Helmholtz Smoluchowski Research Articles

Interface conditions for a polyelectrolyte gel in a salt solution with a thin electrical double layer

Polyelectrolyte gels are electro-active soft materials that are often surrounded by a viscous salt solution. The formation of an electrical double layer (EDL) at the interface between the gel and the solution can give rise to fluid-structure interactions driven by Coulomb forces. However, these interactions are usually neglected in modelling studies due to the difficulty of resolving the EDL, which has a thickness that is orders of magnitude smaller than the characteristic gel size. In this paper, an asymptotically consistent model of a polyelectrolyte gel that is surrounded by a salt solution is derived in the thin-EDL limit. The model consists of bulk equations describing an electrically neutral gel and bath, along with interfacial conditions that capture the electro-mechanical impact of the EDL. A key result is the derivation of a Helmholtz–Smoluchowski slip condition that describes how shear stresses in the EDL lead to a bulk flow in the surrounding salt solution. The impact of this flow on gel-gel friction and swelling-driven instabilities is discussed. The model derived here will lead to a better understanding of how polyelectrolyte gels interact with an external salt solution, which is crucial for applications.

Mathematical modeling of electroosmotically driven peristaltic propulsion due to transverse deflections of two periodically deformable curved tubes of unequal wavelengths

The present study aims to investigate the viscid fluid propulsion due to the electroosmosis and transverse deflections of the sinusoidally deformable tubes of unequal wavelengths in the presence of electro-kinetic forces. This situation is estimated from the physical model of physiological fluid flow through a tubular structure in which an artificial flexible tube is being inserted. In this model, both peristaltically deforming tubes are taken in a curved configuration. The flow-governing momentum equations are simplified by the approximation of the long wavelength as compared to the outer tube's radius, whereas the Debye–Hückel approximation is used to simplify the equations that govern the electric potential distribution. Here, the authors have used the DSolve command in the scientific computing software MATHEMATICA 14 to obtain the expressions for electric potential and axial velocity of viscid fluid. In this work, the authors have analyzed the impact of various controlling parameters, such as the electro-physical parameters, curvature parameter, radius ratio, wavelength ratio, and amplitude ratios, on the various flow quantities graphically during the transport of viscid fluid through a curved endoscope. Here, contour plots are also drawn to visualize the streamlines and to observe the impacts of the control parameters on fluid trapping. During the analysis of the results, a noteworthy outcome extracted from the present model is that an increment in electro-physical parameters, such as Helmholtz–Smoluchowski velocity and the Debye–Hückel parameter, are responsible for enhancement in the shear stress at the inner tube's wall and the axial velocity under the influence of electro-kinetic forces. This is because of the electric double layer (EDL) thickness, which gets reduced on strengthening the Debye–Hückel parameter. This reduced EDL thickness is responsible for the enhancement in the axial velocity of the transporting viscid fluid. The present model also suggests that the axial velocity of viscid fluid can be reduced by enhancing the ratio of wavelengths of waves that travel down the walls of the outer curved tube and the inner curved tube. The above-mentioned results can play a significant role in developing and advancing the endoscopes that will be useful in many biomedical processes, such as gastroscopy, colonoscopy, and laparoscopy.

Electro osmotic flow of nanofluids within a porous symmetric tapered ciliated channel

AbstractIn this investigation, a comprehensive study has been made to reveal electro osmosis flow through a tapered ciliated symmetric porous channel. The flow is initiated due to metachronal dynamics of cilia. Axial electric field is deployed and thermal radiation phenomenon is scrutinised by applying convective conditions. The equations tackling the flow are non dimensionalized and simplified by capitalizing the low Reynolds number and long wave length approximations. Analytical solution is presented for well reputed Poissson equation and the axial velocity. Whereas, traverse velocity, temperature and nanofluids concentration profiles are examined numerically in MATHEMATICA. Variation of emerging crucial parameters on the velocity profile, temperature and concentration profiles, pressure gradient, pressure rise per wavelength, and on the velocity distribution inside the micro ciliated are exhibited with the aid of graphical deliberations. It worth to mention in this work that in case of tapered channel transverse velocity also has significant contribution in the flow, which is observed trivial in symmetric and non‐symmetric channel flows. Temperature of the nanofluid in the ciliated tapered channel is raised with permeability and thermal radiations phenomena and can be controlled with Helmholtz Smoluchowski velocity, and electroosmotic parameter. Pumping phenomena is affected with increase in Helmholtz Smoluchowski velocity and permeability. Reported investigation cover a informative insight about biological fluid system, may be beneficial for the understanding the flow through ductus efferentes of human reproductive tract since it assumed that cilia are responsible for the transport of sperm from rete testis to the epididymis, also have worth in cilia designed bio‐sensors and in certain drug delivery systems.

Generalizing electroosmotic-flow predictions over charge-modulated periodic topographies: tuneable far-field effects

The classical Helmholtz–Smoluchowski (HS) model of electroosmosis holds for homogeneously charged interfaces in contact with a fluid layer bearing an equal and opposite net charge. However, inhomogeneities in the surface charge and topography are inevitable, either as practical materials and fabrication artefacts, or at times as deliberately introduced modulations for flow control. In an effort to arrive at an analytically tractable theoretical framework for addressing the underlying electro-mechanical coupling, here, we generalize the traditional HS theory to an extent where both the surface charge and topographies may bear arbitrary and independent periodic forms. Using a spectral-asymptotic approach, we further arrive at closed-form expressions for describing the resulting electroosmotic pumping for topographic features with small characteristic amplitude to pattern period ratio, as relevant for most practical scenarios. We subsequently execute full-scale numerical simulations without any restrictions on the surface charge and topography variations to assess the efficacy of the theoretical framework. The corresponding test beds include distinctive signature patterns – for example, a square-wave surface charge distribution on trapezoidal pit topographies. Our results reveal that the charge–topography interplay induces an anisotropic flow drift, deviating from the classical HS paradigm. This, in turn, provides new quantitative insights into highly selective electroosmotic flow control via judicious design of the charge and topographical patterns, resulting in controllable accentuation, attenuation, nullification, deflection and even complete reversal of the flow. Our analysis further establishes a provision of estimating the zeta potentials of naturally ‘contaminated’ surfaces, as well as explaining the electrophoresis of large inhomogeneous particles; a paradigm that remained to be explored thus far.

ELECTRO-OSMOTIC EFFECT ON PERISTALTIC FLOW OF PHAN–THIEN–TANNER FLUID IN A PLANAR CHANNEL

Abstract There are several factors that can cause the excessive accumulation of biofluid in human tissue, such as pregnancy, local traumas, allergic responses or the use of certain therapeutic medications. This study aims to further investigate the shear-dependent peristaltic flow of Phan–Thien–Tanner (PTT) fluid within a planar channel by incorporating the phenomenon of electro-osmosis. This research is driven by the potential biomedical applications of this knowledge. The non-Newtonian fluid features of the PTT fluid model are considered as physiological fluid in a symmetric planar channel. This study is significant, as it demonstrates that the chyme in the small intestine can be modelled as a PTT fluid. The governing equations for the flow of the ionic liquid, thermal radiation and heat transfer, along with the Poisson–Boltzmann equation within the electrical double layer, are discussed. The long-wavelength ( $\delta \ll 1$ ) and low-Reynolds-number approximations ( $Re \to 0$ ) are used to simplify the simultaneous equations. The solutions analyse the Debye electronic length parameter, Helmholtz–Smoluchowski velocity, Prandtl number and thermal radiation. Additionally, streamlines are used to examine the phenomenon of entrapment. Graphs are used to explain the influence of different parameters on the flow and temperature. The findings of the current model have practical implications in the design of microfluidic devices for different particle transport phenomena at the micro level. Additionally, the noteworthy results highlight the advantages of electro-osmosis in controlling both flow and heat transfer. Ultimately, our objective is to use these findings as a guide for the advancement of lab-on-a-chip systems.

Numerical investigation of electroosmotic driven transport of MHD Casson fluid over an exponential stretching sheet using quartic B-spline method

The stretching surface flow phenomenon plays novel significance in the differential industrial sectors like manufacturing processes, food processing, polymers, petroleum engineering, glass fiber production, metal spinning, and many others. This research focuses on investigating the flow of electroosmotic-driven transport of magnetic Casson fluid over an exponential stretching sheet. The study employs mathematical modeling based on partial differential equations which are further truncated into ordinary system under the implementation of dimensionless variables. The quartic B-spline method is applied for obtaining solutions. Key findings indicate that electroosmotic and Helmholtz–Smoluchowski effects enhance fluid velocity, while magnetic forces reduce velocity. Additionally, an increase in the Hartmann parameter leads to a decrease in the skin friction coefficient.

Complex cilia-generated flow of hybrid nanofluid with electroosmosis, viscous dissipation and slippage

Complex cilia-generated flow of hybrid nanofluid with electroosmosis, viscous dissipation and slippage

Modulated complexed stenosed region consequences under the electroosmotic stimulation

The present study analyzes the theoretical consequences of slip effects in a complex stenosed region. The flow of blood in a stenosed region is incorporated with hybrid nanofluid features which are being prepared with copper and copper oxide nanoparticles. The flow is also intensified by applying an electric field in the axial direction. The governing equations for the proposed paradigm are solved and the corresponding closed-form solutions are obtained for the cases of mild stenosis. Parameters such as Electro-osmotic, velocity slip and Helmholtz–Smoluchowski are specially focused in this study. The heat transfer, hemodynamic velocity, wall shear stress and resistance impedance for the flow are precisely determined. The various parameters that influence the physical characteristics of flow are plotted, and their effects are discussed in detail. The present model has the potential application in medical pumps for drug delivery systems.

Numerical treatment and global error estimation for thermal electro-osmosis effect on non-Newtonian nanofluid flow with time periodic variations

The essential purpose of this study is to discuss the impact of time-periodic variations on mixed convection heat transfer for MHD Eyring-Powell nanofluid. The fluid flows through a non-Darcy porous medium over an infinite vertical plate. The effects of viscous dissipation, Ohmic dissipation, electro-osmosis force, heat source, thermal radiation, Dufour feature, and chemical reaction are presumed. The system of partial differential equations which governs the problem is transformed into a system of non-linear algebraic equations and then an explicit finite difference approach is espoused to solve these nonlinear algebraic equations. The numerical results for the velocity, temperature, and nanoparticles concentration distributions are computed and displayed through a set of graphs. Also, the skin friction coefficient, reduced Nusselt number, and Sherwood number are computed numerically for various values of the physical parameters. It is found that the velocity becomes greater with an elevation in the value of the Helmholtz–Smoluchowski velocity. Meanwhile, it enlarges with rising in the value of the electro-osmotic parameter. The rise in the value of the thermal radiation parameter causes a dwindling influence on both temperature and nanoparticles concentration. Investigations of these effects together are very useful due to their important vital applications in various scientific fields, especially in medicine and medical industries, such as endoscopes, respirators, and diverse medical implementations, as nanoparticles can be utilized in the remedy of cancer tumors. Additionally, electroosmotic flow is important due to its ability to control fluid movement and enhance mass transport, making it valuable in various application such as sample separation, drug delivery, and DNA analysis, offering enhanced efficiency and sensitivity.

A multiphase ciliary flow of Casson fluid in a porous channel under the effects of electroosmosis and MHD: Exact solutions

A multiphase ciliary flow of Casson fluid in a porous channel under the effects of electroosmosis and MHD: Exact solutions

Modified Darcy’s law and couple stress effects on electro-osmotic flow of non-Newtonian nanofluid with peristalsis

In this study, we focused on the heat transfer through a uniformly inclined rectangular duct caused by the electro-osmotic peristaltic flow of an unsteady non-Newtonian nanofluid. With couple stress, the fluid obeys the Papanastasiou model. The flow is through a porous medium that follows Darcy’s law in a modified form. In addition, Dufour and Soret effects, mixed convection, the impacts of a chemical reaction, and the effects of viscous couple stress dissipation are all considered. The governing equations that explain the velocity, temperature, and concentration of nanoparticles are simplified when wave transformation is used. The homotopy perturbation method was used to solve these equations analytically. Additionally, a collection of figures is used to discuss and visually illustrate the consequences of the physical characteristics. In fact, the modified Darcy’s law makes the velocity gradient appear in the momentum equation, which increases the contribution of the velocity gradient to the velocity profile. In addition, the electro-osmotic parameter and Helmholtz-Smoluchowski velocity have a significant impact on the velocity gradient’s direction, as well as the velocity gradient’s ability to be either positive or negative, depending on their values. In addition, in the case of forced convection, the values of the Nusselt number and the Sherwood number are highly affected by the value of Helmholtz–Smoluchowski velocity. The current findings have applications in biology and medicine, particularly in cancer therapy, which involves peristaltic blood pumps(arteries) and suspended gold nanoparticles (nanofluid). According to our knowledge, no prior studies have merged the couple stress Papanastasiou model and the modified Darcy’s law.

Heat and mass transfer analysis of non-Newtonian radiative nanofluid flow driven by the combined action of peristalsis and electroosmosis

The present numerical investigation uncovers the flow, heat, and mass transfer of ethylene glycol-based nanofluid (BN-EG) led by the combined impacts of peristaltic pumping and electroosmosis. The thermal conductivity of carrier liquid is deemed to change with temperature. Further, the features of electric field, Ohmic heating, radiative heat flux, mixed convection, and magnetic field are taken. Mathematical modeling is conducted under the assumption of lubrication theory. The nonlinear-coupled equations are addressed numerically by implementing Shooting method. The effective results of all physical constraints linked with the flow model are vigilantly studied and highlighted through various curves. The observations obtained from current analysis reveal that temperature augments with higher electroosmotic parameter. However, the Joule heating parameter increases the rate of heat transmission, while a lower rate of heat transfer is perceived for the radiation parameter. An increment in Helmholtz–Smoluchowski velocity increases the velocity profile. Pressure gradient rises in a forward way when electrical field favors peristaltic flow. Concentration of nanomaterial considerably declines when concentration Grashoff number is assigned higher values.

Towards an approximate solution of highly viscous electro-osmotic flows in inclined Channel: Applications in petroleum and gas engineering

Electro-osmotic flows (EoF) through an inclined channel are investigated in this article. Differential type non-Newtonian fluid model has been used to form highly viscous bi-phase fluid flows. Two different kinds of metallic particles are considered for disperse phase. The application of external electric fields and gravitational force are the main sources of multiphase flows under the magnetic environment. Nonlinear differential equations are formulated taking the stress tensor of fourth-grade fluid, Poisson equation, and Debye length approximation into account, respectively. An approximate solution is obtained for the nonlinear flow problems. Comparative analyses have also been performed with the previous investigations via tables and graphs and, were found to be in great agreement. A comprehensive parametric study is carried out, as well, which reveals that there is a definite difference between the momentum of the fluid phase and the particles phase. The contribution of the Helmholtz–Smoluchowski parameter in each phase is quite supportive. However, the effects of the transverse magnetic field and electro-osmotic parameter are in full coherence. Moreover, traditional multiphase flows can be derived for the limiting case. Finally, this theoretical study explicitly describes the applications in petroleum and gas engineering.

Electroosmosis oriented flow of Jeffrey viscoelastic model through scraped surface heat exchanger

Scraped surface heat exchanger (SSHE) is vibrantly utilized in various industries. The heat exchanger devices play pivotal role in the energy transfer phenomenon between the fluids, and it improves the efficient use of energy. The steady isothermal flow of a rheological aqueous ionic solution in a narrow gap scraped surface heat exchanger is explored. The mathematical model is established with the aid of lubrication approximation theory and extracted analytical solution. Exact analytical expressions are gathered for velocities, volume fluxes, pressure gradients, pressure at the edges of the blades and stream functions in the various stations of SSHE. Strength of significant flow parameters on the velocities, volume fluxes, pressure gradients, pressure at the edges of the blades and stream functions is revealed with aid of various plots. Significant enhancement in the velocity profile is observed in main domain of the flow with increasing ratio of relaxation to retardation time parameter and Helmholtz Smoluchowski velocity. While, it is observed that electro-osmotic parameter creates impediments to the flow.

Computational and theoretical model of electro-osmotic flow pumping in a microchannel with squeezing walls

Numerical simulation and theoretical solution for the electro-osmotic pumping flow of electrolyte solution in a microchannel with squeezing and charged walls are developed in this study. The mathematical model is derived based on using a strong coupling between the nonlinear Poisson–Boltzmann equation and the flow lubrication theory. The governing equations are integrated numerically using the finite difference method. Moreover, an analytical solution to the problem is also obtained using the lubrication theory and is used to solve the Poisson–Boltzmann equation without any approximation technique. The effects of various parameters such as the wall zeta potential, Debye length, and electric field on the fluid pressure distribution, velocity field, and the net flow rate are investigated in detail. The results show that the induced pumping rate depends strongly on the combined effects of the Helmholtz–Smoluchowski, zeta potential, and electrical double layer. Moreover, the produced net flow directionality can be controlled efficiently by manipulating the Helmholtz–Smoluchowski and/or the wall zeta potential. The results obtained from the numerical simulation are then compared with the theoretical analysis and have shown to be in agreement with the proposed mathematical model.

Effect of Debye length scale surface features on electro-osmosis and its use to devise a novel electro-microfluidic separation

Numerically validated analytical predictions for electro-osmosis over a charged surface decorated with a nanoscale groove pattern are developed for the situation when the electrical double layer thickness is comparable to the spatial period of the grooves. For the analytical predictions, the groove shape can be specified by any continuous periodic function, such as the triangular, trapezoidal, and sinusoidal waveforms, which are investigated as special cases. We discover that the classical Helmholtz–Smoluchowski expression for electrokinetic mobility, notwithstanding its widespread use in measurements, is rendered invalid by the presence of Debye-length-scale unevenness in the surface topography. Furthermore, we use the depth-resolved anisotropic response of oblique grooves to design and optimize a novel electro-microfluidic strategy for separating constituents of a nano-particulate mixture.

Hall currents and EDL effects on multiphase wavy flow of Carreau fluid in a microchannel having oscillating walls: A numerical study

In this analysis, the authors reveal the effects of electro-osmosis on the multiphase flow of Carreau fluid in a microchannel in the presence of Hall currents and solid particles. Moreover, the compliant channel walls are subject to oscillation occurring at the surface. To investigate the problem quantitatively, mathematical models for fluid phase and particulate phase have been structured. A lubrication approach is adopted due to laminar flow and the small dimensions of the channel. To produce the data, a system of differential equations is produced with the help of a numerical process performed on Mathematica through a built-in NDSolve tool. The results are presented graphically to examine the effects of various physical factors on the flow quantities. From pictorial discussion, it is gathered that the Helmholtz–Smoluchowski velocity parameter and the presence of an increasing amount of solid particles increasing the heat exchange while producing electro-kinetic energy. It is also found that velocity is a direct function of solid particles and compliant walls, but an inverse link is seen in the presence of electro-kinetic energy. Such studies can be employed with microfluidic devices and \may also be productive in medical and mechanical research.

Electroosmosis Augmented MHD Third-Grade Fluid with Slip and Variable Properties: An Application for Blood Flow in Arteries

The electroosmotic force effect on the peristaltic motion of the third-grade fluid is considered in a uniform channel. The governing equations that supplement the flow are designed for long wavelengths and low Reynolds numbers. Solutions are obtained for velocity, temperature, concentration, and trapping by considering the variable liquid properties for analyzing the various parameter effects. These effects are depicted through graphs and the relevance is discussed. The variable fluid properties have a declining impact on the velocity and temperature fields. Increasing the Helmholtz–Smoluchowski velocity values decreases the velocity field. Temperature decreases as the Deborah number increases. The velocity slip characteristics rise, and the trapping bolus’s size shrinks. The results of this paper may be beneficial in understanding the control of microvascular transport in the time of fractionation of blood into plasma and erythrocytes.

EMHD ciliated pumping of viscoelastic liquid in a complex convergent/divergent channel

Because of the diverse applications of cilia propulsions and viscoelastic fluids in biochemical engineering, we consider the flow of a non-Newtonian fluid through both convergent and divergent channels. The transportation of fluid through the flow geometry takes place due to metachronal waves that exist at both walls. Expressions of the Jeffrey fluid in the presence of magnetic and electric fields are used in the development of rheological equations. The mathematical analysis is carried out under the valid physiological assumption of creeping phenomena and long-wavelength approximation in the wave frames. The solutions of rheological equations are obtained analytically with an integration scheme. The transport features are discussed for numerous sundry variables via the Mathematica Software 11.0. Comparative studies among viscous and viscoelastic liquids also argued for both complex convergent and divergent channels. The special nature of metachronal waves (i.e. wavy pattern) is used in both boundary walls. The viscoelastic and magnetic parameters have played an important role to boost of the velocity profile magnitude. The impacts of the cilia length and Helmholtz–Smoluchowski velocity parameters have a significant act in the augmentation of pressure gradient and flow rate under the influence of magnetic and electric fields (EMHD phenomena) for the divergent channel. The current mathematical formulation gives information to the medical students about how to handle the transportation of viscoelastic fluids in both convergent and divergent channels via ciliated walls and electro-magnetic devices. The consequences of the present analysis have remarkable applications in micro-ciliated pumps, blood pumping in uniform and non-uniform arteries, and the rheology of viscoelastic liquids in different chemical processes.

Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry- Breaking of a Polarizable Dimer in Non-Uniform Fields.

A theoretical framework is presented for calculating the polarization, electro-rotation, travelling-wave dielectrophoresis, electro-hydrodynamics and induced-charge electroosmotic flow fields around a freely suspended conducting dimer (two touching spheres) exposed to non-uniform direct current (DC) or alternating current (AC) electric fields. The analysis is based on employing the classical (linearized) Poisson–Nernst–Planck (PNP) formulation under the standard linearized ‘weak-field’ assumption and using the tangent-sphere coordinate system. Explicit expressions are first derived for the axisymmetric AC electric potential governed by the Robin (mixed) boundary condition applied on the dimer surface depending on the resistance–capacitance circuit (RC) forcing frequency. Dimer electro-rotation due to two orthogonal (out-of-phase) uniform AC fields and the corresponding mobility problem of a polarizable dimer exposed to a travelling-wave electric excitation are also analyzed. We present an explicit solution for the non-linear induced-charge electroosmotic (ICEO) flow problem of a free polarized dimer in terms of the corresponding Stokes stream function determined by the Helmholtz–Smoluchowski velocity slip. Next, we demonstrate how the same framework can be used to obtain an exact solution for the electro-hydrodynamic (EHD) problem of a polarizable sphere lying next to a conducting planar electrode. Finally, we present a new solution for the induced-charge mobility of a Janus dimer composed of two fused spherical colloids, one perfectly conducting and one dielectrically coated. So far, most of the available electrokinetic theoretical studies involving polarizable nano/micro shapes dealt with convex configurations (e.g., spheres, spheroids, ellipsoids) and as such the newly obtained electrostatic AC solution for a dimer provides a useful extension for similar concave colloids and engineered particles.