non-Newtonian Flow Research Articles

Important of Slip Effects in Non-Newtonian Nanofluid Flow with Heat Generation for Enhanced Heat Transfer Devices

In various fields such as engineering, nanotechnology, and biomedical sciences, the study of non-Newtonian nanofluid flow with heat generation is becoming increasingly important. However, it is challenging to accurately model such flows due to their complex behavior and slip effects at the fluid-solid interface. This research investigates the impact of first and second-order slip conditions on the flow and heat transfer properties of a non-Newtonian nanofluid using a power law model to describe the fluid's non-Newtonian behavior and numerical methods to solve the resulting equations. To determine the influence of various parameters such as slip parameters, Brinkman number, power law index, and Eckert number on the velocity, temperature, and concentration profiles, which this study examines. The study shows that slip parameters significantly determine the flow and heat transfer properties of non-Newtonian nanofluids, the study also reveals that slip parameters are a crucial factor in understanding the flow and heat transfer characteristics of nanofluids, with the second-order slip condition having a greater impact on velocity and temperature profiles than the first-order slip condition. These findings are valuable for developing and optimizing heat transfer devices that involve non-Newtonian nanofluids with heat generation, which is essential for technological advancements in today's industry.

Time-dependent stagnation point flow of nano Casson fluid with Joule heating over an elongated surface subjected to viscous heating and exponential space-based heat source/sink: Boungiorno model

The unsteady two-directional flow of a non-Newtonian magneto-Casson nanoliquid flow over an elongated flat surface with the porous matrix is investigated. The flow is subjected to space-based exponential heat generation/absorption (ESHS), thermophoresis, Brownian motion of nanoparticles, and transverse magnetic field. Within the base fluid, the diffusion of chemically reactive nanoparticles is assumed to be highly significant; hence considered. The governing equations of the flow model admit self-similar equations and are numerically solved by employing the Runge-Kutta-based shooting technique (RKSM). The significance of key parameters on the temperature, velocity, friction factor at the surface, heat transfer rate, and mass transfer rate distributions is analyzed. The use of high-Prandtl number base fluid and nanoparticles of high thermal conductivity could be of practical use to increase the heat transfer rate and avoid nanoparticle accumulation. The occurrence of nanoparticles in the operating liquids reduces the shearing stress at the plate surface to avoid backflow.

Hemodynamic Study of Cerebral Arteriovenous Malformation: Newtonian and Non-Newtonian Blood Flow

ObjectiveArteriovenous malformation is a disease of the vascular system that occurs mainly in the cerebral arteries and spine. Numerical simulation as a powerful method is used to investigate the Cerebral Arteriovenous Malformation hemodynamic after occlusion of the abnormality step by step by embolization. MethodsThe CT Angiographic imaging data of two patients are used and a geometric model is extracted by the Mimics software. Numerical simulation of blood flow is performed in both Newtonian and non-Newtonian models. The Navier–Stokes, and continuity governing equations are solved by finite element method using the COMSOL Multiphysics software (the commercial computational fluid dynamics (CFD) simulation software). To validate the numerical results, the real data on blood flow rate in the feeding artery and draining veins are used, as well as angiographic images at different times. ResultsRegarding the comparison of pressure contours for different occlusions of 0, 30, 50, and 90%, by increasing the amount of occlusion in the nidus, there is an increase in the blood pressure. Regarding the comparison of the blood flow velocity in the feeding artery, draining veins, and inside the AVM nidus for Newtonian and non-Newtonian models, there is a significant difference between these two simulations in vessels with smaller dimensions (such as vessels inside the nidus). ConclusionBy increasing the amount of nidus occlusion, the blood pressure is increased, so the blood supply process is better. According to a significant difference between the Newtonian and non-Newtonian simulations in vessels with smaller dimensions (such as vessels inside the nidus), therefore, non-Newtonian simulation should be done for different occlusions of 30, 50, and 90%.

Non-Newtonian magnetic nanofluid flow with activation energy: Updated nanofluid model

The study of the stagnation flow of magnetic nanofluids is an appealing topic of research with various applications in magnetic resonance imaging (MRI) machinery, rotatory seals in hard drives, and the polymer industry. The presence of some kind of chemical reaction in the chemical species makes it essential to measure the activation energy effects on the mass transfer of the magnetic nanofluids. The current study presents a novel idea of examining the activation energy effects on the flow field of a tangent hyperbolic ferrofluid that is driven by a quadratically stretchy sheet. The updated non-homogeneous nanofluid model of Buongiorno is adopted to emphasize the non-Newtonian aspects of the nanoparticle sliding processes. The Kelvin body forces are produced by a magnetic source dipole at a set distance away from the sheet. Additionally, the equation of energy is modified by the thermophoretic heat diffusion caused by the magnetic dipole. The mathematical model is transformed to a set of connected non-linear ODEs with the application of relevant dimension-free variables and tackled numerically using the bvp4c algorithm. According to the graphical display, the velocity profiles are elevated with the power law index (n) in the vicinity of ( 0 ≤ Y ≤ 0.7 ). However, above this area, an inflection point in the velocity profiles causes the velocity to decrease with n. The skin friction is hampered with activation energy ( E * ), whereas the heat transfer rate is augmented with it. The rate of mass is augmented with the quadratic stretching parameter (B).

On the strong solution for a diffuse interface model of non-Newtonian two-phase flows

The main purpose of this paper is to study the well-posedness of a diffuse interface model of non-Newtonian two-phase flows in [Formula: see text]. First, we prove the local existence of a unique strong solution provided that the initial velocity and initial concentration of two phases are sufficiently regular. Then, by using the energy estimates and standard continuity argument, we show that there exists a unique global strong solution provided that the initial velocity and initial concentration are sufficiently small.

The statistical characteristics and auto-regeneration of backflow in non-Newtonian turbulent pipe flow

The backflow phenomenon in shear-thinning and shear-thickening fluids is investigated in pipe flows at friction Reynolds number Reτ=180 via direct numerical simulations. Conditional average results show that the extreme fluctuation of wall shear stress around the backflow regions is more abrupt under the shear-thinning effect. The statistical characteristics of the backflow at different flow indices from 0.5 to 1.5 show remarkable differences. The probability of the backflow events at the wall increases in both the shear-thinning and the shear-thickening fluids under different mechanisms. The backflow occurs more frequently and exists further away from the wall in the shear-thinning fluids owing to the suppressed near-wall turbulent structures and the laminarization at low flow indices. The increase in the probability of the backflow events in the shear-thickening fluids is caused by increased Q2 and Q4 events in the near-wall region. The variation in the size and the lifespan of the backflow regions with the flow index is very prominent which both increase with the shear-thinning effect and decrease as the flow becomes dilatant. In the weakly turbulent flow of shear-thinning fluid, large backflow regions appear near the leading edge of the turbulent spots where the off-axial turbulent fluctuations are significantly lowered. Observations show the linked evolution between the hairpin vortices and the backflow regions induced underneath the strong spanwise rotations. The backflow follows the auto-regeneration process of the hairpin vortices in a packet which results in coherent streamwise-aligned backflow regions under the hairpin packets confined closer to the wall.

A numerical exploration of the comparative analysis on water and kerosene oil-based Cu–CuO/hybrid nanofluid flows over a convectively heated surface

The fluid flow over an extending sheet has many applications in different fields which include, manufacturing processes, coating, thin film decomposition, heat and mass transfer, biomedical applications, aerospace engineering, environmental science, energy production. Keeping in mind these applications, the non-Newtonian hybrid nanofluid flow comprising of Cu and CuO nanoparticles over an extending sheet is analyzed in this work. Two different base fluids called kerosene oil and water have been incorporated. The sheet is considered to be thermally convective along with zero mass flux condition. The main equations of modeled problem have been transformed to dimensionless form by using similarity variables. The designed problem is evaluated computationally by using bvp4c Matlab function. Validation of the present results is also performed. The impacts of magnetic, Brownian motion, chemical reaction, suction and thermophoresis factors are analyzed and discussed in details. The outcomes of the present investigation declare that the kerosene oil-based hybrid nanofluid flow has greater velocity and concentration profiles than that of the water-based hybrid nanofluid flow. The water-based hybrid nanofluid has greater temperature distribution than that of kerosene oil-based hybrid nanofluid flow. The streamlines of the kerosene oil-based Newtonian and non-Newtonian hybrid nanofluid flows are more stretched than water-based Newtonian and non-Newtonian hybrid nanofluid flows.

Examining the interfacial behavior of non-Newtonian gas-liquid two-phase flow in horizontal square microchannels

This study investigates the flow dynamics of a non-Newtonian two-phase flow within a horizontal square microchannel. The experimental apparatus employed an acrylic channel with a side length of 8 × 10−4 m. Nitrogen served as the test gas, and the test liquids consisted of water and CMC (Carboxymethylcellulose) aqueous solutions at concentrations of 0.2 and 0.4 %wt. The superficial velocity of working gas and liquids spanned ranges of 0.26–20 m/s and 0.05–1.0 m/s, respectively. A differential pressure transducer was used to measure pressure gradients, while a high-speed video camera captured the flow behavior for analysis with a custom image processing technique. Nonlinear regression statistical analysis was applied to establish the criteria for flow pattern transition. Furthermore, signal processing techniques such as PSD (Power Spectral Density), DWT (Discrete Wavelet Transform), ANN (Artificial Neural Network), and Kolmogorov entropy were employed to analyze the flow behavior based on the pressure gradient signals from the differential pressure transducers. Our findings reveal the occurrence of bubbly, slug, slug-annular, and churn flow patterns. The nonlinear regression statistical analysis offered ambiguous results for determining flow pattern transitions. However, an empirical coefficient was obtained for the nondimensional liquid height. Moreover, the empirical nondimensional liquid height aligned well with the experimental values. The PSD and DWT analysis corroborated each other in clearly characterizing the flow pattern based on pressure gradient fluctuation and wavelet energy. The Kolmogorov entropy effectively captured the chaotic flow pattern. Furthermore, the ANN analysis demonstrated a solid agreement between the predicted and actual flow patterns.

Unsteady non-Newtonian fluid flow past an oscillating vertical plate with temperature-dependent viscosity: A numerical study

The analysis of non-Newtonian fluid flow over an oscillating surface often involves numerical simulations and experimental investigations. Computational fluid dynamics method including finite difference or finite element techniques can be used to crack the governing equations of the fluid flow. In this work, we used the Crank–Nicolson numerical technique to analyze the numerical behavior of unsteady boundary layer flow of Casson fluid with natural convection past an oscillating vertical plate. The temperature-dependent viscosity is assumed for the flow analysis. The impact of chemical reaction and heat generation coefficient is used to examine the mass and heat transferal rates. The investigation of non-Newtonian fluid flow over an oscillating surface is crucial for a wide range of industrial, biomedical, and scientific applications. The governing model of equations occurs in the form of nondimensional PDEs and then we use the dimensionless variables in order to achieve the dimensional PDEs. These equations are numerically solved by using the Crank–Nicolson technique. The Crank–Nicolson scheme is used because it has the ability to provide accurate and stable solutions and make it a valuable numerical technique in various scientific and engineering disciplines. The findings indicate the significance of numerous parameters on the mass, velocity and energy regions. The numerical outcomes of skin friction are observed due to fluid parameter, viscosity parameter, Grashof numbers of heat and solutal rates.

2-D Modeling of hyperconcentrated fluid flow in curvilinear coordinates: dam break study

The hyperconcentrated fluid flow occurs as a result of heavy rainfall, during which a large amount of sediments from the upstream basin is washed away and suddenly increases the flow concentration of the alluvial channels. The stresses exerted by this type of fluid on the bed and body of the stream/river and related structures such as dam lead to the failure them and cause many human and financial losses. One of the important topics in the simulation of dam break caused by non-Newtonian fluid flow is the modeling of frictional stresses. In this research, after collecting several relationships to model the coefficient of friction loss of non-Newtonian fluid, a two-dimensional model was developed based on the numerical solution of shallow water equations in curvilinear coordinates to simulate hyperconcentrated flow. The results of the validation of the model were presented by comparing the measurement data of the suddenly complete dam break caused by the non-Newtonian fluid flow in the form of graphs, which all emphasize the accuracy of the developed model. It was also shown that for a suddenly complete dam break, with an increase in fluid volume concentration from 13.8 to 36.4%, the flow depth at the failure site increases by 18.8%. Next, asymmetric two-dimensional partial dam break of non-Newtonian fluid was simulated and compared with the results of Newtonian fluid. The results showed that the maximum flow velocity in the center of the fracture wall for the non-Newtonian fluid with a concentration of 32.2% is less than half of the maximum velocity of the Newtonian fluid.

Entropy generation of thermophysical properties on heat and mass transfer pulsatile flow of non-Newtonian nanofluid

Purpose The purpose of this study is to investigate the effects of entropy generation of some embedded thermophysical properties on heat and mass transfer of pulsatile flow of non-Newtonian nanofluid flows between two porous parallel plates in the presence of Lorentz force are taken into account in this research. Design/methodology/approach The governing partial differential equations (PDEs) were nondimensionalized using suitable nondimensional quantities to transform the PDEs into a system of coupled nonlinear PDEs. The resulting equations are solved using the spectral relaxation method due to the effectiveness and accuracy of the method. The obtained velocity and temperature profiles are used to compute the entropy generation rate and Bejan number. The influence of various flow parameters on the velocity, temperature, entropy generation rate and Bejan number are discussed graphically. Findings The results indicate that the energy losses can be minimized in the system by choosing appropriate values for pertinent parameters; when thermal conductivity is increasing, this leads to the depreciation of entropy generation, and while this increment in thermal conductivity appreciates the Bejan number, the Eckert number on entropy generation and Bejan number, the graph shows that each time of increase in Eckert will lead to rising of entropy generation while this increase shows a reduction in Bejan number. To shed more light, these results were further demonstrated graphically. The current research was very well supported by prior literature works. Originality/value All results are presented graphically, and the results in this article are anticipated to be helpful in the area of engineering.

Numerical Simulations of Chemically Dissipative MHD Mixed Convective Non-Newtonian Nanofluid Stagnation Point Flow over an Inclined Stretching Sheet with Thermal Radiation Effects

The study of non-Newtonian nanofluid stagnation point flow over an inclined stretching sheet with thermal radiation effects aims to understand how the fluid's non-Newtonian behavior, nanoparticles, the inclined sheet, and thermal radiation affect velocity profiles, temperature distribution, shear stress, and heat transfer rates. It might be used in materials processing, chemical engineering, and energy systems, where understanding fluid behavior in complicated settings is essential for process optimization and system efficiency. The flow problem is reflected in a set of partial differential equations (PDEs) that serve as the governing equations. After appropriate reformatting into Ordinary Differential Equations (ODEs). Mathematica's NDSolve technique is implemented to do a numerical treatment of the dimensionless equations once they have been translated. The upsides of this strategy lie in its ability to automatically track errors and select the best algorithm. Various dimensionless parameters effects on velocity, temperature, and nanoparticle concentration have been studied, and the results are graphically shown. These include the Casson parameter, Brownian motion and thermophoresis, chemical reaction parameter, thermal radiation, viscous dissipation, and mixed convection parameter. The Casson parameter slows down the velocity and speeds up the distributions of temperature and concentration. The skin friction coefficient increases rapidly with increasing tilt and thermophoretic impact amplitudes. The insights were cross-referenced with previous inquiries in order to validate their veracity. All indications are that it complies rigorously and is highly accurate.

Power-law fluid annular flows between concentric rotating spheres subject to hydrodynamic slip

ABSTRACT We report analytical solutions to the problem of non-Newtonian power-law fluid flows in the annular space between a pair of concentric spherical surfaces rotating at distinct angular velocities with the inner and outer wall boundaries subject to general asymmetric hydrodynamic slip conditions. Analytical solutions are possible because of assuming constant valued apparent hydrodynamic slip lengths in the linearized kinematic slip conditions, and our solutions can be validated against the limiting results of Newtonian fluids, no-slip conditions or a single rotating sphere reported in previous literature. Comprehensive systematic parametric studies show that (additional to the power-law fluid flow behavior index) the degree of hydrodynamic slip at the inner surface is the dominant factor that determines the limiting values of the viscous torque exerted on the inner sphere as the outer-to-inner radius ratio assumes significantly large values. Nonetheless, the flow behavior index and outer slip length prove to be the crucial key parameters responsible for a variety of torque responses, which can be categorized by a compact analytical expression, as the outer-to-inner radius ratio is increased in the small to moderate regime. We propose a criteria which identifies the proper slip length and outer-to-inner radius ratio combinations for a given power-law flow behavior index such that the hydrodynamic slip wall effects of the outer surface can be minimized or eliminated. A simple method is also presented to characterize and quantify the apparent hydrodynamic slip effects by use of the concentric rotating spheres viscometer.

Computational analysis of Carreau fluid flow in presence with buoyancy force, viscous dissipation, and chemical reaction

Abstract This study presents the computational analysis of heat and mass transfer of an in-compressible electrically conducting non-Newtonian Carreau fluid flow in the presence of buoyancy forces, viscous dissipation, chemical reaction, and heat source/sink. Mathematical modeling is made subject to an inclined disk, which is linearly stretching and also momentum, thermal, and concentration boundary layer are used to simplify laws of fluid mechanics. A similarity transformation is implied to convert the governing system of PDE's into system of ordinary differential equations and numerical solution is obtained by the Software Matlab. The impact of pertinent parameters involved in the velocity, temperature, and concentration profiles are shown through graphs and the results for Sherwood number, skin friction, and Nusselt number are demonstrated through the tables.

Cattaneo-Christov heat and mass fluxes model of Casson fluid employing non-Fourier double diffusion theories with ion slip and Hall effects

It is vitally important to comprehend non-Newtonian fluids flow behaviour from an industrial perspective. Non-Newtonian fluids are needed for a number of industrial and technical processes, including the production of paper, photographic films, and polymer sheet extrusion. There are several industrial applications for heat and mass transfer. However, thermal and solute relaxation time phenomena can't be predicted by classical heat and mass transfer laws (Fourier and Fick laws). This work aims to apply the modified Ohm law to thermal and mass transfer which are based on Fick's and generalized Fourier principles, respectively. The current study examines the three-dimensional Darcy-Forchheimer flow of non-Newtonian fluid “Casson fluid” via porous medium with ion slip and Hall effects across a stretched sheet. Thermophoresis and Brownian motion aspects are also considered. The diffusion phenomena are modelled using the Boungrino model. The modified Buongiorno model (MBM) for nanofluids is used to investigate the enhancement of heat transport. Together with practical nanofluid properties, it covers the mechanics of thermo-migration and random motion of nanoparticles. By adding the appropriate similarity transformations, this collection of (PDEs) that reflect the mathematical model is transformed into an (ODEs) system, which it is subsequently resolved utilizing the Lobatto IIIA technique's powerful computing capabilities via Matlab software. Data visualisations and numerical examples are presented considering many physical restrictions affect the fluctuation of velocities, temperatures, concentration, dimensionless shear stress, Nusselt number, and Sherwood number. According to certain findings, the existence of Hall and ion slip effects, and other factors contribute to an increase in horizontal velocity distribution. Furthermore, Fourier's law produces a wider temperature distribution than the Cattaneo-Christov heat flux model.

Numerical and optimal thermal analysis of non-Newtonian fluids flow over a wedge by using response surface methodology (RSM): Sensitivity analysis

The numerical and optimal thermal analysis of Casson fluid flow over a wedge is investigated in this work. To perform numerical and sensitivity analysis we have convert third order non-linear PDEs into set of ODEs by using similarity transformations along with boundary conditions. These equations are simulated using numerical technique Keller Box method and then compared the values of skin friction coefficient and Nusselt number with those existing in literature and found excellent agreement. Then by using these numerical data we have performed sensitivity analysis by using Response Surface Methodology (RSM). ANOVA tables are generated and developed a correlations between input parameters and output responses. The coefficient of determination for output responses skinfriction coefficient and Nusselt number are 96.50% and 99.51% respectively which shows the best fitted correlations. Finally, we have performed sensitivity analysis which reflects that skinfriction coefficient is more sensitive to Falkner–Skan exponent (m) and there is no effect of Prandtl number (Pr) on skinfriction coefficient. It is also concluded that Nusselt number is sensitive to Prandtl number (Pr) and obtains its optimum value with variation in Prandtl number (Pr).

Comparative Analysis of Non-Newtonian and Newtonian Fluid Flow with Dual Slip in the Presence of Motile Microorganisms and Nanoparticles

Comparative Analysis of Non-Newtonian and Newtonian Fluid Flow with Dual Slip in the Presence of Motile Microorganisms and Nanoparticles

Study on the hydrodynamics of the helix with non-uniform helical radius in viscoelastic fluid at low Reynolds number

Bacteria exhibit swimming behavior through the rotation of helical flagellar filaments, a common mode of propulsion for artificial micro-swimmers as well. However, most studies have focused on helical filaments with a uniform radius, leaving limited research on filaments with nonuniform radii. This paper presents a simulation of the hydrodynamics of a rotating helical filament with a nonuniform radius. The degree of nonuniformity is characterized by a radius ratio,β. Using the OpenFOAM platform, a simulation module for non-Newtonian Stokes flow was developed and integrated. The kinematics of the filament were predefined by fixing either the pitch or the helical angle, in addition to the helical radius. Thrust acting on the filament were analyzed with regards to the helical radius ratio, rotation speed, and fluid viscoelasticity. Certain modifications were made to apply the resistive force theory (RFT) to filaments with nonuniform helical radii. Results revealed that when the pitch was fixed, the thrust of the helix varied non-monotonically with the radius ratio, with excessive ratios leading to increased resistance. In contrast, helices with fixed helical angles exhibited an increasing thrust with a higher radius ratio. The thrust was greater in Newtonian fluids compared to non-Newtonian fluids, and the propulsion of the helix was influenced by the Deborah number. As the Deborah number approached zero, the thrust of viscoelastic swimming speed approached that of viscous swimming speed. For small, β(,β=1 and 4), the ratio of the thrust in viscoelastic fluid to the thrust in viscous fluid was always less than unity. At, β = 9, elasticity hindered propulsion at low De (De<1.5), while the opposite was observed at high De (De>1.5). A new approach was proposed to evaluate the impact of viscoelasticity on the helix, it was expected that the new calculation method can reduce the computational effort effectively.

Modeling of Shear Flows over Superhydrophobic Surfaces: From Newtonian to Non-Newtonian Fluids.

The design and use of superhydrophobic surfaces have gained special attentions due to their superior performances and advantages in many flow systems, e.g., in achieving specific goals including drag reduction and flow/droplet handling and manipulation. In this work, we conduct a brief review of shear flows over superhydrophobic surfaces, covering the classic and recent studies/trends for both Newtonian and non-Newtonian fluids. The aim is to mainly review the relevant mathematical and numerical modeling approaches developed during the past 20 years. Considering the wide ranges of applications of superhydrophobic surfaces in Newtonian fluid flows, we attempt to show how the developed studies for the Newtonian shear flows over superhydrophobic surfaces have been evolved, through highlighting the major breakthroughs. Despite the fact that, in many practical applications, flows over superhydrophobic surfaces may show complex non-Newtonian rheology, interactions between the non-Newtonian rheology and superhydrophobicity have not yet been well understood. Therefore, in this Review, we also highlight emerging recent studies addressing the shear flows of shear-thinning and yield stress fluids in superhydrophobic channels. We focus on reviewing the models developed to handle the intricate interaction between the formed liquid/air interface on superhydrophobic surfaces and the overlying flow. Such an intricate interaction will be more complex when the overlying flow shows nonlinear non-Newtonian rheology. We conclude that, although our understanding on the Newtonian shear flows over superhydrophobic surfaces has been well expanded via analyzing various aspects of such flows, the non-Newtonian counterpart is in its early stages. This could be associated with either the early applications mainly concerning Newtonian fluids or new complexities added to an already complex problem by the nonlinear non-Newtonian rheology. Finally, we discuss the possible directions for development of models that can address complex non-Newtonian shear flows over superhydrophobic surfaces.

Heat transfer in the melting of ice slurry during flow in a vertical slit channel

The article concerns the heat transfer process of ethanol ice slurry during downward and upward flow in a vertical slit channel (3x35.8x700mm) under constant heat flux density conditions. Experimental studies were conducted for three initial concentrations of ethanol (Xvai = 10.5 %, 13.2 %, 15.8 %) and mass fractions of ice of 0 ≤ xs ≤ 30 %. For upward flow, heat transfer coefficients were 6–14 % higher than for downward flow. For laminar flow, a greater effect of the melting process was observed, and for turbulent flow, the convection process had a greater effect on heat transfer coefficients. For laminar flow, the heat transfer coefficients of the ice slurry can be 200 % higher than the corresponding values for the carrier fluid. In turbulent flow, the corresponding increase in heat transfer coefficients did not exceed 15 %. For generalised non-Newtonian flow of ice slurry, criterion relationships were proposed for calculating heat transfer coefficients, taking into account the effect of the concentration of the carrier fluid, mass fractions of ice, its melting process, as well as the nature and direction of the flow.