Physics of Fluids

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Table of Contents for Physics of Fluids. List of articles from both the latest and ahead of print issues.
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Stirring anisotropic turbulence with an active grid

Fri, 07/31/2020 - 11:26
Physics of Fluids, Volume 32, Issue 7, July 2020.
We study spectra and high-order structure functions in anisotropic wind tunnel turbulence, which is generated using an active grid. In the first experiment, we impose homogeneous shear turbulence with a constant gradient of the mean flow and (approximately) homogeneous turbulent fluctuations. We measure mixed structure functions of order 2, 3, 4, 6, and 10 using an array of two-component hotwires. These structure functions, which vanish for isotropic turbulence, display scaling with scaling exponents that highlight intermittency: the return to isotropy at small scales of large fluctuations is much slower than expected on the basis of a simple Kolmogorov-like scaling argument [J. L. Lumley, “Similarity and the turbulent energy spectrum,” Phys. Fluids 10, 855 (1967)]. In the second experiment, we impose anisotropy in otherwise homogeneous turbulence through the time modulation of the active grid. This is done by driving the grid using signals from a turbulence (shell) model, which acts as a convenient turbulent random signal generator. In this way, different statistical properties of different velocity components could be imposed. Similar to the first experiment, our interest is in the return to isotropy of the small-scale turbulent fluctuations, which is quantified using second-order quantities such as spectra and correlation functions. Also, in this case, the strongly anisotropic correlations induced by the forcing at large scales tend to return to isotropy at small, inertial-range scales, but with the imprint of large-scale anisotropy retained.

Hydrodynamic driven dissolution in porous media with embedded cavities

Wed, 07/29/2020 - 11:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
Hydrodynamics characterization and analysis is an essential part in studying mineral dissolution in porous media with complex heterogeneous pore structures including embedded cavities. Cavities affect the pore-scale pressure and flow distribution in the surrounding porous matrix. Transport of the dissolved solute, concentration gradient, and thermodynamic driving forces in that area will be affected as a result of local flow features. Given the properties of cavities and porous media, vorticities may form, and the cavity may partially or fully contribute to the overall flow. Depending on the shape and alignment of the cavity with respect to the direction of general flow, fluid flow will be focused at certain locations on the cavity boundary. Reaction hotspots can form as a result of the facilitated mineral dissolution at those locations. A rigorous flow modeling approach that preserves the flow features inside the cavity and in the porous matrix is used. Stokes flow and seepage flow are applied as two different physics governing the fluid flow in a fluid-filled cavity and a highly permeable sediment-filled cavity consecutively. The analytical model framework permits capturing the detailed flow structure of a single-phase fluid at the curved interface of a prolate spheroidal cavity. The solutions for flow are used within a fully coupled, fully implicit reactive transport simulator to investigate the mineral dissolution in the porous host matrix. The cavity aspect ratio and slip parameter at the border are investigated as the two parameters that affect the dissolution. The simulation results showed that the reaction hotspots are mainly located on the border of the cavity where the influent enters and leaves the cavity. The midpoint between them is where the minimum mineral dissolution was placed. Approximating the cavity as a highly permeable sediment-filled porous zone showed a higher effective reaction rate compared to the fluid-filled cavity. The cavity aspect ratio showed to have a significant impact on the effective reaction rate of the investigated cases. The cavities with a shape closer to a sphere show a higher effective reaction rate.

Interactions of multiple three-dimensional nonlinear high frequency magnetosonic waves in magnetized plasma

Wed, 07/29/2020 - 11:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
The interaction of three-dimensional nonlinear high frequency magnetosonic waves in a magnetized plasma is investigated theoretically via the nonlinear Kadomtsev–Petviashvili equation. Though such wave patterns are commonly observed in the solar system and can be generated by magnetic resonance generators, only limited theoretical studies have been performed. We examined the existence of both periodic and solitary solutions of magnetosonic waves by using the modulation instability analysis. The Phillips wave resonance criterion is employed for capturing the periodic wave interaction whose energy conversion is analyzed via Fourier spectra. It is found that more energy is carried by the primary wave relative to that by the higher-order harmonic wave. In addition, it is noted that the rhodonea curve is smooth and closed for rational wavenumbers, but it becomes chaotic to form a dense set for irrational ones. We believe that this work can fill the blanks in the research of magnetosonic wave behaviors in the magnetized plasma.

Effects of tilt on the orientation dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection

Tue, 07/28/2020 - 11:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
We experimentally test the effects of tilting a turbulent Rayleigh–Bénard convection cell on the dynamics of the large-scale circulation (LSC) orientation θ0. The probability distribution of θ0 is measured and used to obtain a tilt-induced potential acting on θ0, which is used in a low-dimensional model of diffusion of θ0 in a potential. The form of the potential is sinusoidal in θ0 and linear in tilt angle for small tilt angles, which is explained by a simple geometric model of the vector direction of the mean buoyancy force acting on the LSC. However, the magnitude of the tilt-induced forcing is found to be two orders of magnitude larger than previously predicted. When this parameter is adjusted to match the values obtained from the probability distribution of θ0, the diffusive model can quantitatively predict the effects of tilt on θ0. In particular, tilt causes a change in potential barrier height between neighboring corners of a cubic cell, and changes in the barrier-crossing rate for θ0 to escape a corner are predicted with an accuracy of ±30%. As a cylindrical cell is tilted, the tilt-induced potential provides a restoring force that induces oscillations when it exceeds the strength of damping; this critical tilt angle is predicted within 20%, and the prediction is consistent with the measured oscillation frequencies. These observations show that a self-consistent low-dimensional model can be extended to include the dynamics of θ0 due to tilt. However, the underprediction of the effect of tilt on θ0 warrants revisiting the predicted magnitude.

Rivulet flow down a slippery substrate

Mon, 07/27/2020 - 10:33
Physics of Fluids, Volume 32, Issue 7, July 2020.
A detailed analysis of small-scale locally unidirectional gravity-driven rivulet flow with prescribed volume flux down an inclined slippery substrate for a rivulet with either constant width (i.e., pinned contact lines) or constant contact angle is undertaken. In particular, we determine the effect of varying the Navier slip length λ (i.e., the strength of the slip at the solid–fluid interface) on the rivulet. The present analysis shows that the shape and size of the rivulet and the velocity within it depend strongly on the value of λ. Increasing the value of λ reduces the viscous resistance at the substrate and, hence, leads to a larger velocity within the rivulet, and so the prescribed flux is achieved with a smaller rivulet. In particular, in the limit of strong slip, λ → ∞, for a rivulet of a perfectly wetting fluid and a rivulet with constant width, the velocity becomes large and plug-like like O(λ1/2) ≫ 1, and the rivulet becomes shallow like O(λ−1/2) ≪ 1, while for a rivulet with positive constant contact angle, the velocity becomes large and plug-like like O(λ2/3) ≫ 1, and the rivulet becomes narrow like O(λ−1/3) ≪ 1 and shallow like O(λ−1/3) ≪ 1.

Non-Newtonian rheology in inertial suspensions of inelastic rough hard spheres under simple shear flow

Mon, 07/27/2020 - 10:33
Physics of Fluids, Volume 32, Issue 7, July 2020.
Non-Newtonian transport properties of an inertial suspension of inelastic rough hard spheres under simple shear flow are determined by the Boltzmann kinetic equation. The influence of the interstitial gas on rough hard spheres is modeled via a Fokker–Planck generalized equation for rotating spheres accounting for the coupling of both the translational and rotational degrees of freedom of grains with the background viscous gas. The generalized Fokker–Planck term is the sum of two ordinary Fokker–Planck differential operators in linear v and angular ω velocity space. As usual, each Fokker–Planck operator is constituted by a drag force term (proportional to v and/or ω) plus a stochastic Langevin term defined in terms of the background temperature Tex. The Boltzmann equation is solved by two different but complementary approaches: (i) by means of Grad’s moment method and (ii) by using a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to inelastic rough hard spheres. As in the case of smooth inelastic hard spheres, our results show that both the temperature and the non-Newtonian viscosity increase drastically with an increase in the shear rate (discontinuous shear thickening effect) while the fourth-degree velocity moments also exhibit an S-shape. In particular, while high levels of roughness may slightly attenuate the jump of the viscosity in comparison to the smooth case, the opposite happens for the rotational temperature. As an application of these results, a linear stability analysis of the steady simple shear flow solution is also carried out showing that there are regions of the parameter space where the steady solution becomes linearly unstable. The present work extends previous theoretical results (H. Hayakawa and S. Takada, “Kinetic theory of discontinuous rheological phase transition for a dilute inertial suspension,” Prog. Theor. Exp. Phys. 2019, 083J01 and R. G. González and V. Garzó, “Simple shear flow in granular suspensions: Inelastic Maxwell models and BGK-type kinetic model,” J. Stat. Mech. 2019, 013206) to rough spheres.

Study of the vortex structure of a subsonic jet in an axisymmetric transonic nozzle

Mon, 07/27/2020 - 10:33
Physics of Fluids, Volume 32, Issue 7, July 2020.
The jet interaction flow field generated by a subsonic circular jet exhausting into a transonic cross-flow over a convergent–divergent nozzle is investigated using numerical simulations. The simulations use the three-dimensional large eddy simulation and Reynolds-averaged Navier–Stokes equations coupled with the standard k-ε turbulence model. The numerical method is verified via cold-flow and schlieren experiments. The vortex structures are identified via the Liutex–Omega method, and the flow details of various pressure ratios and injection angles are studied. The numerical results capture the main vortex structures of a jet in cross-flow, such as the trailing upper vortex and trailing major vortex. The trailing top vortex, which is difficult to capture, and a new vortex structure, named the longitudinal shear vortex, are both observed when the momentum flux is sufficiently large. This study identifies the longitudinal shear vortex for restricted flow, which to some extent facilitates the mixing of the cross-flow and the jet. The results presented in this paper indicate that the vortex structure distribution of a subsonic jet and transonic cross-flow in the restricted region can be optimized. The main vortex structures are analyzed in detail.

Heat-transfer analysis of a transitional boundary layer over a concave surface with Görtler vortices by means of direct numerical simulations

Mon, 07/27/2020 - 10:33
Physics of Fluids, Volume 32, Issue 7, July 2020.
This work studies the development of a thermal boundary layer during the laminar-to-turbulent transition process over a concave surface. Direct numerical simulations are performed where the temperature variable is treated as a passive scalar. The laminar flow is perturbed with wall-roughness elements that are able to produce centrifugal instabilities in the form of Görtler vortices with a maximum growth rate. It is found that Görtler vortices are able to greatly modify the surface heat-transfer by generating a spanwise periodic distribution of temperature. Similar to the Görtler momentum boundary layer, elongated mushroom-like structures of low-temperature are generated in the upwash region, whereas in the downwash region, the thermal boundary layer is compressed. Consequently, temperature gradients are increased and decreased in the downwash and upwash regions, respectively, thereby generating an overall enhancement of the heat-transfer rate of ∼400% for the investigated Prandtl numbers (Pr = 0.72, Pr = 1, and Pr = 7.07). This enhancement surpasses the turbulent heat-transfer values during the transitional region, characterized by the development of secondary instabilities. However, downstream, the heat-transfer rate decays to the typical turbulent values. Streamwise evolution of several thermal quantities such as temperature wall-normal distribution, thermal boundary layer thickness, and Stanton number is reported in different regions encountered in the transition process, namely, linear, nonlinear, transition, and fully turbulent regions. These quantities are reported locally at upwash and downwash regions, where they present minima and maxima, as well as globally as spanwise-averaged quantities. Furthermore, it is found that the Reynolds analogy between streamwise-momentum and heat-transfer holds true throughout the whole transition process for the Pr = 1 case. Moreover, the turbulent thermal boundary layer over a concave surface is analyzed in detail for the first time. The viscous sub-layer and the log-law region are described for each investigated Pr. Besides, the root-mean-squared temperature fluctuations are computed, finding that its wall-normal distribution exhibits a higher peak when Pr is increased.

Exact solutions to steady radial flow in a porous medium with variable permeability

Mon, 07/27/2020 - 10:33
Physics of Fluids, Volume 32, Issue 7, July 2020.
A wide class of exact solutions to equations of steady flow of ideal gas in a porous medium is obtained in a planar annular domain comprising sectors of distinct permeabilities. The formulation involves an unconventional Sturm–Liouville problem. The class also admits solutions to configurations with respective distinct generation rates. The solutions are expected to be applicable to landfill gas or natural gas extraction. The breaking of axial symmetry enables realistic modeling of the radius of influence for both horizontal and vertical wells. The analysis proves that the well’s reach would have an azimuthal dependence in any problem with a heterogeneous medium. Therefore, it is suggested that the concept of radius of influence as perceived today be revised.

Bistable states in the wake of a wavy cylinder

Mon, 07/27/2020 - 03:45
Physics of Fluids, Volume 32, Issue 7, July 2020.
Wavy cylinders have been recognized as a type of effective flow control device in previous studies. In this paper, we investigate the wake dynamics of an optimally designed wavy cylinder that completely suppresses the Kármán vortex shedding. Such a wavy cylinder is forced to oscillate with a sinusoidal motion in the crossflow direction. Examination of the lift force spectrum reveals that for a fixed forcing amplitude, a critical forcing frequency exists, below which the flow control effectiveness of the wavy cylinder is retained and beyond which the inherent vortex shedding resurrects. The resurrected unsteady vortex shedding can persist even without sustained forcing. This indicates that in addition to the steady state developed from uniform initial condition, an oscillatory state exists in the wake of a wavy cylinder if the initial state is sufficiently perturbed. The newly revealed unsteady flow features comparable hydrodynamic performance with the benchmark two-dimensional cylinder. The discovery of the bistable states calls for re-examination of the flow control effectiveness of the wavy cylinder in more complicated inflow conditions.

Coupling analysis for sway motion box with internal liquid sloshing under wave actions

Mon, 07/27/2020 - 03:45
Physics of Fluids, Volume 32, Issue 7, July 2020.
This work investigates the coupling effects of internal sloshing flow on the sway motion response of rectangular box sections. The impulse-response-function method is employed for external wave action, while the viscous two-phase flow model with the volume of fluid interface capturing technique based on the OpenFOAM® package is adopted for the internal sloshing flow. A new lower critical frequency is defined to understand the coupling effects of the internal sloshing flow, which is the corresponding frequency of the minimal sway motion amplitude. The external wave and internal sloshing forces are out-of-phase at the lower critical frequency. The numerical simulations show that the lower critical frequency is equivalent to the sloshing natural frequency when the internal sloshing flow is in the non-breaking pattern. The non-breaking sloshing-induced force approaches the same magnitude as the external wave force, which leads to a zero-amplitude sway motion. When the internal sloshing exhibits the breaking phenomenon, a phase transition of the internal sloshing force can occur, which causes the lower critical frequency to be smaller than the sloshing natural frequency. The increased incident wave amplitude or decreased tank breadth can strengthen the nonlinear behavior of the sloshing coupling action. That is, the sway motion response deviates more from the linear sloshing flow results, including the smaller lower critical frequency and the larger minimal sway motion amplitude. However, with the increased breaking-sloshing-induced nonlinearity, the difference in the sway motion response between the coupling and uncoupling results reduces, which implies a lower coupling effect.

Flow-induced fractionation effects on slip of polydisperse polymer melts

Fri, 07/24/2020 - 02:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
The slip behavior of several high-density polyethylenes with a broad range of molecular weights (MWs) including bimodal is studied as a function of MW and its distribution (MWD). A formulation inspired by the reptation theory is used to predict the slip velocity of the studied polymers as a function of MWD coupled with a model of surface MW fractionation that includes (i) the entropy driven migration of short chains toward the die wall due to the concentration gradient and (ii) the flow (stress)-induced migration effects. While surface fractionation has a minor effect on slip of narrow to moderate MWD polymers (particularly unimodal), its role is significant for broad bimodal MWD polymers. The inclusion of both effects (concentration and flow gradients) accurately captures the slip velocity of broad MWD polymers.

Vortex dynamics and acoustic sources in the wake of finned cylinders during resonance excitation

Fri, 07/24/2020 - 02:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
The flow–sound interaction mechanism and its effect on the vortex dynamics in the wake of circular finned cylinders are experimentally investigated using phase-locked particle image velocimetry at Reynolds numbers between 7 × 104 and 9.5 × 104. In addition, a hybrid experimental–numerical technique using the theory of vortex sound is employed to quantify the acoustic sources and sinks in the vicinity of finned cylinders with different fin-to-root diameter ratios, Df/Dr = 1.5, 2.0, and 2.5. The results show that changing the diameter ratio of the fins induces fundamental changes in the wake structure and the vortex shedding process downstream of the cylinder. Finned cylinders induce stronger vortex cores with a shorter formation length compared to their equivalent bare cylinders. Moreover, the flow topology over the spanwise direction shows that acoustic resonance results in uniform cylindrical vortex cores with less three-dimensional distortion, which demonstrates that the flow field becomes highly two-dimensional during resonance excitation. Quantification of the energy transfer between the flow and the sound fields reveals an enhancement in the acoustic energy production closer to the cylinder with a significant dependence on its fin-to-root diameter ratio.

Wet chemical etching of single-bore microstructured silicon dioxide fibers

Fri, 07/24/2020 - 02:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
We model the process of wet chemical etching of the external surface of a single-bore microstructured silicon dioxide fiber in hydrofluoric acid (HFA) while water is pumped through the internal channel to prevent etching of it. The model uses the Stokes flow for the velocity throughout the system and the advection–diffusion equation for the concentration of HFA. We determine the etch rate as a function of HFA concentration using data from experiments designed for this purpose, from which we calculate the change in the fiber surface. We solve our equations using a time-stepping finite-element method and verify our model by comparing to results found experimentally. We investigate the effects of different water flow rates, diffusivity, buoyancy, and bore radius. We find the water being pumped through the bore does not fully protect it and there is some etching of the internal channel, which is difficult to see in experimental images. We also obtain an estimate of the diffusivity of high-concentration HFA in water.

Micro-plasma actuator mechanisms in interaction with fluid flow for wind energy applications: Physical parameters

Fri, 07/24/2020 - 02:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
Plasma actuator is a flow control device to improve the aerodynamic performance of wind turbine blades at low airspeeds. One of the most robust numerical models for simulation of plasma actuator interaction with the fluid flow is the electrostatic model. This model is improved recently and is extensively verified by the authors. Due to the high cost of performing experimental optimizations, the optimized geometrical dimensions and materials of a plasma actuator may be sought by this numerical model. The aim of the present study is the aerodynamic enhancement of a DU21 wind turbine blade airfoil in which the effect of geometric parameters and the dielectric material is examined separately. The examined parameters include the dielectric thickness and material, the electrode thickness, and the embedded electrode length. This study shows that for performance improvement, there is a certain limit for each parameter. The length of the embedded electrode and the dielectric permittivity have a maximum limit, after which increasing the values of these parameters does not significantly affect the performance of the actuator. The increase in both the electrode thickness and the dielectric thickness reduces the effect of the actuator, and after increasing to a certain extent, no significant extra effect on the actuator performance is seen. These results also show that the improved electrostatic model can be used as a powerful tool to model the effects of different parameters to find an optimum blade design.

Effects of gravity and surface tension on steady microbubble propagation in asymmetric bifurcating airways

Fri, 07/24/2020 - 02:14
Physics of Fluids, Volume 32, Issue 7, July 2020.
Mechanical ventilation is nowadays a well-developed, safe, and necessary strategy for acute respiratory distress syndrome patients to survive. However, the propagation of microbubbles in airway bifurcations during mechanical ventilation makes the existing lung injury more severe. In this paper, finite element and direct interface tracking techniques were utilized to simulate steady microbubble propagation in a two-dimensional asymmetric bifurcating airway filled with a viscous fluid. Inertial effects were neglected, and the numerical solution of Stokes’s equations was used to investigate how gravity and surface tension defined by a Bond (Bo) number and capillary (Ca) number influence the magnitudes of pressure gradients, shear stresses, and shear stress gradients on the bifurcating daughter airway wall. It is found that increasing Bo significantly influenced both the bubble shape and hydrodynamic stresses, where Bo ≥ 0.25 results in a significant increase in bubble elevation and pressure gradient in the upper daughter wall. Although for both Bo and Ca, the magnitude of the pressure gradient is always much larger in the upper daughter airway wall, Ca has a great role in amplifying the magnitude of the pressure gradient. In conclusion, both gravity and surface tension play a key role in the steady microbubble propagation and hydrodynamic stresses in the bifurcating airways.

Mitigation of aerodynamic sound for a laminar flow past a square cylinder using a pair of cowl plates

Thu, 07/23/2020 - 13:50
Physics of Fluids, Volume 32, Issue 7, July 2020.
A new arrangement of splitter plates has been proposed for the mitigation of aeroacoustic noise generated by the two-dimensional laminar flow over a square cylinder at the Reynolds number Re = 100 and the Mach number M = 0.2. The proposed arrangement involves a pair of cowl plates (arc-shaped splitter plates) symmetrically positioned on either side of the wake center-line near the rear corners of the square cylinder. Direct numerical simulations have been carried out to analyze the nature of flow and flow induced sound fields. Unsteady, two-dimensional, compressible fluid flow equations are solved using high-resolution, space–time accurate, dispersion relation preserving schemes. Simulations have been performed for various radial locations of the cowl plates. It is observed that the maximum reduction in sound pressure level of around 24 dB is possible using the proposed cylinder and cowl plate arrangement. Based on the observed directivity patterns, we have classified the sound fields into three different regions.

Mpemba effect in molecular gases under nonlinear drag

Wed, 07/22/2020 - 03:39
Physics of Fluids, Volume 32, Issue 7, July 2020.
We look into the Mpemba effect—the initially hotter sample cools sooner—in a molecular gas with nonlinear viscous drag. Specifically, the gas particles interact among them via elastic collisions and with a background fluid at equilibrium. Thus, within the framework of kinetic theory, our gas is described by an Enskog–Fokker–Planck equation. The analysis is carried out using the first Sonine approximation, in which the evolution of temperature is coupled to that of excess kurtosis. This coupling leads to the emergence of the Mpemba effect, which is observed at an early stage of relaxation and when the initial temperatures of the two samples are close enough. This allows for the development of a simple theory, linearizing the temperature evolution around a reference temperature, namely, the initial temperature closer to the asymptotic equilibrium value. The linear theory provides a semiquantitative description of the effect, including expressions for crossover time and maximum temperature difference. We also discuss the limitations of our linearized theory.

Numerical study on immersed granular collapse in viscous regime by particle-scale simulation

Wed, 07/22/2020 - 02:08
Physics of Fluids, Volume 32, Issue 7, July 2020.
Mixed fluid–particle flows are commonly found in nature and exhibit complex particle–particle and particle–fluid interactions. In this paper, a typical small-scale case of immersed granular collapse under the viscous regime is numerically investigated using computational fluid dynamics coupled with the discrete element method (CFD-DEM), which provide particle-scale information of the collapse. The input parameters for the coupled CFD-DEM model are carefully calibrated from experimental results, and the simulation results achieve good agreement with the experiments in terms of the front evolution and final deposition. The collapse processes for different aspect ratios exhibit similarities and propagate in a three-stage mode that includes acceleration, steady propagation, and deceleration. The propagation velocity, runout distance, and the energy evolution of both fluid and particles are presented. The final runout is linearly proportional to the densimetric Froude number in our high-column cases. The transition of particles’ motion from vertical to horizontal and the drag of the fluid are found to be responsible for the constant velocity in the steady propagation stage. We also show that a small energy bump during the initial stage is the result of particle destabilization and rearrangement.

Electrowetting of power-law fluids in microgrooved channels

Wed, 07/22/2020 - 02:08
Physics of Fluids, Volume 32, Issue 7, July 2020.
Studying the dynamic behavior of droplets is of great importance in the electrowetting phenomena. However, despite the widespread use of non-Newtonian fluids in industry and daily life including medicine, food, petroleum, environmental biomass, and lab on a chip, most studies have focused on Newtonian fluids. In this study, a power-law fluid is considered as a typical example of non-Newtonian fluids and its dynamic behavior is investigated within a microchannel, and the results are compared with those of the Newtonian fluids. Both the grooved and non-grooved substrates are considered. For this purpose, the governing equations for the two phase fluid flow are solved using the finite element method, and the phase field method is used for interface tracking. We show that for four types of the considered grooves in the microchannel, different changes in the fluid dynamics are observed. When the droplets pass over the grooves, the velocity decreases and the pressure drop increases. These behaviors are intensified when the size of the grooves increases. In the shear thinning fluids, the velocity reduction is larger and even causes the drop to stop. However, in the shear thickening fluids, the velocity reduction is smaller, and the droplets can cross the grooves. After the grooves, the velocity of the droplets increases suddenly. Finally, it is shown that the time of separation of a droplet in the splitting process completely depends on the fluid type, which is much less in the shear thinning fluids compared to the shear thickening types.

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