3):1250048

    Article  Google Scholar 

  74. Barnes HA (1995) A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure. J Non-Newtonian Fluid Mech 56:221–251

    Article  Google Scholar 

  75. Tuinier R, Taniguchi T (2005) Polymer depletion-induced slip near an interface. J Phys Condens Matter 17:L9–L14

    Article  Google Scholar 

  76. Olivares ML, Vera-Candioti L, Berli CLA (2009) The EOF of polymer solutions. Electrophoresis 30:921–929

    Article  Google Scholar 

  77. Das S, Chakraborty S (2006) Analytical solutions for velocity, temperature and concentration distribution in electroosmotic microchannel flows of a non-Newtonian bio-fluid. Anal Chim Acta 559:15–24

    Article  Google Scholar 

  78. Herschel WH, Bulkley R (1926) Konsistenzmessungen von Gummi-Benzollosungen. Kolloid Zeitschrift 39:291–300

    Article  Google Scholar 

  79. Scott-Blair GW, Spanner DC (1974) An introduction to biorheology. Elsevier, Amsterdam

    Google Scholar 

  80. Nallapu S, Radhakrishnamacharya G (2016) A two-fluid model for Herschel–Bulkley fluid flow through narrow tubes. J Appl Sci Eng 19(3):241–248

    Google Scholar 

  81. Moreno E, Larese A, Cervera M (2016) Modelling of Bingham and Herschel–Bulkley flows with mixed P1/P1 finite elements stabilized with orthogonal subgrid scale. J Non-Newtonian Fluid Mech 228:1–16

    MathSciNet  Article  Google Scholar 

  82. Das B (1991) Entrance region flow of the Hershel–Bulkley fluid in a circular tube. Fluid Dyn Res 10:39–53

    Article  Google Scholar 

  83. Minatti L, Pasculli A (2011) SPH numerical approach in modelling 2D muddy debris flow. In: International conference on debris-flow hazards mitigation: mechanics, prediction, and assessment, pp 467–475

  84. Remaitre A, Malet JP, Maquaire O, Ancey C, Locat J (2005) Flow behaviour and runout modelling of a complex debris flow in a clay-shale basin. Earth Surf Process Landf 30(4):479–88

    Article  Google Scholar 

  85. Iida N (1978) Influence of plasma layer on steady blood flow in microvessels. Jpn J Appl Phys 17(1):203

    Article  Google Scholar 

  86. Vajravelu K, Sreenadh S, Devaki P, Prasad K (2011) Mathematical model for a Herschel–Bulkley fluid flow in an elastic tube. Cent Eur J Phys 9(5):1357–1365

    Google Scholar 

  87. Merrill EW (1969) Rheology of blood. Physiol Rev 49:863

    Article  Google Scholar 

  88. Nguyen QD, Boger DV (1983) Yield stress measurement for concentrated suspensions. J Rheol 27:321

    Article  Google Scholar 

  89. Bird RB, Dai GC, Yarusso BJ (1983) The rheology and flow of viscoplastic materials. Rev Chem Eng 1:1

    Article  Google Scholar 

  90. Frigaard I (2019) Simple yield stress fluids. Curr Opin Coll Sci 22:638–663

    Google Scholar 

  91. Balmforth N, Frigaard IA, Ovarlez G (2014) Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu Rev Fluid Mech 46:121146

    MathSciNet  MATH  Article  Google Scholar 

  92. Coussot P (2014) Yield stress fluid flows: a review of experimental data. J Non-Newton Fluid Mech 211:31–49

    Article  Google Scholar 

  93. Bonn D, Denn M, Berthier B, Divoux B, Manneville S (2017) Yield stress materials in soft condensed matter. Rev Mod Phys 89:035005

    Article  Google Scholar 

  94. Huilgol RR (2015) Fluid mechanics of viscoplasticity. Springer, Berlin

    MATH  Google Scholar 

  95. Ovarlez G, Hormozi S (2019) Lectures on visco-plastic fluid mechanics. Springer, Cham

    MATH  Book  Google Scholar 

  96. Cloitre M, Bonnecaze RT (2017) A review on wall slip in high solid dispersions. Rheol Acta 56:283–305

    Article  Google Scholar 

  97. Coussot P (2017) Binghams heritage. Rheol Acta 56:163–176

    Article  Google Scholar 

  98. Ewoldt R, McKinley G (2017) Mapping thixo-elasto-visco-plastic behavior. Rheol Acta 56:195–210

    Article  Google Scholar 

  99. Frigaard I, Paso G, de Souza Mendes PR (2017) Binghams model in the oil and gas industry. Rheol Acta 56:259–282

    Article  Google Scholar 

  100. Malkin A, Kulichikhin V, Ilyin S (2017) A modern look on yield stress fluids. Rheol Acta 56:177–188

    Article  Google Scholar 

  101. Mitsoulis E, Tsamopoulos J (2017) Numerical simulations of complex yield-stress fluid flows. Rheol Acta 56:231–258

    Article  Google Scholar 

  102. Saramito P, Wachs A (2017) Progress in numerical simulation of yield stress fluid flows. Rheol Acta 56:211–230

    Article  Google Scholar 

  103. de Souza Mendes P, Thompson R (2019) Time-dependent yield stress materials. Curr Opin Colloid Interfaces Sci 43:15–25

    Article  Google Scholar 

  104. Fusi L (2018) Channel flow of viscoplastic fluids with pressure-dependent rheological parameters. Phys Fluids 30:073102

    Article  Google Scholar 

  105. Putz A, Frigaard IA, Martinez DM (2009) On the lubrication paradox and the use of regularization methods for lubrication flows. J Non-Newton Fluid Mech 163:62–77

    MATH  Article  Google Scholar 

  106. Lipscomb G, Denn M (1984) Flow of Bingham fluids in complex geometries. J Non-Newton Fluid Mech 14:337–346

    MATH  Article  Google Scholar 

  107. Frigaard IA, Ryan DP (2004) Flow of a visco-plastic fluid in a channel of slowly varying width. J Non-Newton Fluid Mech 123:67–83

    MATH  Article  Google Scholar 

  108. Muravleva L (2015) Squeeze plane flow of viscoplastic Bingham material. J Non-Newton Fluid Mech 220:148–161

    MathSciNet  Article  Google Scholar 

  109. Wilson S (1993) Squeezing flow of a Bingham material. J Non-Newton Fluid Mech 47:211–219

    MATH  Article  Google Scholar 

  110. Fusi L, Farina A, Rosso F, Roscani S (2015) Pressure-driven lubrication flow of a Bingham fluid in a channel: a novel approach. J Non-Newton Fluid Mech 221:66–75

    MathSciNet  Article  Google Scholar 

  111. Muravleva L (2018) Squeeze flow of Bingham plastic with stick-slip at the wall. Phys Fluids 30:030709

    Article  Google Scholar 

  112. Fusi L, Farina A (2003) An extension of the Bingham model to the case of an elastic core. Adv Math Sci Appl 13(1):113–163

    MathSciNet  MATH  Google Scholar 

  113. Fusi L, Farina A, Rosso F (2012) Flow of a Bingham-like fluid in a finite channel of varying width: a two-scale approach. J Non-Newton Fluid Mech 177:76–88

    Article  Google Scholar 

  114. Fusi L, Farina A, Rosso F (2016) Squeeze flow of a Bingham-type fluid with elastic core. Int J Nonlinear Mech 78:59–65

    Article  Google Scholar 

  115. Pinkus O, Sternlicht B (1961) Theory of hydrodynamic lubrication. McGraw-Hill, New York

    MATH  Google Scholar 

  116. Walton IC, Bittleston SH (1991) The axial flow of a Bingham plastic in a narrow eccentric annulus. J Fluid Mech 222:39–60

    MATH  Article  Google Scholar 

  117. Liu Y, Liu Y, Guo Q, Yang J (2009) Modeling of electroosmotic pumping of nonconducting liquids and biofluids by a two-phase flow method. J Electroanal Chem 636:86

    Article  Google Scholar 

  118. Tang GH, Ye PX, Tao WQ (2010) Pressure-driven and electroosmotic non-Newtonian flows through microporous media via lattice Boltzmann method. J Non-Newton Fluid Mech 165:1536

    MATH  Article  Google Scholar 

  119. Ng CO (2013) Combined pressure-driven and electroosmotic flow of Casson fluid through a slit microchannel. J Non-Newton Fluid Mech 198:1

    Article  Google Scholar 

  120. Bharti RP, Harvie DJE, Davidson MR (2009) Electroviscous effects in steady fully developed flow of a power-law liquid through a cylindrical microchannel. Int J Heat Fluid Flow 30:804

    Article  Google Scholar 

  121. Zhao C, Yang C (2010) Nonlinear Smoluchowski velocity for electroosmosis of power-law fluids over a surface with arbitrary zeta potentials. Electrophoresis 31:973

    Article  Google Scholar 

  122. Zhao C, Yang C (2011) Electro-osmotic mobility of non-Newtonian fluids. Biomicrofluidics 5:014110

    Article  Google Scholar 

  123. Vasu N, De S (2010) Electroosmotic flow of power-law fluids at high zeta potentials. Colloids Surf A Physicochem Eng Asp 368:44

    Article  Google Scholar 

  124. Babaie A, Sadeghi A, Saidi MH (2011) Combined electroosmotically and pressure driven flow of power-law fluids in a slit microchannel. J Non-Newton Fluid Mech 166:792

    MATH  Article  Google Scholar 

  125. Vakili MA, Sadeghi A, Saidi MH, Mozafari AA (2012) Electrokinetically driven fluidic transport of power-law fluids in rectangular microchannels. Colloids Surf A Physicochem Eng Asp 414:440

    Article  Google Scholar 

  126. Chakraborty S (2007) Electroosmotically driven capillary transport of typical non-Newtonian biofluids in rectangular microchannels. Anal Chim Acta 605:175–184

    Article  Google Scholar 

  127. Zhao C, Zholkovskij E, Masliyah J, Yang C (2008) Analysis of electroosmotic flow of power-law fluids in a slit microchannel. J Colloid Interfaces Sci 326:503–510

    Article  Google Scholar 

  128. Tang GH, Li XF, He YL, Tao WQ (2009) Electroosmotic flow of non-Newtonian fluid in microchannels. J Non-Newton Fluid Mech 157:133–137

    MATH  Article  Google Scholar 

  129. Zhao C, Yang C (2009) Analysis of power-law fluid flow in a microchannel with electrokinetic effects. Int J Emerg Multidiscip Fluid Sci 1:37–52

    Google Scholar 

  130. Zhao C, Yang C (2013) Electroosmotic flows of non-Newtonian power-law fluids in a cylindrical microchannel. Electrophoresis 34:662–667

    Article  Google Scholar 

  131. Ng CO, Qi C (2013) Electroosmotic flow of a viscoplastic material through a slit channel with walls of arbitrary zeta potential. Phys Fluids 25:103102

    Article  Google Scholar 

  132. Ajdari A (1996) Generation of transverse fluid currents and forces by an electric field: electro-osmosis on charge-modulated and undulated surfaces. Phys Rev E 53:4996–5005

    Article  Google Scholar 

  133. Ajdari A (2001) Transverse electrokinetic and microfluidic effects in micropatterned channels: lubrication analysis for slab geometries. Phys Rev E 65:016301

    Article  Google Scholar 

  134. Long D, Stone HA, Ajdari A (1999) Electroosmotic flows created by surface defects in capillary electrophoresis. J Colloid Interfaces Sci 212:338–349

    Article  Google Scholar 

  135. Ng CO, Zhou Q (2012) Electro-osmotic flow through a thin channel with gradually varying wall potential and hydrodynamic slippage. Fluid Dyn Res 44:055507

    MathSciNet  MATH  Article  Google Scholar 

  136. Ng CO, Zhou Q (2012) Dispersion due to electroosmotic flow in a circular microchannel with slowly varying wall potential and hydrodynamic slippage. Phys Fluids 24:112002

    Article  Google Scholar 

  137. Ng CO, Qi C (2014) Electroosmotic flow of a power-law fluid in a non-uniform microchannel. J Non-Newton Fluid Mech 208–209:118–125

    Article  Google Scholar 

  138. Dey P, Shit GC (2020) Electroosmotic flow of a fractional second-grade fluid with interfacial slip and heat transfer in the microchannel when exposed to a magnetic field. Heat Transf 50(3):2643–2666

    Article  Google Scholar 

  139. Ranjit NK, Shit GC (2019) Entropy generation on electromagnetohydrodynamic flow through a porous asymmetric micro-channel. Euro J Mech B Fluids 77:135–147

    MathSciNet  MATH  Article  Google Scholar 

  140. Ranjit NK, Shit GC, Tripathi D (2019) Entropy generation and Joule heating of two layered electroosmotic flow in the peristaltically induced micro-channel. Int J Mech Sci 153–155:430–444

    Article  Google Scholar 

  141. Mondal A, Shit GC (2018) Electro-osmotic flow and heat transfer in a slowly varying asymmetric micro-channel with Joule heating effects. Fluid Dyn Res 50:065502

    MathSciNet  Article  Google Scholar 

  142. Ranjit NK, Shit GC, Tripathi D (2018) Joule heating and zeta potential effects on peristaltic blood flow through porous micro vessels altered by electrohydrodynamic. Microvasc Res 117:74–89

    Article  Google Scholar 

  143. Ranjit NK, Shit GC (2017) Joule heating effects on electromagnetohydrodynamic flow through a peristaltically induced micro-channel with different zeta potential and wall slip. Physica A Stat Mech Appl 482:458–476

    MathSciNet  MATH  Article  Google Scholar 

  144. Khan AI, Dutta P (2019) Analytical solution of time-periodic electroosmotic flow through cylindrical microchannel with non-uniform surface potential. Micromachines 10:498

    Article  Google Scholar 

  145. Lee JSH, Ren CL, Li D (2005) Effects of surface heterogeneity on flow circulation in electroosmotic flow in microchannels. Anal Chim Acta 530:273–282

    Article  Google Scholar 

  146. Kim H, Khan AI, Dutta P (2019) Time-periodic electro-osmotic flow with nonuniform surface charges. ASME J Fluids Eng 141(8):081201

    Article  Google Scholar 

  147. Green NG, Ramos A, Gonzalez A, Morgan H, Castellanos A (2000) Fluid flow induced by nonuniform AC electric fields in electrolytes on microelectrodes. I. Experimental measurements. Phys Rev E 61(4):4011–4018

    Article  Google Scholar 

  148. Potoček B, Gaš B, Kenndler E, Štědrý M (1995) Electroosmosis in capillary zone electrophoresis with non-uniform zeta potential. J Chromatogr A 709(1):51–62

    Article  Google Scholar 

  149. Horiuchi K, Dutta P, Ivory CF (2007) Electroosmosis with step changes in zeta potential in microchannels. AIChE J 53(10):2521–2533

    Article  Google Scholar 

  150. Chang CC, Yang RJ (2006) A particle tracking method for analyzing chaotic electroosmotic flow mixing in 3D microchannels with patterned charged surfaces. J Micromech Microeng 16(8):1453–1462

    MathSciNet  Article  Google Scholar 

  151. Green NG, Ramos A, Gonzalez A, Morgan H, Castellanos A (2002) Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation. Phys Rev E 66:026305

    Google Scholar 

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Acknowledgements

We thank the Reviewers and Associate Editor, Professor I. A. Frigaard for their useful comments, which helped to improve the paper. One of the authors, S. Maiti, is thankful to the University Grants Commission (UGC), New Delhi for awarding the Dr. D. S. Kothari Post Doctoral Fellowship during this investigation.

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Maiti, S., Pandey, S.K. & Misra, J.C. Electroosmotic flow of a rheological fluid in non-uniform micro-vessels. J Eng Math 135, 8 (2022). https://doi.org/10.1007/s10665-022-10234-7

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Keywords

  • Depletion layer
  • Electroosmotic flow
  • Helmholtz–Smoluchowski slip
  • Non-Newtonian Fluid