Infinite Viscosity Research Articles

Specific Mixing Energy of Cemented Paste Backfill, Part II: Influence on the Rheological and Mechanical Properties and Practical Applications

The rheological properties (yield stress, flow index and infinite dynamic viscosity) and mechanical properties (unconfined compressive strength, UCS) of different cemented paste backfill (CPB) recipes must be determined during the laboratory optimization phase. However, the influence of the mixing procedure on these properties has scarcely been studied so far. The objective of this paper is to assess to what extent these properties depend on the specific mixing energy (SME) for a given type of mixer. CPB recipes were prepared based on two types of tailing (CPB-T1 and CPB-T2, also referred to as T1 and T2) at a fixed solid percentage for each type of tailing using the Omcan laboratory mixer. A mixture of 80% slag and 20% GU was used as a binder. The mixing time and the rotation speed of the mixer were successively varied. For each recipe prepared, we determined the SME, the rheological properties of fresh CPB (at the end of mixing) and the UCS at 7, 28 and 90 days of curing. The results show that yield stress and infinite viscosity decreased when SME increased in an interval going from 0.3 to 3.8 Wh/kg and 0.6 to 6 Wh/kg for CPB-T1 and CPB-T2, respectively. An increasing trend in UCS with increasing SME was also observed. Empirical equations describing the change of the rheological properties with the SME are used to estimate the change in rheological properties of CPB along the distribution system, considering the specific energy dissipation during CPB transportation. A mixing procedure for obtaining CPB mixtures that are representative of CPB deposited in underground mine stopes is suggested for laboratories who currently use a same mixing procedure, irrespective of the variable field specific energy.

Lattice Boltzmann simulation of transient blood flow in arterial geometries using a regularised, viscoplastic and shear‐thinning fluid

This paper presents a lattice Boltzmann framework for the transient simulation of blood flow using biologically inspired geometries and pressure boundary conditions. The Kuang-Luo rheological model is used to represent blood as a homogeneous continuum. This model includes the two primary non-Newtonian characteristics of blood, namely viscoplasticity and pseudoplasticity. This paper makes two contributions. First, the numerical challenges associated with zero strain rates and infinite viscosity, as a consequence of the yield stress in the Kuang-Luo model, were addressed by regularising the constitutive equation so that the viscosity tends towards a finite value at low strain rates. A two-relaxation-time operator, which exhibits improved performance over the single-relaxation-time operator and lower computational overhead than the multiple-relaxation-time operator, is employed in the collision process. The recursive relationship between the local strain rate and relaxation rate was addressed by use of an implicit solver for these two quantities. The implemented model was benchmarked against analytic solutions for Poiseuille flow between parallel plates in two dimensions and in a cylindrical tube in three dimensions. More importantly, the transient performance of the implemented model was demonstrated by matching the predicted start-up flow of the Poiseuille flow of a Bingham fluid with the corresponding analytical solution. Second, the numerical developments were applied in the simulation of transient blood flow in complex configurations. The development and implementation of physically inspired pressure profiles highlighted the shortcomings of using a sinusoidal pressure profile in the prediction of velocity and stress distributions. Finally, the simulation of blood flow in a section of a carotid artery indicated a number of flow characteristics that will be of interest to future investigations of clinical problems.

Interpretation of infinite shear rate viscosity and a nonuniform heat sink/source on a 3D radiative cross nanofluid with buoyancy assisting/opposing flow

AbstractWith respect to bionomical concerns and energy security, the performance of refrigeration systems should be enriched, which can be done by improving the characteristics of working liquids. Nanoliquids have attracted interest in the fields of engineering and industry due to their prominent thermophysical characteristics. Researchers have used nanoliquids as working liquids and noticed significant fluctuations in thermal execution. In this study, our prime aim was to study the impact of thermal radiation and varying thermal conductivity on a cross‐nanofuid with the addition of a nonuniform heat sink–source, chemical process, and activation energy (AE) together with effects of assisting and opposing buoyancy. Furthermore, the relationship of zero‐mass flux together with the mechanism of thermophoresis and Brownian motion is considered. Traditionalistic transformations gave the ordinary differential equations (ODEs), which are further dealt with the approach of the Shooting Scheme to change the boundary value problem (BVP) into an initial value problem (IVP) and a numerical comparison is made with the Matlab solver package bvp4c. Bvp4c is based upon a collocation scheme, which yields numeric outcomes for nonlinear ODEs with IVP. Impacts of the involved parameters on mass transfer profile, heat, and momentum fields are shown through graphs. Mass transfer of the cross nanofluid increases with increasing values of AE parameter. Values of physical quantities like drag forces, rate of transport of heat and mass in the case of assisting/opposing flow are tabulated. The drag force magnitudes are greater for enhancing values of M, a, and n, while on the other hand, the opposing tendency is seen for We1 and We2. The magnitude of the rate of heat transport (Nusselt number) falls for greater values of m, σ, δ, and Nt, but in contrast, it accelerates for E, Pr, and n.

Temperature-Dependent Rheological Behaviors of Binary Eutectic Mixtures of Sugar Alcohols for Latent Heat Storage: A Comparative Study with Pure Sugar Alcohols

The temperature-dependent rheological behaviors of five selected binary eutectic mixture sugar alcohols, with great potential for latent heat storage in the range of 353.15 K to 523.15 K, were investigated. It was found that the rheological behaviors of the mixture sugar alcohols depend on those of the pure compounds as well as their molar ratios. The two mixtures of xylitol (75 mol%)+erythritol and erythritol (84 mol%)+d-mannitol behave like pseudoplastic fluids with typical non-Newtonian shear-thinning behaviors, as indicated by the power law index of 0.99 (<1). The mixture of d-mannitol (70 mol%)+d-dulcitol is a nonlinear Bingham fluid, exhibiting a slight yield stress (0.001 Pa to 0.01 Pa) at low shear rates. The rest two mixtures containing the cyclic-structured inositol behave like Herschel-Bulkley fluids. The infinite shear viscosities of the eutectic mixtures over the entire temperature range appear to be higher than those of their respective pure compounds, except for inositol. The mixture of xylitol (75 mol%)+erythritol at its melting point shows higher dynamic viscosity of about 0.546 Pa·s than the values of about 0.396 Pa·s and 0.035 Pa·s for xylitol and erythritol, respectively. In addition, the activation energies of viscous flow of the mixtures, as determined by fitting the dynamic viscosity-temperature curves using the Arrhenius model, also exhibit higher values than those of their pure compounds. The activation energy of viscous flow of the mixture xylitol (75 mol%)+erythritol was determined to be about 92 400 J/mol in the supercooled liquid state, while the supercooled liquid xylitol and erythritol have much lower values of 83 500 J/mol and 51 900 J/mol, respectively. Both the increased dynamic viscosities and activation energies of viscous flow can result in deteriorated crystallization performance during latent heat retrieval.

Modeling Cuttings Transport without Drillpipe Rotation While Using the Concepts of Static and Dynamic Yield Stresses

Summary Cuttings transport, in connection with drilling fluid rheology, has been extensively studied in the literature. Despite this, contradictory results continue to be reported regarding the effect of yield stress on cuttings transport. This study uses the concepts of static and dynamic yield stresses to investigate the effects of yield stress on cuttings transport. A modified form of an existing rheological function is proposed to model static and dynamic yield stresses while incorporating flow history. Flow equations are based on the mixture approach and are numerically solved using computational fluid dynamics (CFD) methodology. Assuming that the liquid phase is homogenous and drill cuttings are noncolloidal, it is shown that the distinction between static and dynamic yield stresses diminishes as volumetric cuttings concentration increases. The Herschel-Bulkley function predicts infinite viscosity at the limit of zero shear rate and, hence, improved cuttings transport with increasing dynamic yield stress, whereas in line with the majority of experimental studies, the proposed rheological model shows that high dynamic yield stress is detrimental for cuttings transport. Comparing fluids with the same dynamic yield stress, the fluid with a larger difference between static and dynamic yield stresses has better cuttings carrying capacity. However, these results are only valid for simple yield stress fluids in which yield stress is dependent on shear rate only.

Solute–solvent interactions of ibuprofen in the aqueous solutions of 1–propanol: Volumetric, acoustic and viscometric study

Ternary system contains ibuprofen + water +1–propanol in the certain composition have been reported as forming two phases systems. Therefore solute–solvent and solute–solute interactions are necessary for a fundamental understanding of the phase equilibrium in this ternary system. In this work solute–solvent and solute–solute interactions were studied via volumetric, acoustic and viscometric methods. The experimental densities, speed of sounds and viscosities of ibuprofen in mixtures of water +1–propanol with different cosolvent proportions (w1-pro = 1.000, w1-pro = 0.950, w1-pro = 0.900, w1-pro = 0.850, w1-pro = 0.800) were measured at temperatures T = 288.15, 298.15 and 308.15 K. The apparent molar volume (Vϕ), apparent molar volume at infinite dilution (Vϕo), apparent molar volume of transfer (ΔtrVϕo), apparent molar isentropic compressibility (κϕ), apparent molar isentropic compressibility at infinite dilution (κϕo) and viscosity B-coefficient were calculated using the experimental data. The obtained apparent molar volume, apparent molar isentropic compressibility at infinite dilution and viscosity B–coefficient are explained in terms of solute–solvent and solute–solute interactions. The obtained results reveal that solute–solvent interactions decrease with increasing temperature. Also, solute–solvent interactions decrease with decreasing weight fraction of 1–propanol. This behavior may be due to preferential solvation of ibuprofen with 1–propanol.

Lattice Boltzmann method for viscoplastic fluid flow based on regularization of ghost moments

• A viscoplastic fluid flow solver is developed with lattice Boltzmann method (LBM). • “Infinite viscosity” is computed by setting the relaxation frequency to zero. • Regularization of ghost moments keeps the simulations numerically stable. • The numerical scheme is second-order accurate and good yield surfaces are obtained. • With GPU computation typical execution time for 3D simulations is 90 min. In the Lattice Boltzmann Method (LBM) the viscosity is inversely proportional to the relaxation frequency. Hence, it should be possible to represent the singularity of some viscoplastic models by setting the relaxation frequency to zero. In the present paper we take full advantage of the LBM capabilities to propose an efficient and stable numerical scheme for viscoplastic fluid flow simulations invoking the exact Bingham constitutive equation. This scheme is expected to suit the need for more accurate viscoplastic simulations because it does compute the “infinite viscosity”, therefore dismissing the need for any viscosity regularization. Although allowing for a wide range of relaxation frequencies varying in space and time, we demonstrate that the present implementation does not degrade the standard LBM's error order. Numerical stability was promoted by of regularization of ghost moments for the lattice Boltzmann equation with force term. Since the locality of the LBM is preserved, so is the scheme's ability to be highly scalable in large computer clusters. We outline the method and the theory behind it through a detailed Chapman–Enskog expansion. Three laminar test cases are analyzed: parallel channel Poiseuille flow, square duct Poiseuille flow and lid-driven cavity flow. We show conformity with the standard LBM's error order for different wall boundary conditions and yield stress levels. Comparisons are made with other numerical studies employing augmented Lagrangian and viscosity regularization methods.

Evaluation of Thermal Hydrolysis Efficiency of Sewage Sludge via Rheological Measurement

Thermal hydrolysis processes (THP) play an important role in sewage sludge treatment. In this study, both low-temperature (60°C–90°C) and high-temperature (120°C–170°C) thermal hydrolysis experiments were performed on the initial sludge (at a total solid of a 7.03% and 6.45% by weight). The results have shown that the release of organic matter is logarithmic with the duration during low-temperature thermal hydrolysis (LTHP). For high-temperature thermal hydrolysis (HTHP), the release of organic matter is linear with a treated temperature. Good linear relationships were obtained between the release of a soluble chemical oxygen demand (rSCOD) and soluble proteins/polysaccharides. The pH decreased with the increase of the treatment temperature. The apparent viscosity of the sludge decreased after the thermal hydrolysis, especially at a high temperature. The yield stress of the sludge disappeared, and the thixotropy increased. Moreover, the flow behavior index increased, and the consistency coefficient K decreased with the increase of the thermal hydrolyzed temperature. Meanwhile, we found a linear relationship between the rSCOD and increasing infinite viscosity. It demonstrated that the rheological measurement could be used as an indicator to evaluate the extent of organic matter solubilization in THP.

Effect of polymer solutions on the propagation and structure of freely translating vortex rings

We experimentally study the propagation of vortex rings in aqueous solutions of polyethylene oxide (PEO400) and hydrolysed polyacrylamide (PAMH25). We study the canonical vortex ring as a caricature problem to the phenomenon of turbulent polymer drag reduction. PAMH25 has a similar infinite shear viscosity, ten times the zero shear viscosity and a thousand times the relaxation time of PEO400. We study the vortex rings generated by the trapezoidal motion of a piston-cylinder mechanism using PIV and PLIF at constant impulse. We see that PAMH25 drastically affects the propagation properties and the structure of the vortex ring, while PEO400 shows a behaviour similar to water. We conclude that PAMH25 retards the roll-up process of the ring, due to both the large range of plateau viscosities and large relaxation time. We surmise that the behaviour of the vortex ring observed might be used to unravel the mechanism of polymer drag reduction.

Hydrodynamic flow of non-Newtonian power-law fluid past a moving wedge or a stretching sheet: a unified computational approach

A unified mathematical equation that combines two different boundary-layer flows of viscous and incompressible Ostwald-de Waele fluid is derived and studied. The motion of the mainstream and the wedge is approximated in the power-law manner, i.e, in terms of the power of the distance from the leading boundary-layer edge. It is also considered that the wedge can move in the same and opposite direction to that of the mainstream. The governing partial differential equations are transformed into the nonlinear ordinary differential equation using a new set of similarity variables. This transformed equation subjected to the boundary conditions describing the flow is then solved using the Chebyshev collocation method. Further, these numerical results are then validated by determining the flow behaviour at far-field by performing asymptotics. The velocity ratio parameter effectively captures and distinguishes two boundary-layer flows. The boundary layer thickness for shear-thinning fluid is thinner compared to corresponding parameters for shear-thickening fluids and is markedly separated by the Newtonian fluid. Further, the boundary-layer flow of the non-Newtonian fluid predicts an infinite viscosity for shear-thinning fluid quite away from the surface. The hydrodynamics of the obtained results is explained thoroughly.

Bioconvection flow of magnetized Carreau nanofluid under the influence of slip over a wedge with motile microorganisms

This article addresses the time-dependent flow of magnetized rheological Carreau nanoliquid conveying microorganisms over a moving wedge with velocity slip and thermal radiation features. Carreau fluid is auspicious to depict several types of physical issues because this fluid model has the capability of revealing the rheology of multiple specific fluids such as fluids with brief-chain suspension particles, fluid crystals, detergents, and blood in animals and humans. The mathematical formulation is developed by combining the impact of infinite shear rate viscosity. The physical aspects for both static and moving are discussed in detail. At first, relevant similarity transformations are employed to obtain dimensionless form of equations, and then renovated equations have been solved numerically by employing bvp4c via MATLAB based on shooting technique. Both the numerical and graphical results against physical quantities, such as velocity temperature, nanoparticles concentration and density of gyrotactic microorganism, are observed under the influence of physical parameters.

Investigating the effects of plug viscosity on annular pressure drop and cuttings transport in a concentric annulus

The API recommended approach to model the rheology of a drilling fluid is to use the Herschel Bulkley equation. This equation breaks down as shear rate approaches zero and predicts infinite viscosity at zero shear rate limit. In an annulus, shear rate can drop to values approaching zero and the use of Herschel Bulkley equation in flow models can lead to inaccurate prediction of drilling hydraulics. This paper investigates the effects of zero shear rate viscosity, called plug viscosity, on annular pressure drop and cuttings transport predictions in a concentric annulus. Flow is modeled using the mixture model approach and is simulated using computational fluid dynamics (CFD) methodology. The rheology of the drilling fluid is characterized using a modified bi-viscosity function. The first half of this paper examines the effect of common drilling parameters on the existence of plug zones, regions with zero shear rate viscosity, whereas the second half of this paper investigates the effects of plug viscosity on annular pressure profile and cuttings transport. The results show that plug viscosity has a significant impact on annular pressure profile and cuttings transport predictions. The same set of Herschel Bulkley parameters can produce different values of annular pressure drop and volumetric cuttings concentration if plug viscosity is ignored. Therefore, to accurately predict annular pressure profile and cuttings transport, it is imperative that plug viscosity is incorporated in the rheological functions.

Mixed convection flow and heat transfer mechanism for non-Newtonian Carreau nanofluids under the effect of infinite shear rate viscosity

The mechanisms of nanofluids for improve heat transfer features has received great consideration due to their wide applications in industry and chemical engineering. In the light of these facts, numerous theoretical and numerical works have been presented to study their applicability in predicting heat transfer with nanofluids. Therefore, the aim of present article is to perform a numerical study for non-Newtonian Carreau fluid flow induced by an inclined surface with suspended nanoparticles. In modelling the flow equations for Carreau fluid, it is assumed that the infinite shear rate viscosity does not vanish. This article quantifies and qualifies the thermal phenomenon caused by the convective heat transfer in the presence of nonlinear thermal radiation and heat generation/absorption. Basic flow equations are obtained by means of Boussinesq approximations representing the conservations of mass, momentum, energy and nanoparticles concentration. The formulated equations are altered to partially coupled nonlinear first order system by employing the non-dimensional transformations. Finally, numerical simulation is performed via Runge–Kutta Fehlberg scheme coupled with shooting technique. The investigation has been conducted for pertinent physical parameters, such as, viscosity ratio parameter, Weissenberg number, shrinking parameter, suction parameter, Grashof number, inclination angle, Brownian and thermophoresis parameters, Biot number, Prandtl number and Lewis number. The analysis reveals that dual solutions exists for a specific range of leading parameters. It is concluded that an enhancement in Grashof number leads to reduce the skin friction for lower solutions.

Change in internal energy of Carreau fluid flow along with Ohmic heating: A Von Karman application

This article examine the impact of internal energy change in steady, Carreau incompressible electrically conducting fluid flow due to a porous Von Karman problem under the effects of transverse magnetic field and Ohmic heating. The nonlinearities generated by the rheological fluid model declines the numerical convergence of outcome results, that is why infinite shear viscosity is assumed to be zero. The basic mathematical model (system of PDEs) is converted into nonlinear ODEs by applying Von Karman similarity transformations. Numerical solutions of the resultant equations are achieved by utilizing (fifth order Runge–Kutta–Fehlberg method) along with shooting technique. Detailed variations of radial and tangential components of the velocity as function of Ha number, power law index, Reynold number and We number have been computed for further analysis. Phenomena of heat and mass transfer depends on thermal-diffusion and diffusion-thermo aspects. So, Soret and Dufour effects are also added in this analysis.

Boundary-layer flow of the power-law fluid over a moving wedge: a linear stability analysis

We study the stability of the two-dimensional boundary-layer flow of a power-law (Ostwald de Waele) non-Newtonian fluid over a moving wedge. The mainstream velocity is assumed to have a power of distance from the leading boundary layer, such that the system admits to the self-similar solutions. We discuss the problem in question for both shear-thickening and shear-thinning fluids which lead to a non-uniqueness (double solutions) in the base flow solutions. We then address an issue of the stability of the non-unique solutions. A linear eigenvalue analysis of the double solution reveals that the basic flow represented by the first solution is always stable, and this flow is practically encountered. The system becomes unstable to the second solutions which have the mode-two perturbations with larger boundary-layer thickness. The first and second solutions form a tongue-like structure in the solution space. Furthermore, the modification of the viscosity for the power-law fluids reveals that the system predicts an infinite viscosity in the confinement of the boundary-layer region. Extensive comparisons of the solutions with the existing models with Newtonian fluid are made, and a physical explanation behind these solutions is proposed.

Non-Newtonian fluid flow around a Y-shaped fin embedded in a square cavity

This study investigates the free convection of Casson fluid inside a square cavity with intruded Y-shaped fin at the bottom surface. The Casson fluid is a non-Newtonian fluid and has an infinite viscosity at zero rates of shear. This is the first time that this fluid is used in the square cavity containing Y-shaped fin at the bottom surface. The Y-shaped fin helps in understanding the constructal theory and enhancing the heat transfer rate from the fin. The main objective of using this fin is to increase the convective heat transfer surface area. The sidewalls of the cavity are kept at cold temperature, the bottom surface at a hot temperature, and the top surface as adiabatic. The tips of the Y-shaped fin are considered as hot, cold, and adiabatic. The effects of the magnetic field and radiation are included in the momentum and energy equations. The influence of viscous heating is neglected. The buoyancy term is included in the momentum equation using Boussinesq approximation. The dimensionless governing equations are solved numerically using a Galerkin weighted residual technique of the finite element method. The effects of Rayleigh number (Ra = 104–106), radiation parameter (Rd = 0–103), Hartmann number (Ha = 0–103), and Casson parameter (γ = 0.1–1) on streamlines, isotherms, dimensionless velocity components, temperature, and local Nusselt numbers along the fin and the bottom heated wall are investigated and presented graphically. It is demonstrated that, in the presence of Y-shaped fin, all the pertinent parameters and dimensionless numbers help in enhancing the heat transfer rate along the bottom surface.

Volumetric and Viscometric Properties for Binary and Ternary Solutions of Diethylenetriamine, N,N-Diethylethanolamine, and Water

The experimental values of density and viscosity for binary solutions diethylenetriamine (DETA) + H2O, DETA + N,N-diethylethanolamine (DEEA), and ternary solution DETA + DEEA + H2O over the entire concentration range were reported at atmospheric pressure and T = (293.15, 303.15, 313.15, 323.15, 333.15) K. The measured densities and viscosities were used to calculate the thermal expansion coefficient, excess molar volume, partial molar volume, apparent molar volume, partial molar volume at infinite dilution, viscosity deviation, and excess Gibbs free energy of activation for viscous flow. The molecular interactions in mixtures were discussed based on the volumetric and viscometric properties.

A note on activation energy and magnetic dipole aspects for Cross nanofluid subjected to cylindrical surface

Our main emphasis in this manuscript was to scrutinize the aspects of activation energy for magnetized Cross nanofluid subjected to cylindrical surface. Formulation for energy expression is developed through heat sink-source phenomenon. More specifically, Velocity of Cross liquid is deliberated by considering infinite shear rate viscosity and Lorentz’s force. The considered Cross nanoliquid expression (Buongiorno relation) comprises thermophoretic and Brownian movement mechanisms. Moreover, Chemical processes are deliberated subjected to appliance of activation energy. Bvp4c algorithm is implemented to tackle the nonlinear structure. Outcomes for Sherwood number, Nusselt number, skin fraction, temperature, concentration and velocity are presented in this manuscript. Our results revealed that temperature of Cross nanoliquid intensifies for larger thermophoretic parameter. Moreover, nanoliquid concentration dwindles for greater estimation of activation energy parameter.

Fractional creep and relaxation models of viscoelastic materials via a non-Newtonian time-varying viscosity: physical interpretation

Fractional viscoelastic models have been confirmed to achieve good agreement with experimental data using only a few parameters, in contrast to the classical viscoelastic models in previous studies. With an increasing number of applications, the physical meaning of fractional viscoelastic models has been attracting more attention. This work establishes an equivalent viscoelasticity (including creep and relaxation) between the fractional Maxwell model and the time-varying viscosity Maxwell model to reveal the physical meaning of fractional viscoelastic models. The obtained time-varying viscosity functions are used to interpret the physical meaning of the order of the fractional derivative α from the perspective of rheology. When α changes from 0 to 1, the viscosity functions quantitatively exhibit the transformation of viscoelasticity from elastic solid to Newtonian fluid, which can be considered as an extension of the Deborah number. The infinite viscosity coefficient for α = 0 shows the elastic solid property, while the constant viscosity coefficient for α = 1 exhibits the Newtonian fluid property. The sharply decreasing viscosity coefficient (versus α) near α = 0 indicates that the elastic solid property decays rapidly. In addition, similar viscoelastic responses between the Hausdorff and fractional derivative models are found due to a similar time-varying viscosity.

Importance of entropy generation and infinite shear rate viscosity for non-Newtonian nanofluid

This study addresses the novel characteristics of infinite shear rate viscosity and entropy generation in magneto-mixed convective flow of cross-nanomaterial toward a stretched surface. Moreover, analysis of current research work has been prepared for Brownian moment and thermophoresis deposition. Radiation and viscous dissipation aspects are accounted. More specifically, roles of activation energy and Lorentz force on nanofluids transportation are examined. ODEs are acquired from PDEs via implementation of suitable transformations. Numerical algorithm is implemented to tackle the nonlinear system for numerical results. Discussion on rheological parameters involved in current research work is presented through graphs. Results demonstrate the significant rise in temperature and nanoparticles concentration with the intensification of Brownian moment aspects. More specially, we perceived that entropy rate is significantly affected by radiation parameter and Brinkman number. Intensification in entropy rate is observed for rising values of magnetic parameter, radiation parameter and Brinkman number.