Shear-thinning non-Newtonian Fluids Research Articles

Numerical study of the interaction between cylindrical particles and shear-thinning fluids in a linear shear flow

In this paper, the interaction between cylindrical particles and shear-thinning non-Newtonian fluids in a linear shear flow is investigated using particle-resolved direct numerical simulation. The Carreau model is used to represent the rheological properties of shear-thinning fluids, and the numerical method is validated against previously published data. Then, the effects of Reynolds number (Re), aspect ratio (Ar), power-law index (n), Carreau number (Cu), and incident angle (α) on drag coefficient (CD), lift coefficient (CL), and torque coefficient (CT) of cylindrical particles are investigated. The numerical results show that the flow field structure and pressure distribution around the cylindrical particle in a shear flow are different from those in a uniform flow, and the particles in a shear flow generate extra CL and CT. Furthermore, comparing with Newtonian fluids, the shear-thinning properties of the non-Newtonian fluid change the viscosity distribution and significantly decrease the CD, CL, and CT of the particles. The variation laws and influencing mechanisms of CD, CL, and CT under different working conditions are discussed by dividing the total coefficients into pressure and viscous shear contributions. Predictive correlations of CD, CL, and CT are established by considering the effects of Re, Ar, n, Cu, and α. The findings indicate that both the shear flow mode and shear-thinning properties must be considered when evaluating relevant particle–fluid interactions, which provides important guidance for predicting and controlling the orientation and distribution of cylindrical particles in shear-thinning fluids. Meanwhile, the predictive correlations can be used for large-scale simulations of multiphase coupling.

Computational study on interaction between rotating non-spherical particles and shear-thinning fluids

In this paper, the drag coefficient (Cd), lift coefficient (Cl), and torque coefficient (Ct) of rotating non-spherical particles in shear-thinning non-Newtonian fluids are investigated based on particle-resolved direct numerical simulation. The Carreau model is used to describe the rheological behavior of non-Newtonian fluids, and the numerical model is validated against previously published data. Then, the effects of aspect ratio (Ar), spin number (Spa), flow index (n), and Carreau number (Cu) on Cd, Cl, and Ct of rotating non-spherical particles are investigated at different Reynolds numbers (Re). The numerical results show that the closer the particles are to the spherical shape, the smaller the fluctuations of Cd, Cl, and Ct curves. The peaks and valleys of Cd, Cl, and Ct of oblate and prolate ellipsoidal particles are reversely distributed. The fluctuations of Cd and Cl curves increase with increasing Spa. Cd decreases with increasing Spa at low Re, contrary to Newtonian fluids' results. Cd and Ct decrease with increasing shear-thinning properties, Cl increases with increasing shear-thinning properties, and the effect of shear-thinning properties decreases with increasing Re. The variation of viscosity and pressure is the main reason for the variation of Cd, Cl, and Ct under different variables. Predictive correlations of Cd and Ct are established based on Re, Spa, n, Cu, and α. The findings indicate that particle rotation and shear-thinning properties must be considered when evaluating particle-fluid interactions, which provide important guidance for predicting and controlling the orientation and distribution of non-spherical particles in non-Newtonian fluids.

Modelling W/O/W double emulsions preparation in static mixers: Accounting for the shear-thinning behaviour of the dispersed phase

The dispersion of a shear-thinning non-Newtonian fluid using in-line mixers to prepare double emulsions is investigated experimentally and numerically in this study. A two-step method is adopted for the preparation of double emulsions, comprising in a first step the preparation of a reverse water-in-oil (W/O) emulsion using a high-shear rotor-stator device. This inner emulsion is dispersed in a second step into an outer aqueous phase by continuous parallel pumping of both phases through a set of static mixers. The second preparation step is modelled using a population balance equation (PBE) accounting for droplet breakage inside the mixers. Since the inner emulsion behaves as a shear-thinning fluid, the breakage efficiency depends on the local shear rate. Using classical turbulent breakage kernels is therefore inappropriate, hence there is a need for a kernel accounting for the shear-thinning character of the dispersed phase. To tackle this issue, single-phase CFD simulations of the fluid flow through the mixers are carried out showing a high shear rate magnitude near the walls of the pipe and the mixer crossbars. To avoid CFD-PBE direct coupling, the probability density function (PDF) of the shear rate is extracted from the CFD simulations. This PDF is then used to develop a volume-averaged PBE, where a classical Ostwald-de Waele model is employed to relate the apparent viscosity of the inner emulsion to the shear rate. Different experimental parameters influencing the quality of the dispersion are investigated, including the fraction of the inner emulsion phase, the oil viscosity and the inner and outer aqueous phase viscosities. In the different cases, the model captures the transient evolution of the droplet size distribution of the W/O/W double emulsions at the outlet of the mixers while the computational time remains very low.

Influence of Hazelnut and Walnut Oil Cakes Powder on Thermal and Rheological Properties of Wheat Flour.

The aim of the study was to assess the influence of the addition of powdery hazelnut oil cakes (HOC) or walnut oil cakes (WOC) to wheat flour (WF) on its selected thermal and rheological properties. In the research material, part of the wheat flour (5%, 10%, 15%) was substituted with powdery oil cakes based on hazelnuts and walnuts. The control sample was wheat flour (100% WF). In the tested systems with the addition of hazelnut oil cakes (WFHOC) and walnuts (WFWOC), the characteristics of the gelatinization and retrogradation processes were determined using the DSC method, the gelatinization characteristics of 10% pastes using the RVA method, flow curves and viscosity curves, as well as mechanical spectra. Based on the results obtained, it was found that the type of oil cakes and the level of their addition significantly influenced the thermal and rheological properties of the tested systems. Partial replacement of wheat flour with HOC or WOC significantly influenced most DSC parameters. The highest values of gelatinization enthalpy ∆HG and retrogradation ∆HR were characteristic of the WFWOC5% sample (5.9 J/g) and the control sample (1.3 J/g), respectively. All tested systems showed the properties of shear-thinning non-Newtonian fluids, and the partial replacement of wheat flour with HOC or WOC resulted in a significant reduction in the maximum viscosity of pastes, increasing with the increase in the proportion of oil cakes. WFHOC-based pastes were characterized by higher values of the G' and G″ modulus, while their values and the values of the K' and K″ parameters decreased as the share of oil cakes increased. Gels based on all tested systems showed the nature of weak gels (tan δ = G″/G' > 0.1). Replacing part of the wheat flour with nut oil cakes modified the thermal and rheological properties of pastes and gels, and the observed changes were influenced by both the origin and the level of addition of powdered oil cakes. It was found that WFHOC/WFWOC15% systems had reduced viscosity and weakened viscoelastic properties compared to systems with a lower OC content, which is not a favorable feature from the technological point of view. However, these systems were the most stable, which is an advantageous feature. However, for baking purposes, research should be carried out on the rheological properties of dough made from these mixtures.

A Comparison of Newtonian and Non-Newtonian Models for Simulating Stenosis Development at the Bifurcation of the Carotid Artery

Blood is a shear-thinning non-Newtonian fluid in which the viscosity reduces with the shear rate. When simulating arterial flow, it is well established that the non-Newtonian nature is important in the smallest vessels; however, there is no consistent view as to whether it is required in larger arteries, such as the carotid. Here, we investigate the importance of incorporating a non-Newtonian model when applying a plaque deposition model which is based on near-wall local haemodynamic markers: the time-averaged near wall velocity and the ratio of the oscillatory shear index to the wall shear stress. In both cases the plaque deposition was similar between the Newtonian and non-Newtonian simulations, with the observed differences being no more significant than the differences between the selected markers. More significant differences were observed in the haemodynamic properties in the stenosed region, the most significant being that lower levels of near-wall reverse flow were observed for a non-Newtonian fluid.

Design of active double-layer gel coatings based on furcellaran-gelatin and aqueous butterfly pea (Clitoria ternatea) flower extract for prolonging the shelf-life of salmon (Salmo salar)

In the paper, a method for producing innovative and active biopolymer double-layer coatings with a matrix based on furcellaran (FUR) and gelatin obtained from salmon skins (GEL) is presented, as well as an aqueous extract of clitoria flower (Clitoria ternatea) (BP). The first layer was based on an innovative combination of gelatin and furcellaran, enriched with an aqueous extract of clitoria flowers, characterized by antibacterial properties. The second layer was a pure combination of these two polymers, designed to provide protection against undesirable external factors. Hydrogels with different contents (1; 1.5; 2%) of the clitoria flower extract (1BP, 2BP, 3BP) inhibited the growth of gram-positive bacteria, and demonstrated antioxidant activity at the level of 1.80 mM Trolox/mg (FRAP method) and 47.57% (DPPH method). The analysed coatings in the applied deformation conditions showed properties typical of gels, with dominance of elastic properties over viscous ones. The shape of the flow curves of the analysed gels is typical for shear thinning non-Newtonian fluids, for which the apparent viscosity decreases with increasing shear rate. The influence of using hydrogels as active coatings on the quality of salmon fillets (Salmo salar) during their refrigerated storage was assessed. It was found that the coating with added clitoria extract had an inhibitory effect on the development of microorganisms during the whole storage period of the fish. The obtained active coatings show significant high applicative potential, however, further research should be carried out using other model food products.

Nutritional composition and techno-functionality of non-defatted and defatted flour of edible insect Arsenura armida

Edible insects are traditional foods worldwide, and in Mexico, is a prehispanic practice. Nowadays, edible insects can be a food source for the increasing population. This research aimed to evaluate the nutritional profile, physical and techno-functional characteristics of non-defatted (NDF) and defatted (DF) flour of the edible insect Arsenura armida to use as a functional ingredient. The lipid content in NDF was 24.18%. Both flours are high in protein, 20.36% in NDF and 46.89% in DF; their soluble proteins from A. armida were classified according to their molecular weight, which ranged from 12 to 94 kDa. The physical properties suggest that both flours have good flow characteristics. Regarding techno-functional properties, DF had the highest water (275.6%) and oil (121%) holding capacity values. The viscosity values indicate that they behave as a non-Newtonian shear-thinning fluid at a high concentration (20%). Emulsion capacity values range between 78.3 and 100% in both flours, with stability between 92.4 and 100%. These flours could be a good source of nutrients, and their techno-functional properties make them a good option for animal protein substitutes.

Flocculation of Cellulose Microfiber and Nanofiber Induced by Chitosan-Xylan Complexes.

This study aims to provide a comprehensive understanding of the key factors influencing the rheological behavior and the mechanisms of natural polyelectrolyte complexes (PECs) as flocculation agents for cellulose microfibers (CMFs) and nanofibers (CNFs). PECs were formed by combining two polyelectrolytes: xylan (Xyl) and chitosan (Ch), at different Xyl/Ch mass ratios: 60/40, 70/30, and 80/20. First, Xyl, Ch, and PEC solutions were characterized by measuring viscosity, critical concentration (c*), rheological parameter, ζ-potential, and hydrodynamic size. Then, the flocculation mechanisms of CMF and CNF suspensions with PECs under dynamic conditions were studied by measuring viscosity, while the flocculation under static conditions was examined through gel point measurements, floc average size determination, and ζ-potential analysis. The findings reveal that PEC solutions formed with a lower xylan mass ratio showed higher intrinsic viscosity, higher hydrodynamic size, higher z-potential, and a lower c*. This is due to the high molecular weight, charge, and gel-forming ability. All the analyzed solutions behave as a typical non-Newtonian shear-thinning fluid. The flocculation mechanisms under dynamic conditions showed that a very low dosage of PEC (between 2 and 6 mg PEC/g of fiber) was sufficient to produce flocculation. Under dynamic conditions, an increase in viscosity indicates flocculation at this low PEC dosage. Finally, under static conditions, maximum floc sizes were observed at the same PEC dosage where minimum gel points were reached. Higher PEC doses were required for CNF suspensions than for CMF suspensions.

Direct dissolution of unbleached pulp from corncob and wheat straw in N-methylmorpholine-N-oxide

Lignocellulose, as a kind of abundant natural resource, continuously developed to convert high value-added biological products is of great significance. Herein, we report a N-methylmorpholine-N-oxide (NMMO) solvent system to completely dissolve unbleached pulp to prepare a renewable lignin-containing cellulose film. The viscosity of the completely dissolved cellulose solution was measured using a high-pressure rotary rheometer. The shear viscosity exceeded 85 Pa·s at a shear rate of 1.62 s−1. It exhibited shear-thinning non-Newtonian fluid behavior with increasing shear rate. CF-WS had excellent tensile strength (>73 MPa), and exhibited unique optical properties of high transmittance in the visible region and high shielding performance in the ultraviolet region. When the thickness is only 0.016 mm, the UV shielding rate exceeds 75 % (λ < 380 nm). The structure of the bioplastic was revealed by SEM, XPS, and Raman spectroscopy. Directly dissolving lignocellulose in NMMO aqueous solution is expected to yield bioplastics with high strength and biodegradability. It is a potential substitute for petrochemical plastics and provides a possible way for the utilization of agricultural waste.

Hydrodynamics of shear thinning fluid in a square microchannel: a numerical approach

Abstract In this present work, a numerical study was conducted for the formation of a slug bubble for shear thinning non-Newtonian fluid in a cross-junction 2-D square horizontal microchannel. Carboxymethyl cellulose (CMC) of concentration 0.2 (w/w%) percent was used as a continuous phase that shows the shear thinning behavior of non-Newtonian fluid and Nitrogen (N2) was used as the discrete phase. The pressure-based double precision solver was used in ANSYS FLUENT 2021 R2 with the volume of fluid (VOF) method. The finite volume method is applied for the discretization of the continuity and momentum equation. This article also focuses on the fluctuation of static pressure, mechanism of slug, annular, and churn annular flow i.e., obtained by the variation in the inlet velocities. On the other hand, a concept that was applied in this work was also validated with the prior literature data.

Generation of droplets of shear-thinning non-Newtonian fluids in T-junction parallelized microchannels

The study of shear-thinning non-Newtonian fluids in T-junction parallelized microchannels provides valuable insights into the emulsification process and droplet generation synchronicity. Silicone oil is used as the continuous phase, and sodium carboxymethyl cellulose (CMC) aqueous solutions are used as the dispersed phase. Based on the evolution of the dispersed phase head and neck characteristic size, the droplet generation is classified into four stages: the waiting stage, filling stage, necking stage, and pinch-off stage. The effects of the two-phase flow rates and rheological characteristics of shear-thinning fluids on the four stages are investigated. Additionally, the coupling of interface evolution between parallelized microchannels is studied, revealing periodic changes between synchronous and asynchronous droplet generation types. The impact of this phenomenon on droplet monodispersity is analyzed, providing crucial information for the preparation of uniform droplets.

A Non-Newtonian liquid metal enabled enhanced electrography

Biopotential signals, like electrocardiography (ECG), electromyography (EMG), and electroencephalography (EEG), can help diagnose cardiological, musculoskeletal and neurological disorders. Dry silver/silver chloride (Ag/AgCl) electrodes are commonly used to obtain these signals. While a conductive hydrogel can be added to Ag/AgCl electrodes to improve the contact and adhesion between the electrode and the skin, dry electrodes are prone to movement. Considering that the conductive hydrogel dries over time, the use of these electrodes often creates an imbalanced skin-electrode impedance and a number of sensing issues in the front-end analogue circuit. This issue can be extended to several other electrode types that are commonly in use, in particular, for applications with a need for long-term wearable monitoring such as ambulatory epilepsy monitoring. Liquid metal alloys, such as eutectic gallium indium (EGaIn), can address key critical requirements around consistency and reliability but present challenges on low viscosity and the risk of leakage. To solve these problems, here, we demonstrate the use of a non-eutectic Ga–In alloy as a shear-thinning non-Newtonian fluid to offer superior performance to commercial hydrogel electrodes, dry electrodes, and conventional liquid metals for electrography measurements. This material has high viscosity when still and can flow like a liquid metal when sheared, preventing leakage while allowing the effective fabrication of electrodes. Moreover, the Ga–In alloy not only has good biocompatibility but also offers an outstanding skin-electrode interface, allowing for the long-term acquisition of high-quality biosignals. The presented Ga–In alloy is a superior alternative to conventional electrode materials for real-world electrography or bioimpedance measurement.

Ultrasonication for preparing high-performance phase change material nano-emulsions: Optimization and characterization

This work was aimed to optimize the formulation of phase change material nano-emulsions by ultrasonic emulsification that has not been investigated systematically. The optimization was performed using response surface methodology on two response parameters, average droplet size and apparent viscosity, and three independent variables including ultrasonic amplitude (X1), treatment time (X2), and surfactant content (X3). The significance of three variables on both droplet size and apparent viscosity followed the order of X3 > X1 > X2. At the optimal variable levels (X1 = 58 %; X2 = 9 min; X3 = 8 wt%), the droplet size of 118.2 nm and apparent viscosity of 7.3 mPa·s were predicated by the regression model and also well verified by experiments. The 25 wt% phase change nano-emulsion formed by ultrasonication exhibited the best emulsion stability and the lowest apparent viscosity compared with those formulated by high-energy rotor-stator homogenizer and low-energy method of phase inversion temperature. In addition, the rheological behavior associated with the average droplet size and the solid/liquid state of phase change material was illustrated numerically for the first time. In general, a threshold of 170–180 nm was found for the transition of liquid-in-water emulsions from a shear-thinning non-Newtonian fluid to a Newtonian fluid in the shear rate range of 0.6–73 s−1. All solid-in-water suspensions showed shear-thinning fluid behavior, though the shear-thinning degree was dramatically reduced to a flow behavior index of about 0.95 as the particle size was below 100 nm. These findings may be useful for designing high-performance phase change material nano-emulsions with long service life, high energy storage capacity, and low pumping power consumption for applications.

Numerical study of non-Newtonian power-law fluids under low-frequency vertical harmonic vibration

Resonance Acoustic MixingⓇ(RAM) technology applies an external low-frequency vertical harmonic vibration to mix ultrafine granular materials and subsequently non-Newtonian fluids. Although this system is used for various applications, its mechanism is yet not well understood, especially in the mixing of non-Newtonian fluids. To address this gap in knowledge, a phase model of the shear-thinning and shear-thickening non-Newtonian power-law fluid in a low-frequency vertical harmonic vibration container is established in this study, and the different power-law index is also considered. During the initial mixing process, there is Faraday instability at the gas–liquid interface, and Faraday waves are related to the power-law index. With the continuous input of external energy, the flow field is further destabilized, so that the uniform mixing is finally completed. In addition, the rheology of non-Newtonian fluids is consistent with the constitutive relation of power-law fluids. The dynamic viscosity of shear-thinning non-Newtonian fluid decreases rapidly with the increase of mixing time, while the shear-thickening non-Newtonian fluid decreases rapidly with the increase of mixing time. The variation of shear rate for Newtonian and non-Newtonian fluids is identical. Finally, a proper vibration parameter for the high mixing efficiency of RAM is proposed.

Steady rheological behavior and unified strength model for reconstituted deep-sea sediments

Investigation of the rheological characteristics of deep-sea sediments is very important for understanding submarine disasters. Three reconstituted deep-sea soils with different plasticity and activity values from the South Sea China are considered in this study. A series of steady rheological tests are conducted on samples of three soils with different water contents and temperatures by a rate-controlled rotational rheometer operated in a steady-state regime. Rheological results show that shear stress and apparent viscosity can be decreased by a maximum of 65% when the temperature varies from 1 °C to 25 °C, and the three deep-sea soils are typical shear thinning non-Newtonian fluids with yield stress. The state of the samples can be divided into three phases during the shearing process, which implies that with an increase in the shear strain rate, all samples first behave as a solid, then transition from solid to liquid, and last behave as a liquid. The three phases are distinguished by the static yield stress (SYS) and fluidic yield stress (FYS), and a simple method for dividing the three phases is proposed. The SYS and FYS decreased with increasing temperature and water content. The relationships between yield stresses (SYS and FYS) and soil properties (water content and activity) and temperature are established. In addition, a unified strength model for deep-sea sediments is suggested, and the validity of the model is verified by comparing the prediction results of the model and test results. The results of this study could be useful for deep-sea submarine mudflow and landslide modelling.

Experimental Realization of a Colloidal Ratchet Effect in a non-Newtonian Fluid

The transport of matter via a ratchet effect represents a convenient way to extract useful work from a thermodynamic system, and can have direct applications in several microscale fluid-based technologies. Most realizations so far have been centered on using Newtonian fluids. This study shows how to achieve directed transport of magnetic microparticles in a non-Newtonian shear-thinning fluid under a square-wave magnetic force. This technique could be used in active microrheology, to infer the nonlinear properties of viscoelastic media.

Novel high energy media mill produced macadamia butter: Effect on the physicochemical properties, rheology, nutrient retention and application

A novel high energy media mill (HEMM) was used to prepare macadamia butter (MB). The effect of grinding time on the properties of MB was investigated. Results showed that with the grinding time increased from 0 min to 65 min, the D[3,2], D[4,3], D50 and D90 value of the MB samples were gradually decreased from 24.03, 68.48, 59.50, and 124.67 μm to 4.37, 14.25, 8.82, and 34.07 μm, respectively, and the release of volatile flavor compounds gradually increased. Moreover, it did not adversely impact the color and chemical composition. Rheological analysis showed that the viscosity and yield stress of MB gradually decreased with the increasing of grinding time and the non-Newtonian shear thinning fluid behavior (flow behavior index, n < 1) was observed. In addition, HEMM treatment result in more oil separation and poor spreadability of MB. The maximum oil separation rate reached 49.46 g/100 g after 65 min of grinding. However, the addition of 1–3 g rice bran wax/100 g can significantly improve its oil separation and spreadability in toast. The results can provide a support for the application of HEMM in MB processing, and new insights for the development of multi-purpose MB.

Formation of Droplets of Shear-Thinning Non-Newtonian Fluids in Asymmetrical Parallelized Microchannels.

Fluids containing polymers are frequently utilized in the chemical industry and exhibit shear-thinning characteristics. The flow distribution of non-Newtonian fluids in parallelized microchannels is a key issue to be solved during numbering-up. Numbering-up means increasing the number of parallelized microchannels. In this study, a high-speed camera is used to explore the distribution of fluid flow as well as the uniformity and stability of droplets in conceptual asymmetrical parallelized microchannels. Cyclohexane and carboxymethylcellulose sodium (CMC) aqueous solutions are used as the continuous phase and dispersed phase, respectively. The effects of fluctuation of pressure difference around the T-junction, the hydrodynamic resistance in microchannels, and the shear-thinning property of fluids on flow distribution and droplet formation are revealed. The uniformity and stability of droplets in microdevices with various cavity settings are compared, and an optimal configuration is proposed. Finally, prediction models for the flow distribution of shear-thinning fluids in asymmetrical parallelized microchannels are established.

THE EFFECT OF POLYDEXTROSE ON RHEOLOGICAL CHARACTERISTICS AND SENSORY PROPERTIES OF REDUCED SUCROSE STARCH-MILK DESSERTS

Introduction. The aim of the study was to assess the impact of replacing sucrose with polydextrose (E 1200) on the rheological characteristics and sensory properties of starch-milk desserts sweetened with steviol glycosides (E 960). The study materials were desserts made of oxidized potato starch (E 1404), skimmed milk powder, curcumin, vanilla flavorsucrose and/or a mixture of polydextrose (E 1200) and steviol glycosides (E 960). As part of the rheological analyses, the pasting characteristics, viscosity curves and mechanical spectra of the desserts were determined. As part of the sensory analysis, palatability and texture were assessed using a quantitative descriptive analysis. The acceptability of the desserts was also assessed. Results and discussion. Based on the results, it was found that the partial or complete elimination of sucrose from the composition of desserts and replacing it with a mixture of polydextrose and steviol glycosides led to a reduction in their maximum viscosity determined at the heating stage in the pasting test, but did not change the starch pasting temperature. The desserts showed shear-thinning non-Newtonian fluid properties, and the modification of the recipe resulted in a reduction of the consistency coefficient and a reduction in the share of elastic properties without a significant impact on the viscous properties of the system. The desserts in which sucrose was completely, or in a greater degree, replaced with polydextrose and steviol glycosides had a cooling effect and had a licorice aftertaste in the sensory assessment, while retaining sweetness similar to the sweetness of the dessert sweetened with sucrose only. The modification of the dessert recipe did not affect the adhesiveness and mouth melting of the finished products, but contributed to a reduction in their viscosity and malleability assessed by the sensory analysis. Both the texture and palatability of the desserts prepared to the recipe, in which two-third of the amount of sucrose or its total amount was replaced with polydextrose and steviol glycosides, turned out to be less sensorily acceptable than the dessert which did not contain polydextrose.

Impact of Melting Heat Transfer and Variable Characteristics on an MHD Non-Newtonian Shear-Thinning Fluid Flow with Gyrotactic Microorganisms over a Nonlinear Stretched Surface

Impact of Melting Heat Transfer and Variable Characteristics on an MHD Non-Newtonian Shear-Thinning Fluid Flow with Gyrotactic Microorganisms over a Nonlinear Stretched Surface