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Journal ArticleDOI

Steady electro-osmotic flow of a micropolar fluid in a microchannel

TL;DR: In this paper, the boundary value problem of steady, symmetric and one-dimensional electro-osmotic flow of a micropolar fluid in a uniform rectangular microchannel, under the action of a uniform applied electric field, was formulated and solved.
Abstract: We have formulated and solved the boundary-value problem of steady, symmetric and one-dimensional electro-osmotic flow of a micropolar fluid in a uniform rectangular microchannel, under the action of a uniform applied electric field. The Helmholtz–Smoluchowski equation and velocity for micropolar fluids have also been formulated. Numerical solutions turn out to be virtually identical to the analytic solutions obtained after using the Debye–Huckel approximation, when the microchannel height exceeds the Debye length, provided that the zeta potential is sufficiently small in magnitude. For a fixed Debye length, the mid-channel fluid speed is linearly proportional to the microchannel height when the fluid is micropolar, but not when the fluid is simple Newtonian. The stress and the microrotation are dominant at and in the vicinity of the microchannel walls, regardless of the microchannel height. The mid-channel couple stress decreases, but the couple stress at the walls intensifies, as the microchannel height increases and the flow tends towards turbulence.
Citations
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Journal ArticleDOI
TL;DR: The outcomes indicate that the axial velocity of Cu-CuO/blood nanoparticles strongly depends on applied electromagnetic field and microrotation, which will be applicable in designing the smart electromagnetic micro pumps for the hemodialysis and lungs-on-chip devices for the pumping of the blood.
Abstract: A thermal analysis of Cu-CuO/ blood nanofluids flow in asymmetric microchannel propagating with wave velocity is presented in this study. For the blood, a micropolar fluid model is considered to investigate the microrotation effects of blood flow. Thermal radiation effects and the influence of nanoparticle shape, electric double layer thickness, and electromagnetic fields on the flow are studied. Three types of nanoparticles shapes namely cylinder, bricks and platelets are taken into account. Governing equations are solved under the approximations of long wavelength, low Reynolds number, and Debye-Huckel linearization. Numerical computations are performed for the axial pressure gradient, axial velocity, spin velocity and temperature distribution. The effects of various physical parameters on flow and thermal characteristics are computed and their physical interpretation is also discussed. The outcomes indicate that the axial velocity of Cu-CuO/blood nanoparticles strongly depends on applied electromagnetic field and microrotation. The model's finding will be applicable in designing the smart electromagnetic micro pumps for the hemodialysis and lungs-on-chip devices for the pumping of the blood.

72 citations

Journal ArticleDOI
TL;DR: In this paper, a mathematical model for electro-osmotic peristaltic pumping of a non-Newtonian liquid in a deformable micro-channel is developed for the linearized transformed dimensionless boundary value problem.
Abstract: A mathematical model is developed for electro-osmotic peristaltic pumping of a non-Newtonian liquid in a deformable micro-channel. Stokes' couple stress fluid model is employed to represent realistic working liquids. The Poisson-Boltzmann equation for electric potential distribution is implemented owing to the presence of an electrical double layer (EDL) in the micro-channel. Using long wavelength, lubrication theory and Debye-Huckel approximations, the linearized transformed dimensionless boundary value problem is solved analytically. The influence of electro-osmotic parameter (inversely proportional to Debye length), maximum electro-osmotic velocity (a function of external applied electrical field) and couple stress parameter on axial velocity, volumetric flow rate, pressure gradient, local wall shear stress and stream function distributions is evaluated in detail with the aid of graphs. The Newtonian fluid case is retrieved as a special case with vanishing couple stress effects. With increasing the couple stress parameter there is a significant increase in the axial pressure gradient whereas the core axial velocity is reduced. An increase in the electro-osmotic parameter both induces flow acceleration in the core region (around the channel centreline) and it also enhances the axial pressure gradient substantially. The study is relevant in the simulation of novel smart bio-inspired space pumps, chromatography and medical micro-scale devices.

66 citations

Journal ArticleDOI
TL;DR: In this article, the influence of micropolar nature of fluids in fully developed flow induced by electrokinetically driven peristaltic pumping through a parallel plate microchannel is analyzed.
Abstract: An analysis is presented in this work to assess the influence of micropolar nature of fluids in fully developed flow induced by electrokinetically driven peristaltic pumping through a parallel plate microchannel. The walls of the channel are assumed as sinusoidal wavy to analyze the peristaltic flow nature. We consider that the wavelength of the wall motion is much larger as compared to the channel width to validate the lubrication theory. To simplify the Poisson Boltzmann equation, we also use the Debye-Huckel linearization (i.e. wall zeta potential ≤ 25mV). We consider governing equation for micropolar fluid in absence of body force and couple effects however external electric field is employed. The solutions for axial velocity, spin velocity, flow rate, pressure rise and stream functions subjected to given physical boundary conditions are computed. The effects of pertinent parameters like Debye length and Helmholtz-Smoluchowski velocity which characterize the EDL phenomenon and external electric field, coupling number and micropolar parameter which characterize the micropolar fluid behavior, on peristaltic pumping are discussed through the illustrations. The results show that peristaltic pumping may alter by applying external electric fields. This model can be used to design and engineer the peristalsis-lab-on-chip and micro peristaltic syringe pumps for biomedical applications.

51 citations

Journal ArticleDOI
TL;DR: The Jefferys' non-Newtonian constitutive model is employed to characterize rheological properties of the fluid and the influence of these parameters and also time on axial velocity, pressure difference, maximum volumetric flow rate and streamline distributions is visualized graphically and interpreted in detail.
Abstract: Analytical solutions are developed for the electro-kinetic flow of a viscoelastic biological liquid in a finite length cylindrical capillary geometry under peristaltic waves. The Jefferys' non-Newtonian constitutive model is employed to characterize rheological properties of the fluid. The unsteady conservation equations for mass and momentum with electro-kinetic and Darcian porous medium drag force terms are reduced to a system of steady linearized conservation equations in an axisymmetric coordinate system. The long wavelength, creeping (low Reynolds number) and Debye-Huckel linearization approximations are utilized. The resulting boundary value problem is shown to be controlled by a number of parameters including the electro-osmotic parameter, Helmholtz-Smoluchowski velocity (maximum electro-osmotic velocity), and Jefferys' first parameter (ratio of relaxation and retardation time), wave amplitude. The influence of these parameters and also time on axial velocity, pressure difference, maximum volumetric flow rate and streamline distributions (for elucidating trapping phenomena) is visualized graphically and interpreted in detail. Pressure difference magnitudes are enhanced consistently with both increasing electro-osmotic parameter and Helmholtz-Smoluchowski velocity, whereas they are only elevated with increasing Jefferys' first parameter for positive volumetric flow rates. Maximum time averaged flow rate is enhanced with increasing electro-osmotic parameter, Helmholtz-Smoluchowski velocity and Jefferys' first parameter. Axial flow is accelerated in the core (plug) region of the conduit with greater values of electro-osmotic parameter and Helmholtz-Smoluchowski velocity whereas it is significantly decelerated with increasing Jefferys' first parameter. The simulations find applications in electro-osmotic (EO) transport processes in capillary physiology and also bio-inspired EO pump devices in chemical and aerospace engineering.

49 citations

References
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Book
01 Jan 1981

2,324 citations


"Steady electro-osmotic flow of a mi..." refers background in this paper

  • ...10K3 V, which is about the upper limit for the Debye–Hückel approximation to be valid at approximately room temperature (Hunter 1988, p. 25; Li 2004, p. 19), but some results are also presented for higher magnitudes of jo in order to transcend the limitations of the Debye–Hückel approximation....

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Journal ArticleDOI
TL;DR: In this article, the authors present an approach for ODE's Phase Plane, Qualitative Methods, and Partial Differential Equations (PDE's) to solve ODE problems.
Abstract: PART A: ORDINARY DIFFERENTIAL EQUATIONS (ODE'S). Chapter 1. First-Order ODE's. Chapter 2. Second Order Linear ODE's. Chapter 3. Higher Order Linear ODE's. Chapter 4. Systems of ODE's Phase Plane, Qualitative Methods. Chapter 5. Series Solutions of ODE's Special Functions. Chapter 6. Laplace Transforms. PART B: LINEAR ALGEBRA, VECTOR CALCULUS. Chapter 7. Linear Algebra: Matrices, Vectors, Determinants: Linear Systems. Chapter 8. Linear Algebra: Matrix Eigenvalue Problems. Chapter 9. Vector Differential Calculus: Grad, Div, Curl. Chapter 10. Vector Integral Calculus: Integral Theorems. PART C: FOURIER ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS. Chapter 11. Fourier Series, Integrals, and Transforms. Chapter 12. Partial Differential Equations (PDE's). Chapter 13. Complex Numbers and Functions. Chapter 14. Complex Integration. Chapter 15. Power Series, Taylor Series. Chapter 16. Laurent Series: Residue Integration. Chapter 17. Conformal Mapping. Chapter 18. Complex Analysis and Potential Theory. PART E: NUMERICAL ANALYSIS SOFTWARE. Chapter 19. Numerics in General. Chapter 20. Numerical Linear Algebra. Chapter 21. Numerics for ODE's and PDE's. PART F: OPTIMIZATION, GRAPHS. Chapter 22. Unconstrained Optimization: Linear Programming. Chapter 23. Graphs, Combinatorial Optimization. PART G: PROBABILITY STATISTICS. Chapter 24. Data Analysis: Probability Theory. Chapter 25. Mathematical Statistics. Appendix 1: References. Appendix 2: Answers to Odd-Numbered Problems. Appendix 3: Auxiliary Material. Appendix 4: Additional Proofs. Appendix 5: Tables. Index.

2,257 citations

Book
01 Jan 1992
TL;DR: In this article, the Taylor series is used to model the wave equation and the Laplace equation in the context of linear algebraic equations, eigenproblems, polynomial approximation and interpolation, and difference formulas numerical integration.
Abstract: Part I Basic tools of numerical analysis: systems of linear algebraic equations eigenproblems solution of nonlinear equations polynomial approximation and interpolation numerical differention and difference formulas numerical integration. Part II Ordinary differential equations: solution of one-dimensional initial-value problems solution of one-dimensional boundary-value problems. Part III Partial differential equations: elliptic partial differential equations - the Laplace equation finite difference methods for propagation problems parabolic partial differential equations - the convection equation coordinate transformations and grid generation parabolic partial differential equations - the convection-diffusion equation hyperbolic partial differential equations - the wave equation. Appendix: the Taylor series.

1,202 citations

Book
01 Jan 1989
TL;DR: In this article, the authors introduce the Transport in Fluids Equations of Change (TUE) model for the transport of uncharged molecules and particles in a fluid and discuss its application in the field of particle capture.
Abstract: Preface to the Paperback Edition Preface to the Second Edition Preface to the First Edition Acknowledgments for the First Edition Introduction Transport in Fluids Equations of Change Solutions of Uncharged Molecules Solutions of Uncharged Macromolecules and Particles Solutions of Electrolytes Solutions of Charged Macromolecules and Particles Suspension Stability and Particle Capture Rheology and Concentrated Suspensions Surface Tension Appendix A SI Units and Physical Constants Appendix B Symbols Author Index Subject Index

1,062 citations


"Steady electro-osmotic flow of a mi..." refers background or methods in this paper

  • ...The boundary conditions on u(y) and j(y) are uðG1ÞZ 0 and jðG1ÞZ 1: ð2:19Þ In addition, the condition jð0ÞZ 0 ð2:20Þ is engendered by the assumption h[lD (Probstein 1989, p. 187)....

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  • ...…Sciences, Bahauddin Zakariya versity, Multan 60800, Pakistan (abidzero0@yahoo.co.uk). eived 2 September 2008 epted 30 September 2008 501 This journal is q 2008 The Royal Society the injection of detoxifying agents and the control of leakage at toxic-waste sites (Probstein 1989, p. 191; Keane 2003)....

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  • ...On setting cZ0, we get k1Zk2Zk4Z0, which implies from equation (3.15) that N(y)h0; simultaneously, from equation (3.17), we recover uðyÞZ 1KcoshðmoyÞ=coshðmoÞ ð3:18Þ for a simple Newtonian fluid (Probstein 1989)....

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  • ...…) Micropolar Helmholtz–Smoluchowski equation and velocity As the counterparts of the Helmholtz–Smoluchowski equation and the Helmholtz–Smoluchowski velocity for simple Newtonian fluids (Probstein 1989, p. 192) are not available for steady flows of micropolar fluids, let us derive both in this…...

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  • ...Setting cZ0 in equations (2.8) and (3.6), we revert to the Helmholtz–Smoluchowski velocity and equation, respectively, for simple Newtonian fluids (Probstein 1989, p. 192)....

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Book
01 Jan 1974
TL;DR: In this article, Stokes' potential flow is used to describe the potential potential of a fluid flow in the presence of a single source and a single sink, and the potential can be expressed as a function of the velocity potential of the potential flow.
Abstract: GOVERNING EQUATIONS Basic Conservation Laws . Statistical and Continuum Methods . Eulerian and Lagrangian Coordinates . Material Derivative . Control Volumes . Reynolds' Transport Theorem . Conservation of Mass . Conservation of Momentum . Conservation of Energy . Discussion of Conservation Equations . Rotation and Rate of Shear . Constitutive Equations . Viscosity Coefficients . Navier-Stokes Equations . Energy Equation . Governing Equations for Newtonian Fluids . Boundary Conditions Flow Kinematics . Flow Lines . Circulation and Vorticity . Stream Tubes and Vortex Tubes . Kinematics of Vortex Lines Special Forms of the Governing Equations . Kelvin's Theorem . Bernoulli Equation . Crocco's Equation . Vorticity Equation IDEAL-FLUID FLOW Two-Dimensional Potential Flows . Stream Function . Complex Potential and Complex Velocity . Uniform Flows . Source, Sink, and Vortex Flows . Flow in a Sector . Flow Around a Sharp Edge . Flow Due to a Doublet . Circular Cylinder Without Circulation . Circular Cylinder With Circulation . Blasius' Integral Law . Force and Moment on a Circular Cylinder . Conformal Transformations . Joukowski Transformation . Flow Around Ellipses . Kutta Condition and the Flat-Plate Airfoil . Symmetrical Joukowski Airfoil . Circular-Arc Airfoil . Joukowski Airfoil . Schwarz-Christoffel Transformation . Source in a Channel . Flow Through an Aperture . Flow Past a Vertical Flat Plate Three-Dimensional Potential Flows . Velocity Potential . Stokes' Stream Function . Solution of the Potential Equation . Uniform Flow . Source and Sink . Flow Due to a Doublet . Flow Near a Blunt Nose . Flow Around a Sphere . Line-Distributed Source . Sphere in the Flow Field of a Source . Rankine Solids . D'Alembert's Paradox . Forces Induced by Singularities . Kinetic Energy of a Moving Fluid . Apparent Mass Surface Waves . The General Surface-Wave Problem . Small-Amplitude Plane Waves . Propagation of Surface Waves . Effect of Surface Tension . Shallow-Liquid Waves of Arbitrary Form . Complex Potential for Traveling Waves . Particle Paths for Traveling Waves . Standing Waves . Particle Paths for Standing Waves . Waves in Rectangular Vessels . Waves in Cylindrical Vessels . Propagation of Waves at an Interface VISCOUS FLOWS OF INCOMPRESSIBLE FLUIDS Exact Solutions . Couette Flow . Poiseuille Flow . Flow Between Rotating Cylinders . Stokes' First Problem . Stokes' Second Problem . Pulsating Flow Between Parallel Surfaces . Stagnation-Point Flow . Flow in Convergent and Divergent Channels . Flow Over a Porous Wall Low-Reynolds-Number Solutions . The Stokes Approximation . Uniform Flow . Doublet . Rotlet . Stokeslet . Rotating Sphere in a Fluid . Uniform Flow Past a Sphere . Uniform Flow Past a Circular Cylinder . The Oseen Approximation Boundary Layers . Boundary-Layer Thickness . The Boundary-Layer Equations . Blasius Solution . Falkner-Skan Solutions . Flow Over a Wedge . Stagnation-Point Flow . Flow in a Convergent Channel . Approximate Solution for a Flat Surface . General Momentum Integral . Karman-Pohlhausen Approximation . Boundary-Layer Separation . Stability of Boundary Layers Buoyancy-Driven Flows . The Boussinesq Approximation . Thermal Convection . Boundary-Layer Approximations . Vertical Isothermal Surface . Line Source of Heat . Point Source of Heat . Stability of Horizontal Layers COMPRESSIBLE FLOW OF INVISCID FLUIDS Shock Waves . Propagation of Infinitesimal Disturbances . Propagation of Finite Disturbances . Rankine-Hugoniot Equations . Conditions for Normal Shock Waves . Normal Shock-Wave Equations . Oblique Shock Waves One-Dimensional Flows . Weak Waves . Weak Shock Tubes . Wall Reflection of Waves . Reflection and Refraction at an Interface . Piston Problem . Finite-Strength Shock Tubes . Nonadiabatic Flows . Isentropic-Flow Relations . Flow Through Nozzles Multi-Dimensional Flows . Irrotational Motion . Janzen-Rayleigh Expansion . Small-Perturbation Theory . Pressure Coefficient . Flow Over a Wave-Shaped Wall . Pandtl-Glauert Rule for Subsonic Flow . Ackert's Theory for Supersonic Flows . Prandtl-Meyer Flow Appendix A. Vector Analysis Appendix B. Tensors Appendix C. Governing Equations Appendix D. Complex Variables Appendix E. Thermodynamics Index

732 citations


"Steady electro-osmotic flow of a mi..." refers background in this paper

  • ...As the magnitude of the velocity gradient must be large near the walls owing to the no-slip boundary condition (2.19)1 (Currie 1974, p. 276), and because figure 3 indicates that the velocity gradient does have maximum magnitude at the walls, it is not surprising that the maximum value of js012ðyÞj…...

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