{'fraction_non_alphanumeric': 0.026516789162703558, 'fraction_numerical': 0.011384925781812941, 'mean_word_length': 4.995248380129589, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Electrothermally Modulated Dispersion in Viscoelastic Fluids on Patterned Surfaces

Augmenting the dispersion of a solute species remains a challenging topic in electroosmotically-actuated flows due to the inherent plug-like feature of its velocity profile. In this study, we improvise on electrothermal flow of viscoelastic fluids through a parallel-plate microchannel with patterned surface charge to depict augmented dispersion coefficient of a solute. Here we consider the physical properties of the fluid to be temperature-dependent, thus, bringing out the combinatorial effect of both the thermal perturbation as well as the surface modulation on the resulting dynamics. This analysis unveils that by making a judicious choice of pertinent parameters one can control the flow rate with the desired account of dispersion. We envisage that the intricate coupling between the electrothermal perturbation and surface charge modulation combined with the fluid rheology may have tremendous implications towards the efficient designing of microfluidic devices.


I. INTRODUCTION
The miniaturization of microfluidic components has attracted significant attention in recent years owing to the emergence of "Lab on a chip" (LOC) devices 1,2 and subsequently the fluid flow within the narrow confinements is modulated with a target to achieve dedicated applications that primarily involve mixing [3][4][5][6][7][8] and separation. [9][10][11] Diffusion and dispersion are the two mechanisms which are highly advantageous for mixing purposes while for separation of chemical species they are undesirable. Hydrodynamic dispersion is a process in which band broadening of a neutral solute occurs because of the non-uniformity in the velocity distribution. 12,13 Because of its wide range of applications in several LOC processes, dispersion holds tremendous significance in the domain of microfluidics. [14][15][16][17] In recent years, non-Newtonian fluids has been employed in several micro and nanofluidic applications because of the close resemblance with biological fluids thus bearing enormous potentials in modern day medical and biotechnological research. [18][19][20][21][22][23] The behavior of these biofluids like blood plasma, saliva, synovial fluid, vitreous humor 24-28 exhibit viscoelasticity upon certain flow conditions, and hence, the use of viscoelastic models to predict their dynamics becomes prominent lately. [29][30][31][32][33][34] Among various flow actuating mechanisms, electroosmosis stands as one of the most viable choices which finds several microfluidic uses ranging from medical diagnostics [35][36][37][38] to enhanced chip cooling. 39 In sharp contrast to pressure-driven flows, electroosmotic flow (EOF) typically exhibits plug-type velocity profile when the electrical double layer (EDL) thickness is small as compared to the channel dimension. Although more energy efficient, this uniformity in the velocity profile of EOF results in decline of solute dispersion in comparison with the pressure-driven flow. Thus, in order to improve the extent of mixing, the usual approach is to bring non-uniformity in the channel geometry or to alter the surface charge distribution, 40,41 thereby bringing out interesting flow visualizations with enhanced mixing 42,43 . Such surface charge modulation has applications in several fields like electrothermal flow, 44 thin film patterning, 45 AC electroosmosis, [46][47][48] induced-charge electroosmosis. 49,50 Consequently, the electrokinetically driven flow of complex viscoelastic fluids have also drawn special attention because of their huge applications ranging from species separation 51,52 to energy conversion in microfluidic devices. 53,54 In addition, earlier reports of electro-hydrodynamic instabilities involving the interplay between the temperature gradients and concentration over small length scales may be exploited to further enhance solute dispersion in a medium. 55,56 Temperature gradients may occur either due to intrinsic Joule heating or via an external heat source. This developed non-isothermal condition results in the variation of the physical properties like viscosity, permittivity, electrical and thermal conductivity. For instance, the variation in the fluid conductivity interacting with the electric field can be used as a flow actuation mechanism, commonly termed as electroconvection flow. 56 This temperature dependence of the physical properties gives rise to drastic alteration in the flow field. Besides, the materials widely used for microchannel fabrication like polymethylmethacrylate (PMMA), polydimethylsiloxane (PDMS) have lower specific heat capacities and the lesser dissipation of heat from these materials may further contribute to augment the temperature gradients, thus, affecting the flow physics. 57 While most of the studies are directed towards the efficient transportation of fluid using the temperature-dependent property variation, the contribution of the electrothermal perturbation on the rotational behavior of the flow field remains relatively unexplored. More specifically, the combined effect of the electrothermal interaction and the fluid rheology along with the charge modulated surface condition, which may give rise to interesting flow controllability and augment dispersion of solutes, remains yet-to-be investigated, to the best of our knowledge.
Here we attempt to delineate the effect of electrothermal perturbations on the flow field and the associated dispersion coefficient in a patterned electrothermal flow of viscoelastic fluids through a parallel plate microchannel. The property variation coupled with the charge modulated surface significantly alters the flow field as well as the dispersion coefficient and it gets augmented when combined with the fluid rheology. We have developed an asymptotic approach following the classical lubrication approximation theory to take into account both the effect of electrothermal perturbation as well as the viscoelasticity on the dispersion coefficient. The results reported herein holds practical relevance in proper manipulation of flow in presence of finite temperature gradient, and thus, can be beneficial in the thermal management of microfluidic devices.

II. PROBLEM FORMULATION
The representative diagram of patterned electrothermal flow of a viscoelastic fluid through a parallel plate microchannel is depicted in Fig. 1. The half-width of the microchannel is denoted by h , which is much smaller as compared to the length ( ) l of the microchannel, i.e. presenting the governing equations for the above system, it is necessary to discuss all assumptions required for this analysis.
In this study, we consider a symmetric binary electrolyte (z:z) being actuated by combined electrothermal and electroosmotic forces wherein the flow is assumed to be steady, After taking into account the foregoing assumptions, the continuity, momentum and energy equations take the following form where v is the velocity vector, τ is the stress tensor, p is the hydrodynamic pressure and T is the temperature field. eff k is the thermal conductivity of the fluid obeying the following relationship ( )  63,66 where e ρ is the free charge density and E is the applied electric field written as  59,65 To determine the electric field, we employ the charge distribution equation along with the current continuity equation given by ( ) The charge distribution subjected to the assumption of Poisson-Boltzmann description yields In the energy equation, one can neglect the relative contribution of the viscous dissipation term with respect to the other two terms since this effect becomes insignificant due to the thin EDL assumption 2 2 Here we introduce a new variable p  as In this study, we have considered the simplified Phan-Thien Tanner model (sPTT) 67 where F is the stress coefficient defined by ( )

Governing equations
For solving the velocity, temperature and potential distribution, we non-dimensionalise the equation using the following dimensionless variables: Energy Equation : In Eqs. Current Continuity : After using the relevant scales, the dimensionless forms of the stress components for a viscoelastic fluid takes the following form where Deborah number ( where t l ω ω = is the patterning frequency of modulation. To obtain the flow and temperature fields from the set of above dimensionless forms, we have performed an asymptotic approach followed by the classical lubrication approximation theory. 71 From Eq. (17), it is clear that 0 yy τ = and a relationship between the stress components xx τ and yx τ can be established. Meanwhile in the energy equation, the conduction terms cannot be neglected even in the limit of 0 χ → because of their relative strengths with respect to the other terms. To determine the relative contributions of the convective components, we compare the respective scales of the characteristic temperature differences which are Additionally, one can assume that the surface potential is very small as compared to the applied potential, i.e., Now we compare the relative contributions of the terms of the charge distribution described by Eq. (3). Choosing appropriate scales of the respective parameters, i.e.
Since the axial variation of the temperature in the x co-ordinate is more significant compared to the y co-ordinate, one can expand the temperature distribution in an asymptotic series in the following manner where ν characterizes the rate of heat loss to the surrounding. Now, we utilize this expansion along with the two thermal boundary conditions and integrate the energy equation over the entire Now, the potential distribution for the patterned electrothermal flow is given by Using typical values of the involving parameters, one can show that ( ) The simplified stress components (after substituting In order to solve the Eqs. (26)- (27), we have used an asymptotic approach which is described in detail in Appendix B.

LIMITING CASES
On the basis of the present asymptotic analysis, we investigate some limiting cases.
Case 1: If we substitute 0 ξ = , the velocity profile reduces to the following form cos sin cos 8cosh dp dp y u y x y dx dx 1 cosh dp y u y x dx dp dp d y which is the velocity distribution for patterned electrothermal flow of a Newtonian fluid which is further reduced on substitution of 2 1 0, 1 The expressions for 0,0,0 T and 1,0 d dx φ can be found in Appendix F.

Stream function
For σ and x  are the two parameters which are associated with the band broadening phenomenon where the velocity of the centre of mass is given by Now, combining these two equations one can rewrite the dispersion coefficient ( ) eff D as where u  also represents the cross-sectional averaged flow velocity through the microchannel and the plate height h  is evaluated by where D is the molecular coefficient and * h is the minimum plate height which is given by 73 where eff D is the dimensionless dispersion coefficient, D Pe is the Peclet number for dispersion and u  is the dimensionless average velocity.

III. RESULTS AND DISCUSSIONS
For presenting the results of the velocity, temperature and potential fields, we have chosen some typical values of the pertinent parameters: for (iii),(iv) 2 1 α = . Also (i),(iii) accounts for viscoelastic fluids while (ii),(iv) shows its Newtonian counterpart.
pronounced in viscoelastic fluid. For example, with higher degree of thermal perturbation, the increment of pressure is about ~65% for Newtonian fluid while it is ~45% for viscoelastic fluids.
The enhancement in the volumetric flow rate is manifested in Fig. 3a where  The proper interpretation of the flow physics can be done from the streamline contour plots of Figs. 5a-5c where the variation of the stream function is shown. Fig. 5a shows the effect of viscoelasticity on the flow field where the profile for 0 De = represents the Newtonian behavior. As we increase the value of Deborah number ( ) De , deviation from Newtonian like behavior is clearly visible and it becomes exaggerated at higher values of De .This is attributed to the fact that the stresses are very high near the walls where the effect of EDL on the flow field is observed whereas in the bulk, stresses are relatively lower. Intuitively, the effect is viscoelasticity is observed at its highest level near the wall while very weak influence can be noticed near the channel centreline.
For constant surface potential, the streamlines are straight lines which shows some wavy nature when the potential is axially modulated (Fig. 5b).    cos cosh y u dp y x y dx y dp To solve this non-linear equation, one needs to use the same asymptotic approach and the variables are expanded in a similar fashion using * De as the perturbation parameter defined as  cos cosh dp y y x dx u dp dp y y y x y dx dx dp y y x dx Now the solution of Eq. (48) subjected to the no-slip boundary condition is given by where the pressure gradient The solution for the set of equations of ( )       u dp dp dp y T y T y y dx dx dx d y y x T x dx dp dp y T y dp dx dx y dx x y x φ κ κ κ α α ω κ α α ω κ κ δ κ α α ω κ κ κ α α ω κ ∂ = + + ∂ The solution of Eq. (63) is given by Here (i) (ii) (iii) (iv) the region of excess charge distribution is very less as compared to the channel dimension. In presence of axially modulated surface potential, the favorable pressure gradient for 2 ω π = occurs at the middle of the channel while adverse pressure appears to be present at the channel ends the effect of which is clearly reflected in the velocity distribution of Fig. 6a. Similarly, the distribution of the dispersion coefficient shows bimodal behavior with maximum augmentation occurring in the middle and minimum at the two ends. Also, the degree of fluctuation gets amplified on imposition of non-isothermal condition ( ) ξ as well as fluid viscoelasticity ( ) De .