Electrokinetically Forced Turbulence in Microfluidic Flow

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 1 doi:10.20944/preprints201811.0462.v1 Article 3 Electrokinetically Forced Turbulence in Microfluidic Flow 4 Willy L. Duffle 1 and Evan C. Lemley 2,* 2 5 6 7 University of Central Oklahoma; wduffle@uco.edu University of Central Oklahoma; elemley@uco.edu * Correspondence: elemley@uco.edu; Tel.: +01-405-974-5473 1 2 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Abstract: While laminar flow heat transfer and mixing in microfluidic geometries has been investigated experimentally, as has the effect of geometry-induced turbulence in microfluidic flow (it is well documented that turbulence increases convective heat transfer in macrofluidic flow), little literature exists investigating the effect of electrokinetically-induced turbulence on heat transfer at the micro scale. Using recently observed experimental data, this work employed computational fluid dynamics coupled with electromagnetic simulations to determine if electrokinetically-forced, low-Reynolds number turbulence could be observed in a rectangular microchannel with using Newtonian fluids. Analysis of the results was done via comparison to the experimental criteria defined for turbulent flow. This work shows that, even with a simplified simulation setup, computational fluid dynamics (CFD) software can produce results comparable to experimental observations of low-Reynolds turbulence in microchannels using Newtonian fluids. In addition to comparing simulated velocities and turbulent energies to experimental data this work also presents initial data on the effects of electrokinetic forcing on microfluidic flow based on entropy generation rates. 23 24 25 Keywords: micro-fluidics, micro-mixer, entropy generation, micro-turbulence, electrokinetic mixer 26 1. Introduction 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Fluid flow in microscale devices (microfluidics) has become an important area to study for applications in recent years. With an expanding set of techniques to create microchannels of increasingly complex geometries and recent biomedical and security applications, the interest in microfluidics continues to increase. Two important applications are micro-mixing and micro heat exchangers. The difficulty in mixing and heating fluids at the microscale is that the fluid flow is almost always laminar. The chaotic nature of turbulent flow at the macroscale is useful for both mixing and heating, but is an unusual phenomenon at the microscale. Many efforts have been made to create chaotic advection in microscale applications [1], which in some sense can mimic turbulence. Although turbulence in the laminar flow of Newtonian fluids at the microscale is an unusual occurrence, some recent experimental reports have claimed turbulent behavior at Reynolds numbers far below typical accepted values [2]. Computational fluid dynamics (CFD) simulations can provide a benefit to researchers as a tool to either design experiments or improve validation of experiments. Simulations require benchmarks to validate their results and these benchmarks may be numerical or experimental in nature. Previous efforts to validate laminar microscale flow with CFD have been successful. Unlike Passive mixing, which uses the geometry of microchannels to increase mixing efficiency, Active mixing uses external forces such as acoustically driven vibration [3] or external electric and magnetic fields [4,5] to force mixing. Attempts to model active mixing applications are ongoing, but researchers often choose to perform experiments because of the complexities involved in the modeling of these microscale phenomena. An additional level of complexity to add to the modeling of these active © 2018 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 2 of 15 47 48 49 50 mixers would be to include the existence of low Reynolds number, electrokinetically-induced turbulence, or μEKT [6]. 51 52 53 54  55 2. Methodology 56 57 58 59 60 61 62 63 64 To build the simulation model based on the Wang et al. experiments it first had to be determined how to model the force due to the applied electric field. The model used the laminar flow equations of Navier-Stokes with a body force, ⃗ , included (see Equation 1). 65 66 67 68 69 70 71 72 73 74 75 76 77 The objectives of this work were:   To determine the feasibility of observing turbulence in an electrokinetically-forced microfluidic mixer using CFD. To quantify the effects of electrokinetic forcing in microfluidic mixing using CFD. To quantify the entropy generation in an electrokinetically-forced microfluidic mixer using CFD. ⃗ =− Wang et al. defined this force as ⃗= ⃗− ⃗ + ⃗, + ⃗⋅ ⃗ ⃗⋅ ⃗ + (1) , (2a) The RHS terms represent the contributions due to the Coulomb force, the dielectric force and the force due to thermal expansion, respectively. is the fluid permittivity, and is the fluid density [6,7] and represents the free charge density: =− ⃗⋅ ⃗ , (2b) The partial derivative in Equation (2a) disappears for an incompressible fluid and the second term is negligible compared to the first when dealing with fluids of different conductivities [6], the following simplification can be made for the electric body force ⃗= ⃗, (3) 78 2.1 The Model 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 The simulation model was built using SolidWorks® 3D design software for the modeling of the physical geometry and COMSOL Multiphysics® for the computational modeling. The bulk of the CFD runs utilized the University of Central Oklahoma’s (UCO) BUDDY supercomputer cluster. BUDDY is a 38-node Linux cluster with one control node, 31 compute nodes (20 CPUs with 64GB total memory), 4 high memory compute nodes (20 CPUs with 128GB total memory) and 2 GPU nodes. The initial conditions for this work were taken from the experimental conditions [2,6]. The side walls of the microchannel were defined as gold foil with all other walls being acrylic. The fluids defined were de-ionized water and a phosphate buffer solution though this work used a saline solution instead of phosphate; the important factors in choosing a working fluid was to ensure it was Newtonian and that the electrical conductivity gradient was 5000:1. The AC electric field range investigated was 0-20Vpp with a phase difference between electrodes of 180 degrees. This potential was defined as a DC voltage on the electrodes. The geometry was split into two fluid domains. At time t = 0s the two fluids are completely unmixed resulting in a virtual boundary along the centreline of the microchannel (see Figure 1) with a conductivity gradient of 5000:1. At the point where the two fluids meet the concentration gradient remains at a maximum while the downstream concentration gradient moves toward equilibrium. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 3 of 15 95 96 97 98 99 100 101 102 Because the electrodes are non-parallel, they are closest at the entrance creating a maximum electric field value there [2] and from Equation 3 it can be seen that the greatest electric field yields the greatest body force. The inlet geometry was designed for the model to ensure laminar flow at the channel entrance and the channel length was approx. 5mm. The flow parameters shown in Table were used to match the inputs to the physical experiments by Wang et al. The calculated outputs provided a reference for the values expected if the simulations were valid. Figure 1. Two fluid domains considered in this work. 103 2.2 Simulation Software 104 105 106 107 108 COMSOL Multiphysics® couples multiple physics interfaces together automatically but also allows the user the manually couple if needed. This work makes use of the Electric Currents, Laminar Flow and Transport of Diluted Species (TDS) interfaces with the Electric Currents & Laminar Flow automatically coupled while the Laminar Flow and Transport of Diluted Species were manually coupled. Because the problem to be solved was complex, simplifications were required. The primary 109 110 111 112 113 114 115 116 117 Table 1. Initial and Boundary Conditions (20Vpp shown) simplification was to disable the migration in an electric field option in the TDS interface; while this interaction would exist in any physical experiment, this work was trying to quantify the specific effects due to enhanced mixing caused by μEKT which would justify ignoring effects that would be present if an electric field was applied but μEKT was absent. To justify this omission, two assumptions were made: 1) the migration due solely to the electric field acting on the charged species would be the same with or without turbulence and 2) the fluid-particle interaction is negligible meaning the additional Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 4 of 15 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 particle motion would not appreciably effect the flow, together these assumptions render the migration inconsequential when comparing total entropies between forced and unforced cases. For the second assumption, the Stokes drag coefficient = can be manipulated to find the drag force ⃗ =3 , = ⃗ , (4) (5) created by the Na+ and Cl- ions moving through the solution in the electric field, a force which was found to be negligible compared to the electric body force ( ~ 10-7 N vs ~ 105 N). In addition, the particle diameters were small enough that the electrophoretic force was deemed negligible as well, on the order of 10-27 N. Figure 2 shows the relationship between the physics interfaces after the problem was simplified. Comsol automatically uses steady state solution steps as the initial conditions for a transient study step when applicable to prevent mismatched boundary conditions at t = 0. The electric field was solved as a steady state DC potential on the electrodes and then the electric field components were multiplied by cos( ) to calculate the electric body force components used as the driving force in Navier-Stokes, seen in Equation 1. The force is defined in COMSOL as a Volume force and it was added into the Laminar Flow interface as shown in Figure 3. Figure 2. Governing physics setup for simulations. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 5 of 15 145 Figure 3. Implementation of Volume Force in COMSOL. 146 147 148 149 150 151 152 153 154 The stationary study steps in the final simulation runs were done using an iterative, segregated solver while the time dependent solver for the Laminar Flow step was done using a fully-coupled, iterative solver and the solver for the Transport of Diluted Species was a fully-coupled, direct solver (MUMPS). The iterative method uses the Newton-Raphson method and the difference between segregated and fully-coupled approaches is how the equations for the different physics interfaces are related. The Transport of Diluted Species elements were cubic and the Electric Currents elements were quadratic. The fluid discretization setting for the Laminar Flow interface was set to P2 + P1; this setting denotes second order elements for the velocity components and linear elements for the pressure field. 155 2.3 Entropy 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 Why study the entropy of a system? When comparing two different physical processes it is first necessary to find a common relationship between them to base the comparison on. For example, when measuring the effectiveness of a heat transfer process, it is accepted practice to look at the Nusselt number as a performance gauge [8]. For internal flow one looks at the pressure differential to determine system losses, which is good for comparing the efficiency of a design as it relates to frictional flow loss. However, if the system involves heat transfer as well, the temperature differential is used to quantify thermal losses. Pressure drop is measured in Pascals while temperature gradients are measured in degrees (Kelvin, Celsius or Fahrenheit), two units that do not add together for the purpose of determining a total system loss without first converting to some unitless expression. To describe the total system loss requires all involved expressions to be comparable in terms of units and order which is where the study of entropy really starts to make sense. Entropy ([W/Km3]) can be calculated from the incompatible variables of each process (fluid flow, heat transfer, species diffusion etc.), typically in post processing, when using finite element software. Once a single equation of similar terms is expressed, cause-effect relationships can be more readily seen. Adding the entropy equations for each of these individual physics processes together gives an equation for the total entropy generation for the system. Because isothermal conditions are assumed throughout the simulation, Equation 6 contains only a viscous dissipation term for laminar flow (in brackets) and a term to cover the entropy generated by diffusion of species. 176 177 = + + + + + + + + + + + ≥∅ , (6) 178 2.4 Mesh Study 179 180 181 182 183 184 185 186 187 188 189 To get accurate results in a finite element analysis, a mesh convergence study can be used to determine the optimal mesh size needed to balance the computational cost with the desired solution accuracy. A laminar flow simulation was completed for 4 meshes of increasing resolution (Coarser, Coarse, Normal and Fine) at a forcing voltage of 20Vpp. In this case many of the parameters of interest are derived from velocity but with both positive and negative velocity values it was difficult to compare volumetric totals. The mesh study looked at the total values for Te and Sgen over the volume shown in Figure 4 at each mesh size. Because the differences were on the order of 10-11 for each mesh, it shows that the choice of mesh in this case is arbitrary as the velocities involved are on the order of 10-3, the values of Te are on the order of 10-6 with Sgen at the channel entrance are as high as 200. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 6 of 15 190 191 192 193 194 195 196 197 198 199 200 201 Figure 4. Volume from trailing edge to x = 0.5mm. For this reason the Coarse Mesh, shown in Error! Reference source not found., was used as it was the best choice in regard to solution time with each voltage run taking approximately 1hr 22min with Transport of Diluted Species included. For comparison, without TDS, a simple forced flow simulation with a Normal mesh resolution took 1hr 47min and the Fine resolution took 2hrs 33min. The elements making up the meshes included tetrahedral, pyramid, prism, triangular, quadrilateral, edge and vertex elements. Figure 5. Course mesh used in mesh resolution study. 202 203 204 205 3. Results 206 207 208 209 210 211 212 213 214 The results reported are for simulation runs conducted over a period of 0.25s, over the voltage range of 0-20Vpp, with the values of interest examined in the volume of fluid from the point where the streams converge to the plane x = 0.5mm (shown in Figure 4). All the figures presented here are at t = 0.25s. In Wang et al., the experimental results are discussed as they relate to six indicators of turbulent flow2: fast diffusion, high dissipation, irregularity, multi-scale eddies, continuity and 3-D flow. Of these six parameters, this work presents results correlating to four: high dissipation, fast diffusion, 3D flow and irregularity. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 7 of 15 215 3.1 Fast Diffusion 216 217 218 219 220 221 From Figure 5 it can be seen that, without forcing, the flow is completely laminar and becomes more turbulent in appearance as the forcing voltage increases. While comparing the simulation stream lines to the LIFPA images of diffusion in the Wang experiment is not completely analogous, the experiment’s visualization images using polystyrene particles as tracing devices shows similar results to the streamlines in Error! Reference source not found.. 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 Figure 5. Streamlines - 0Vpp (Unforced). Figure 6. Streamlines - 20Vpp. The effect of the electric field on dissipation can be seen in Figure 7 & Figure 8 with slices shown at x = 0, 100, 200, 300, 400, & 500μm. At a forcing voltage of 20Vpp the mixture appears to be completely homogenous at the plane x = 0.5mm. Because the two fluid domains are discretely populated at t = 0s with fluids of concentrations shown in the inlets, Figure 8 shows that homogeneity disappears shortly after this plane but it can be assumed that if the simulation was continued past 0.25s, the entire channel would become completely mixed due to the developing secondary flows that can be clearly seen downstream. Figure 7. NaCl concentration (mol/m3) @ 20Vpp, t=0.25s (Slices at 100μm intervals). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 8 of 15 238 239 Figure 8. Downstream NaCl concentration (mol/m3) @ 20Vpp, t=0.25s. 240 3.2 High Dissipation 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 In macro-flows, a rapid, non-linear increase in the turbulent dissipation (another way of denoting pressure drop) signals the release of turbulent energy [2] so if the dissipation rate is high then the rate of change of turbulent energy is also high, indicating turbulence. The turbulent energy, Te, was calculated at the point (100, 0, 0). When using turbulent models, one of the dependent variables calculated by COMSOL is the turbulent energy (Te), however, because the simulations were completed using a Laminar Flow study instead of Turbulent Flow study, an equation was created to derive the value of Te from laminar flow data starting from the definition [2] where and given that, in 3-dimensions, = = √ (7) 〉, = 〈 − 〈 〉, + (8) , + (9) the equation used in COMSOL for post-processing (given 25 time steps) was = (( −( ( , , 0, _ )/25)) ^2, , 0, _ The Electric Rayleigh number was defined as [2] = ( )/25, ) , (10) (11) And in COMSOL for each electrode potential difference was = ((80.2 ∗ . 0_ ) ∗ (2 ∗ ( 0^2)) ∗ (0.0275 − 5.56 − 6))/(5.5 − 6 ∗ 0.001[ ∗ ] ∗ 1.5 − 9[ ^2/ ]), (12) When Te is plotted vs. Rae, as shown in Figure 9 & Figure 10, the rapid increase in turbulent energy after reaching the critical Rayleigh number [2] can be seen. A voltage range of 2-20Vpp was used for the data set and the plots, when compared, show a similar trend, electric Rayleigh number Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 9 of 15 273 and range (within the 274 275 Figure 9. Te vs Rae (from simulated data). 276 277 Figure 10. Te vs Rae (from experimental data). 278 same order of magnitude). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 10 of 15 279 280 281 282 283 Table 1 clearly shows that while the slopes are smaller than the experimental values by one order of magnitude, the relationship between the turbulent and laminar slopes (how much greater the turbulent slope is than the laminar) is closer in value, as is the critical Rayleigh number (Raec) range discovered from the simulations; both are within the same order of magnitude. 284 285 Figure 9. Te vs Rae (from simulated data). 286 287 Figure 10. Te vs Rae (from experimental data). 288 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 11 of 15 289 290 291 292 293 294 295 296 297 298 299 Table 1. Te vs Rae Results Raec (range) Simulation Results Wang Results 1.18e7 7.57e7 1.9e7 4.3e7 log-log log-log Laminar Turbulent Slope Slope Turbulent Slope/ Laminar Slope 0.044 0.690 15.738 0.16 3.03 18.938 300 3.3 Three Dimensional Flow 301 302 303 304 305 306 With inhomogeneous, 3D flow being a basic feature of turbulence [2], evidence can be found visually by looking to Figure 5 and Figure 11 or, more analytically, by referring to Figure 12 which shows a distribution of Te along z similar to the Wang et al. results. It must be noted that the Wang results show unforced values of Te on the order of 10-10 and mean forced values of 10-7 while this thesis reports values of 10-7and 10-6, respectively. 307 308 Figure 11. Transverse view 3D velocity streamlines in channel entrance. 309 310 Figure 12. Te vs z (inhomogeneity of flow in transverse plane) 311 312 313 314 315 For a voltage of 20Vpp, the experimental data showed that at z = 0 μm in the y-direction, the value for Te at y = 0 μm was about 2.7 times larger than the value at y = -100 μm (see Figure 13) which is understandable as the value of Te is greater towards the centerline of the channel as depicted in Figure 12. The simulation results showed a maximum difference of 1.4 times at z = -15 μm. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 12 of 15 316 317 Figure 13. Te vs z (Wang et al. experimental data). 318 3.4 Irregularity 319 320 321 322 323 The irregularity feature of turbulence is plotted as a time trace of the velocity at point (100, 0, 0). While fluctuations in the value of us became greater as the voltage increased in the experimental paper, the results of this work showed relatively constant values over time, Figure 14. However, the mean experimental values are on the same order as those found in these simulations; see Table 2 for a comparison of the experimental-to-thesis values (dashed lines are experimental values. 324 Table 2. Mean us value comparisons. us at 0Vpp Simulation Results Wang Results us at 8Vpp us at 20Vpp 3.96 4.37 7.50 3.23 4.65 11.29 over (0.25s) 325 326 327 Figure 14. us vs time. 328 3.5 Entropy 329 330 331 332 333 334 335 The entropy calculated by the simulations in this work come from two sources, the flow itself and the transport of the NaCl ions using Equation 6. Error! Reference source not found. shows the values for each component of the entropy calculated at time t = 0.25s. Error! Reference source not found. shows the entropy generated by the forced flow and Figure 16 shows the entropy generated by the species transport, both at three different forcing voltages. While the transport component contributes the majority of entropy to the system initially (on the order of 10-9 compared to 10-11), its contributions diminish over time as the concentration gradient Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 13 of 15 336 337 338 decreases from 5000:1 towards equilibrium at which point the contributions are separated by only a single order of magnitude. Table 3. Entropy components (at t = 0.25s). Sgen at 0Vpp Sgen at 8Vpp Sgen at 20Vpp Entropyflow 1.65E-13 9.17E-13 2.70E-11 EntropyTDS 2.89E-10 5.27E-10 6.04E-10 Entropytotal 2.90E-10 5.28E-10 6.31E-10 339 340 341 Figure 15. Entropy generated by flow. 342 343 Figure 16. Entropy generated by species transport. 344 345 346 347 Alternatively, the entropy generated by the flow has a smaller magnitude but is constant over the length of the simulation and it illustrates the impact that the increase in forcing voltage has on the flow entropy. The increase from 0Vpp to 8Vpp is less than 1 order of magnitude while the increase from 8Vpp to 20Vpp increases the entropy value by 2 orders of magnitude. 348 4. Conclusions 349 350 351 352 353 354 355 356 The results of this work show that it is possible to observe the turbulent flow properties witnessed in the electrokinetically-forced microfluidic mixer experiments performed by Wang et al. [2] using CFD simulations. While the calculated values for the quantities of interest were not the same as the experimental data, the values were within the same order of magnitude of those reported by the experiments and showed the same trends in each of the indicators of turbulence looked at: fast diffusion, high dissipation, irregularity and 3-D flow. The results section illustrates the effects of electrokinetic forcing in microfluidic mixing found through the use of CFD software. Figure 7 & Figure 8 depict the fast diffusion in the channel that Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 14 of 15 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 takes place within 0.25s of starting the flow and applying the electric field. The concentration has reached a near-homogeneous state at x = 0.5mm and secondary flows have started downstream which shows that a 5.0 mm channel is more than long enough for complete mixing of the 2 fluids; in fact a 0.75mm channel would likely suffice to completely mix the fluids in 0.25s. From Figure 12 it can be seen that with a 20Vpp forcing voltage the turbulent energy increases by up to two orders of magnitude (108 - 106) which indicates a greater mixing capability and, by extension, greater heat transfer potential; the two-magnitude increase in entropy seen in figures Error! Reference source not found. - Figure 16 would indicate the same. Discrepancies in the range of Te compared to Wang et al. may be due to assuming-out real world phenomena in an effort to simplify the simulations. Adding in the effects from an electroosmotic-wall boundary condition and the drag effects on the fluid due to the movement of the sodium and chlorine atoms in the electric field may help to explain the differences seen between the experimental and simulation data. Also, this study was done using laminar flow equations that are much less complex than the turbulent model equations. While there are many different turbulence models that can be used for incompressible turbulent flow, the standard is the Reynolds averaged Navier-Stokes (RANS), κ-ε model [9]. The RANS equation showing the components of turbulent kinetic energy (k here) [10] using the Einstein summation notation is shown in Equation 13 376 + = − + − − − (13) 377 378 379 380 381 382 The five right-most terms in Equation 13 are not accounted for in this thesis and may account for deviations from the experimentally observed data. 383 384 385 386 387 388 389 390  391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 Future research:    Move the trailing edge of the inlet dividing plate to the entrance of the channel instead of upstream. This may eliminate secondary flows that occur before the fluid enters the channel and confine the entropy generation to the channel (i.e. largest Δσ is at the plane x = 0). Run simulations at the Normal mesh resolution and increase the solution time to determine the time needed to reach a fully-mixed outflow. Incorporate heat transfer into the simulation to quantify the effects of μEKT on heat transfer. Develop simulations using the COMSOL Turbulent Flow interface for comparison to the results presented here and to provide additional validation for Wang et al. experiments. 5. Nomenclature C, c De Deff R ⃗ ⃗ ⃗ , Eo ⃗ k k Sgen Te P, p Δp De concentration, mol/m3 Diffusion coefficient effective diameter, m Rydberg gas constant, 8.314 J/K∙mol drag force, N electrokinetic body force, N electric field, V/m velocity field (u, v, w), m/s thermal conductivity, W/m∙K turbulent kinetic energy (Conclusions) entropy generation rate, W/ m3∙K turbulent kinetic energy, m2/s2 pressure, Pa pressure drop, Pa diffusivity (or diffusion coefficient 1.5 x 10-9 m2/s) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 November 2018 doi:10.20944/preprints201811.0462.v1 15 of 15 407 408 409 410 411 412 413 414 415 416 ρ ρv ρf σ1 σ2 μ ѡ 417 absolute permittivity vacuum permittivity, 8.854 × 10-12 F/m relative permittivity fluid density, kg/m3 charge density, C/m2 free charge density, C/m2 conductivity of distilled water conductivity of NaCl solution dynamic viscosity, kg/ms channel width at entrance, m 418 419 420 Author Contributions: Conceptualization, W.L.D. and E.C.L.; methodology, W.L.D. and E.C.L.; validation, W.L.D.; formal analysis, W.L.D.; writing—original draft preparation, W.L.D.; writing—review and editing, W.L.D. and E.C.L.; visualization, W.L.D.; supervision, E.C.L.; funding acquisition, E.C.L.”. 421 422 Funding: This work was partially sponsored by the National Science Foundation grant ACI-1429702 (funding for UCO’s Buddy Supercomputing Cluster). 423 424 425 Acknowledgments: The authors would like to acknowledge Dr. Guiren Wang from the University of South Carolina and his team for providing both the inspiration for this study as well as his expertise. 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