Transient electro-osmotic flow of generalized Maxwell fluids in a straight pipe of circular cross section

Open Physics ◽  
2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Shaowei Wang ◽  
Moli Zhao ◽  
Xicheng Li
Keyword(s):  
Fractional Derivative ◽  
Integral Transform ◽  
Capillary Tube ◽  
Circular Cross Section ◽  
Maxwell Fluid ◽  
Navier Stokes Equation ◽  
Transform Method ◽  
Osmotic Flow ◽  
Poisson Boltzmann Equation ◽  
Electro Osmotic Flow

AbstractThe transient electro-osmotic flow of a generalized Maxwell fluid with fractional derivative in a narrow capillary tube is examined. With the help of an integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson-Boltzmann equation and the Navier-Stokes equation. It was shown that the distribution and establishment of the velocity consists of two parts, the steady part and the unsteady one. The effects of relaxation time, fractional derivative parameter, and the Debye-Hückel parameter on the generation of flow are shown graphically and analyzed numerically. The velocity overshoot and oscillation are observed and discussed.

Exact Solutions of Electro-Osmotic Flow of Generalized Second-Grade Fluid with Fractional Derivative in a Straight Pipe of Circular Cross Section

2014 ◽  
Vol 69 (12) ◽  
pp. 697-704 ◽  
Author(s):  
Shaowei Wang ◽  
Moli Zhao ◽  
Xicheng Li ◽  
Xi Chen ◽  
Yanhui Ge
Keyword(s):  
Fractional Derivative ◽  
Integral Transform ◽  
Capillary Tube ◽  
Circular Cross Section ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Osmotic Flow ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid ◽  
Electro Osmotic Flow

AbstractThe transient electro-osmotic flow of generalized second-grade fluid with fractional derivative in a narrow capillary tube is examined. With the help of the integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson-Boltzmann equation and the Navier-Stokes equation. It was shown that the distribution and establishment of the velocity consists of two parts, the steady part and the unsteady one. The effects of retardation time, fractional derivative parameter, and the Debye-Hückel parameter on the generation of flow are shown graphically.

scholarly journals Electro-osmotic flow of a third-grade fluid past a channel having stretching walls

2019 ◽  
Vol 8 (1) ◽  
pp. 56-64 ◽  
Author(s):  
Mamata Parida ◽  
Sudarsan Padhy
Keyword(s):  
Differential Equation ◽  
Boltzmann Equation ◽  
Linear Partial Differential Equation ◽  
Third Grade ◽  
Third Grade Fluid ◽  
Osmotic Flow ◽  
Grade Fluid ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Electro Osmotic Flow

Abstract The electro-osmotic flow of a third grade fluid past a channel having stretching walls has been studied in this paper. The channel height is taken much greater than the thickness of the electric double layer comprising of the Stern and diffuse layers. The equations governing the flow are obtained from continuity equation, the Cauchy’s momentum equation and the Poisson-Boltzmann equation. The Debye-Hückel approximation is adopted to linearize the Poisson-Boltzmann equation. Suitable similarity transformations are used to reduce the resulting non-linear partial differential equation to ordinary differential equation. The reduced equation is solved numerically using damped Newton’s method. The results computed are presented in form of graphs.

scholarly journals Analysis of damped vibrations of thin bodies embedded into a fractional derivative viscoelastic medium

2013 ◽  
Vol 21 (5-6) ◽  
pp. 155-159
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Viscoelastic Medium ◽  
Integral Transform ◽  
Viscous Friction ◽  
Middle Surface ◽  
Thin Body ◽  
Governing Equations ◽  
Transform Method ◽  
External Forces ◽  
The Given

AbstractDamped vibrations of elastic thin bodies, such as plates and circular cylindrical shells, embedded into a viscoelastic medium, the rheological features of which are described by fractional derivatives, are considered in the present article. Besides the forces of viscous friction, a thin body is subjected to the action of external forces dependent on the coordinates of the middle surface and time. The boundary conditions are proposed in such a way that the governing equations allow the Navier-type solution. The Laplace integral transform method and the method of expansion of all functions entering into the set of governing equations in terms of the eigenfunctions of the given problem are used as the methods of solution. It is shown that as a result of such a procedure, the systems of equations in the generalized coordinates could be reduced to infinite sets of uncoupled equations, each of which describes damped vibrations of a mechanical oscillator based on the fractional derivative Kelvin-Voigt model.

scholarly journals An Investigation of Fractional Bagley–Torvik Equation

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 28 ◽  
Author(s):  
Azhar Ali Zafar ◽  
Grzegorz Kudra ◽  
Jan Awrejcewicz
Keyword(s):  
Fractional Derivative ◽  
Mechanical System ◽  
Caputo Fractional Derivative ◽  
Integral Transform ◽  
Rolling Bearing ◽  
Degree Of Freedom ◽  
Free Oscillations ◽  
Derivative Operator ◽  
Transform Method ◽  
Integral Transform Method

In this article, we will solve the Bagley–Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.

Transient Electroosmotic Flow in Slit Microchannel Containing Salt-Free Medium

2009 ◽  
Author(s):  
Shih-Hsiang Chang
Keyword(s):  
Surface Charge Density ◽  
Electroosmotic Flow ◽  
Theoretical Study ◽  
Capillary Tube ◽  
Navier Stokes ◽  
Free Medium ◽  
Navier Stokes Equation ◽  
Poisson Boltzmann ◽  
Flow Through ◽  
Poisson Boltzmann Equation

A theoretical study on the transient electroosmotic flow through a slit microchannel containing a salt-free medium is presented for both constant surface charge density and constant surface potential. The exact analytical solutions for the electric potential distribution and the transient electroosmotic flow velocity are derived by solving the nonlinear Poisson-Boltzmann equation and the Navier-Stokes equation. Based on these results, a systematic parametric study on the characteristics of the transient electroosmotic flow is detailed. The general behavior of electroosmotic flow in a planar slit is similar to that in a capillary tube; however, the rate of evolution of the flow in a tube with time is faster by a factor of about 2.4 than that in a slit with its width equal to the tube diameter.

A numerical investigation of transient electro-osmotic flow in rectangular microchannels: A comparison of different models

2010 ◽  
Vol 224 (8) ◽  
pp. 1655-1663 ◽  
Author(s):  
R Kamali ◽  
M Eslami
Keyword(s):  
Zeta Potential ◽  
Transition Period ◽  
Effective Parameters ◽  
Start Time ◽  
Rectangular Microchannels ◽  
Steady State Condition ◽  
Osmotic Flow ◽  
Aspect Ratios ◽  
Poisson Boltzmann Equation ◽  
Electro Osmotic Flow

Transient electro-osmotic flow in rectangular microchannels is investigated numerically in this article. The complete Poisson—Boltzmann equation along with the time-dependent momentum equation is solved using the finite-difference method. Moreover, linearized equations based on the Debye—Huckle assumption are also solved to compare with the available analytical approximate solutions. The effects of different parameters such as wall zeta potential, non-dimensional electrokinetic width, and channel aspect ratio are also studied. It is shown that the Debye—Huckle approximation is not only valid for small values of zeta potential, but also the channel hydraulic diameter should be large enough with respect to electrical double layer (EDL) thickness. In addition, the flow behaviour at higher values of zeta potential is shown to be completely different from what available analytical solutions predict. Effective parameters on the transition period from the start time to the steady-state condition are also discussed. On the other hand, a comparison between the present numerical solution and the results of slip velocity approximation reveals that the slip model could be only used for very large values of non-dimensional electro-kinetic width. Finally, velocity distributions in channels of different aspect ratios are provided and discussed.

Exact solution of an electro-osmotic flow problem in a cylindrical channel of polymer electrolyte membranes

2009 ◽  
Vol 465 (2109) ◽  
pp. 2663-2679 ◽  
Author(s):  
Peter Berg ◽  
Kehinde Ladipo
Keyword(s):  
Polymer Electrolyte ◽  
Cylindrical Channel ◽  
Polymer Electrolyte Membranes ◽  
Osmotic Flow ◽  
Counter Ions ◽  
Electrolyte Membranes ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Analytical Expressions ◽  
Electro Osmotic Flow

The electric potential of counter-ions (protons) in an infinite cylindrical channel is presented as a solution of the Poisson–Boltzmann equation, involving a constant ion charge density along the wall. The distribution of protons is derived and used subsequently to compute the velocity profile and mass flow rate of the corresponding electro-osmotic flow, driven by an electric field. Analytical expressions are derived for all quantities, including the conductivity and water drag coefficient. This analysis relates especially to cylindrical nano-channels of polymer electrolyte membranes such as Nafion and addresses the validity of continuum models for these materials.

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