In this investigation, the transient electroosmotic flow of multi-layer immiscible viscoelastic fluids in a slit microchannel is studied. Through an appropriate combination of the momentum equation with the rheological model for Maxwell fluids, an hyperbolic partial differential equation is obtained and semi-analytically solved by using the Laplace transform method to describe the velocity field. In the solution process, different electrostatic conditions and electro-viscous stresses have to be considered in the liquid-liquid interfaces due to the transported fluids content buffer solutions based on symmetrical electrolytes. By adopting a dimensionless mathematical model for the governing and constitutive equations, certain dimensionless parameters that control the start-up of electroosmotic flow appear, as the viscosity ratios, dielectric permittivity ratios, the density ratios, the relaxation times, the electrokinetic parameters and the potential differences. In the results, it is shown that the velocity exhibits an oscillatory behavior in the transient regime as a consequence of the competition between the viscous and elastic forces; also, the flow field is affected by the electrostatic conditions at the liquid-liquid interfaces, producing steep velocity gradients, and finally, the time to reach the steady-state is strongly dependent on the relaxation times, viscosity ratios and the number of fluid layers.

This work investigates the transient multilayer electro-osmotic flow of viscoelastic fluids through an annular microchannel. The dimensionless mathematical model of multilayer flow is integrated by the linearized Poisson-Boltzmann equation, the Cauchy momentum equation, the rheological Maxwell model, initial conditions, and the electrostatic and hydrodynamic boundary conditions at liquid-liquid and solid-liquid interfaces. Although the main force that drives the movement of fluids is due to electrokinetic effects, a pressure gradient can also be added to the flow. The semi-analytical solution for the electric potential distribution and velocity profiles considers analytical techniques as the Laplace transform method, with numerical procedures using the inverse matrix method for linear algebraic equations and the concentrated matrix exponential method for the inversion of the Laplace transform. The results presented for velocity profiles and velocity tracking at the transient regime reveal an interesting oscillatory behavior that depends on elastic fluid properties via relaxation times. The time required for the flow to reach steady-state is highly dependent on the viscosity ratios and the dimensionless relaxation times. In addition, the influence of other dimensionless parameters on the flow as the electrokinetic parameters, zeta potentials at the walls, permittivity ratios, ratio of pressure forces to electro-osmotic forces, number of fluid layers, and annular thickness are investigated. The findings of this study have significant implications for the precise control of parallel fluid transport in microfluidic devices for flow-focusing applications.

Because the development of techniques for pumping parallel flows in miniaturized systems are required, in the present investigation, a semi-analytical solution based in the matrix inverse method and by Laplace transform for the transient flow of multi-layer immiscible fluids in a narrow capillary, under electroosmotic and pressure driven effects, is obtained. The dimensionless mathematical model to solve the electric potential distribution and the velocity field in the start-up of flow, consist on the Poisson-Boltzmann and momentum equations, respectively. Here, the transported fluids are considered symmetrical electrolytes and because the interfaces between them are polarizable and impermeable to charged particles, interesting interfacial effects appear on the velocity profiles when an external electric field is applied. The results show graphically the influence of the different dimensionless parameters involved in the dynamics of the fluid flow. This study demonstrates that by considering electrical interfacial effects, produce velocity jumps at liquid-liquid interfaces, whose magnitude and direction depend on the concentration and polarity of electric charges in those regions; finally, it is observed that the time to reach the steady-state regime of the fluid flow is only controlled by the dimensionless viscosity ratios. This investigation is a theoretical contribution to simulate transient multi-layer fluid flows under electric interfacial effects, covering different implications that emerge in the design of small devices into the chemical, biological and clinical areas.

The present investigation analyzes the transient multilayer electro-osmotic flow through an annular microchannel with hydrophobic walls. The fluids are considered immiscible and viscoelastic, following the Maxwell rheological model. In the problem examined, the linearized Poisson–Boltzmann and Cauchy momentum equations are used to determine the electric potential distribution and the flow field, respectively. Here, different interfacial phenomena are studied through the imposed boundary conditions, such as the hydrodynamic slip and specified zeta potentials at solid–liquid interfaces, the velocity continuity, the electroviscous stresses balance, the potential difference, and the continuity of electrical displacements at the interfaces between fluids. The semi-analytic solution uses the Laplace transform theory. In the results, the velocity profiles and velocity tracking show the oscillatory behavior of flow, which strongly depends on the dimensionless relaxation time. Furthermore, the hydrodynamic slip on the channel walls contributes to the release of energy stored in the fluids due to elastic effects at the start-up of the flow. Similarly, other dimensionless parameters are also investigated. This research aims to predict the parallel flow behavior in microfluidic devices under electro-osmotic effects.

The present work analyzes the combined viscoelectric and steric effects on electroosmotic flow in a soft channel with polyelectrolyte coating. The structured channel surface, which controls the electric potential, creates two different flow regions: the electrolyte flow within the permeable polyelectrolyte layer (PEL) and the bulk electrolyte. Thus, this study discusses the interaction of various electrostatic effects to predict the electroosmotic flow field. The nonlinear governing equations describing the fluid flow are the modified Poisson–Boltzmann equation for the electric potential distribution, the mass conservation equation, and the modified Navier–Stokes equations for the flow field, which are solved numerically using a one-dimensional (1D) scheme. The results indicate that the flow enhances when increasing the electric potential magnitude across the channel cross-section via the rise in different dimensionless parameters, such as the PEL thickness, the steric factor, and the ratio of the electrokinetic parameter of the PEL to that of the electrolyte layer. This research demonstrates that the PEL significantly enhances control over electroosmotic flow. However, it is crucial to consider that viscoelectric effects at high electric fields and the friction generated by the grafted polymer brushes of the PEL can reduce these benefits.

This paper presents the analytical solution of a combined electroosmotic and pressure driven flow of multilayer immiscible fluids in a narrow capillary. The mathematical model is based in the Poisson-Boltzmann equation and the modified Navier-Stokes equations. In the steady-state analysis, we consider different conditions at the interfaces between the fluids as potential differences, surface charge densities and electro-viscous stresses balances, which are discussed in detail. Results show the transport phenomena coupled in the description of velocity distribution, by the analyzing of the dimensionless parameters obtained, such as: potential differences, surface charge densities, electrokinetic parameters, term involving the external pressure gradient, ratios of viscosity and of dielectric permittivity. Here, the presence of a net electric charges balance at the interfaces breaks the continuity of the electric potential distributions and viscous shear stresses, modifying the flow field; thus, the electrical conditions established at the interfaces play an important role on the flow behavior. The present work, in which the velocity field is described, aims to be an important contribution in the development of theoretical models that provide a better understanding about labs-on-a-chip design.