Variable Prandtl Number Research Articles

Mean temperature profiles in turbulent internal flows

We derive explicit formulas for the mean profiles of temperature (modeled as a passive scalar) in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. The derivation leverages on the observed universality of the inner-layer thermal eddy diffusivity with respect to Reynolds and Prandtl number variations and across different flows, and on universality of the passive scalar defect in the core flow. Matching of the inner- and outer-layer expression yields a smooth compound mean temperature profile. We find excellent agreement of the analytical profile with data from direct numerical simulations of pipe and channel flows under various thermal forcing conditions, and over a wide range of Reynolds and Prandtl numbers.

Numerical study of MHD flow over stretching cylinder with variable Prandtl number and viscous dissipation in ternary hybrid nanofluids with velocity and thermal slip conditions

Industrial applications in domains such as warm rolling, crystal development, thermal extrusion and optical fiber illustration are seeing a significant increase. These applications specifically focus on addressing the challenge of a cylinder in motion inside a fluid environment. Elevated temperatures may affect the viscosity and thermal conductivity of fluids. Understanding the relationship between temperature and the properties of fluids is crucial. In light of these presumptions, the primary goal of this study is to examine, under transverse magnetic field, shape factor, velocity, thermal slip conditions and viscous dissipation, how temperature-dependent fluid properties could enhance the heat transfer efficiency and performance evolution of ternary hybrid nanofluid. In order to study flow fluctuations, the impact of nanoparticle addition and improvements in heat transfer, a variable Prandtl number is also included. The use of similarity variables converts the controlling flow model from partial differential equations (PDEs) to ordinary differential equations (ODEs). Mathematica’s shooting strategy solves ODEs using the fourth-order Runge–Kutta (RK-IV) method. Numerical calculations were done after setting parameters to acquire the desired results. Analytical data are provided in tables and graphs for convenient usage. The results showed that the velocity profile increases as the values of [Formula: see text], Pr, M, Re and S grow, and decreases when the values of [Formula: see text] decrease. Re, Pr and S lower the temperature profile, whereas [Formula: see text], [Formula: see text] and Ec raise it. The skin friction profile steepens as [Formula: see text], S, Re and M increase relative to the stretched cylinder, and flattens as [Formula: see text] and [Formula: see text] decrease. The Nusselt number profile rises as [Formula: see text], Pr, S and Re decrease with [Formula: see text], Ec and [Formula: see text]. When the Prandtl number goes from 3.0 to 6.2 in a ternary hybrid nanofluid with brick-shaped nanoparticles, the Nusselt number goes up by around 55.7%.

Heat transfer in radiative hybrid nanofluids over moving sheet with porous media and slip conditions: Numerical analysis of variable viscosity and thermal conductivity

Our current numerical investigation aims to estimate the thermal transport properties of hybrid nanofluids on a moving surface that is being shrinked horizontally. The hybrid nanoparticles consist of alumina (Al2O3) and copper (Cu), suspended in water as the base fluid. The transportation phenomenon takes place in a Darcy-Forchheimer medium, which is a porous material that exhibits thermal radiation, variable viscosity, variable thermal conductivity, Joule heating, viscous dissipation, and the effects of velocity and thermal slip conditions. The governing partial differential equations (PDEs) are converted using suitable similarity transformations into non-linear ordinary differential equations (ODEs). The equations were solved using the shooting technique with the RK-IV algorithm in the MATHEMATICA program. Dimensionless velocity, temperature, skin friction, and Nusselt numbers all provide numerical outcomes. Graphs and tables show the results of an investigation into the influence of different physical characteristics on these numbers. The velocity profile exhibits an upward trend when the velocity slip and nanoparticles volume fraction rise, but a downward trend when the viscosity and Prandtl number are varied. if the thermal slip and variable Prandtl number grow, the temperature profile falls. Conversely, if the nanoparticles volume fraction and variable thermal conductivity increase, the temperature profile increases. The skin friction and Nusselt number exhibit an upward trend as the volume percentage of nanoparticles and the viscosity of the system rise. At a Prandtl number of 6.2 and a nanoparticle volume fraction of 1 %, the Xue model demonstrates a heat transfer rate increase that is 5.32 % superior to the Yamada-Ota model and 3.18 % superior to the Hamilton-Crosser model for nanofluid. In the case of hybrid nanofluid, the Xue model exhibits a 19.01 % superiority over the Yamada-Ota model and 12.197 % superior to the Hamilton-Crosser 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).

Instability onset of Rayleigh-Bénard convection for diatomic rarefied gases: Effects of molecular interaction models and flow parameters

Studies on the stability of Rayleigh-Bénard convection in rarefied gases mainly focus on monatomic gases, whereas diatomic gases are the prevailing medium in actual flows. This paper conducts a study on the Rayleigh-Bénard convection stability of diatomic rarefied gases based on the Navier-Stokes (NS) equations modified with slip boundary conditions. We aim to elaborate the effects of different molecular interaction models and flow parameters (Knudsen number Kn, Froude number Fr, Prandtl number Pr, etc.). And several interesting phenomena have been discovered: (1) Compared with monatomic gases, the instability region of diatomic gases expands towards higher Knudsen numbers and smaller Froude numbers; (2) It is generally believed that increasing rarefaction leads to flow stabilization. However, we have found that in the case of small wave numbers within the scope of our research, the rarer the gas, the higher the growth rate, which contradicts traditional cognition; (3) Competition mechanism between buoyancy and gravity in the stability of Rayleigh-Bénard convection exists; (4) When considering the variations of Prandtl number and specific heat ratio with temperature, the relative deviation of the growth rate of the most unstable mode is significant.

Magnetohydrodynamic hybrid nanofluid flow through moving thin needle considering variable viscosity and thermal conductivity

In modern science and technology, industrial applications that deal with the problem of continuously moving thin needle, surrounded with fluid in sectors like hot rolling, crystal growing, heat extrusion, glass fiber drawing, etc., are rapidly increasing. Such processes involve high temperatures which may affect the fluid properties that is, viscosity and thermal conductivity. So, it’s crucial to understand temperature-dependent fluid properties. Focused on these assumptions, the main objective of the current research work is to investigate how temperature-dependent fluid properties might improve the heat transfer efficiency and performance evolution of hybrid nanofluid in the presence of transverse magnetic field over a moving thin needle. Variable Prandtl number is also introduced to observe flow fluctuation, the effect of adding nanoparticles, and enhancement in heat transmission. The results are obtained for different needle thicknesses, temperature-dependent viscosity, temperature-dependent thermal conductivity, and heat generation. Moreover, Fe3O4/Graphene nanoparticles are considered to be dispersed in water. The governing partial differential equations of flow and heat transfer are transformed into a system of coupled nonlinear ordinary differential equations using analysis of similarity conversion. Subsequently, the numerical solution of the problem is attained by employing the MAPLE software. The fourth-fifth-order Runge-Kutta-Fehlberg (RFK45) approach is used by default in this MAPLE program to address the numerical problem of boundary value. The velocity and temperature field are pictured for different values of the parameters as well as physical quantities of interest such as skin friction coefficient and rate of heat transfer are visually depicted in graphs and tables. It is found that fluid motion and energy transport are highly regulated by the variation of magnetic field strength. As the volume fraction of Fe3O4 is increased, the heat generation, and thermal conductivity parameter vehemently enhance the temperature profile which leads to a rise in thermal boundary layer. A strong augmentation in the heat transfer rate has been found with the increment in the variable Prandtl number.

Influence of inclined magnetic field and cross diffusion on transient forced convective heat and mass transfer flow along a porous wedge with variable Prandtl number

The unsteady 2-D forced convective mass and heat transfer flow of an incompressible, viscous and electrically conducting fluid along a porous wedge is examined in this article using numerical analysis to survey the effects of a magnetic field and thermophoresis. This is done in the existence of temperature-dependent thermal conductivity, a variable Prandtl number, and the effect of cross diffusion. The governing nonlinear PDEs have been converted into nonlinear ODEs by using a similarity transformation. The modified ODEs are solved for similar solutions using the shooting technique and the Runge–Kutta fourth-order method. The temperature, velocity, concentration, thermophoretic velocity, and thermophoretic particle deposition velocity results for various parameters are provided. The Dufour number’s effect enhances the temperature profile and rate of heat transfer. Moreover, it is noted that the rate of mass transfer is meaningfully influenced by the variation of the thermophoresis parameter and Soret number. Thus, in any physical model where the thermal conductivity of the fluid is temperature dependent, the Prandtl number within the BL have to be treated as variable rather than constant.

Reduce-Order Modeling and Higher Order Numerical Solutions for Unsteady Flow and Heat Transfer in Boundary Layer with Internal Heating

We obtain similarity transformations to reduce a system of partial differential equations representing the unsteady fluid flow and heat transfer in a boundary layer with heat generation/absorption using Lie symmetry algebra. There exist seven Lie symmetries for this system of differential equations having three independent and three dependent variables. We use these Lie symmetries for the reduced-order modeling of the flow equations by constructing invariants corresponding to linear combinations of these Lie point symmetries. This procedure reduces one independent variable of the concerned fluid flow model when applied once. Double reductions are achieved by employing invariants twice that lead to ordinary differential equations with one independent and two dependent variables. Similarity transformations are constructed using these two sets of derived invariants corresponding to linear combinations of the Lie point symmetries. These similarity transformations have not been obtained earlier for this flow model. Moreover, the corresponding reduced systems of ordinary differential equations are different from those which exist in the literature for fluid flow and heat transfer that we have been dealing with. We obtain multiple similarity transformations which lead us to new classes of systems of ordinary differential equations. Accurate numerical solutions of these systems are obtained using the combination of an adaptive fourth-order Runge–Kutta method and shooting procedure. Effects of variation of unsteadiness parameter, Prandtl number and heat generation/absorption on fluid velocity, skin friction, surface temperature and heat flux are studied and presented with the help of tables and figures.

Steady blood flow through a porous microchannel with the influence of an inclined magnetic field

This research deals with steady blood flow through a porous microchannel with the influence of an inclined magnetic field. In the research, we formulate time-independent differential equations which represent the blood momentum, energy equation, and mass concentration in the flood governing the flow. The governing equations were scaled to be dimensionless using some prescribed dimensionless parameters in the work. The reduced dimensionless governing equations were further solved using the Frobenius method (FM), where the blood velocity, mass concentration and blood temperature profiles were obtained. Numerical simulation was carried out using Wolfram Mathematica, version 12, to investigate the effect of the variation of the values of pertinent parameters on the flow profiles. In the investigation, it was revealed that the variation of Grashof number, Schmidt number, and porosity parameters increased the blood velocity before decreasing to zero when the boundary layer thickness was at its peak, while the variation of magnetic field and Prandtl number reduced blood velocity. However, the magnetic field angle of inclination initially increased the blood velocity before decreasing. In a similar vein, as the Schmidt number and chemical reaction parameters decrease the mass concentration level in the fluid, the blood temperature increases for the variation of the Schmidt number. In conclusion, the problem was formulated, solved, and simulation was done successfully, where various pertinent parameters were checked. These results are very useful for clinicians and scientists who are interested in investigating the role of some pertinent factors in understanding blood circulation problems theoretically.

Iterative Method of Thomas Algorithm on The Case Study of Energy Equation

Implicit method is one of the finite difference method and is widely used for discretization some of ordinary or partial differential equations, such like: advection equation, heat transfer equation, burger equation, and many others. Implicit method is unconditionally stable and has been proved with the approximation of Von-Neumann stability criterion. Actually, implicit method is always identical to block matrices (tri-diagonal matrices or penta-diagonal matrices). These matrices can be solved numerically by Thomas algorithm including Gauss elimination using pivot or not, backward or forward substitution. Furthermore, it can be also solved using LU decomposition method with the elimination of lower triangle matrices first and then the elimination of upper triangle matrices. In this research, Thomas algorithm is used to solve numerically for the problem of convective flow on boundary layer, especially for energy equation with the variation of Prandtl number ( ).

Numerical investigation of heat transfer deterioration and the variation of Prandtl number in the main flow area under supercritical conditions

The problem of heat transfer deterioration (HTD) exists in supercritical conditions of the power plant during peak operation. The turbulence model is selected to be applicable to the cross-critical flow field, and the different simulation models results are compared with the Yamagata experimental data. The different inner diameters and heat fluxes have a significant effect on the HTD, accompanied by variation in the Prandtl number change rate. This change can be expressed by the full width at half maximum of the Pr number along the bulk fluid enthalpy. It is analyzed that the HTD is related to the layer difference between the temperature boundary layer and velocity boundary layer, and to the decrease of the thermal conductivity inside the temperature boundary layer near the wall.

Characteristics of scramjet regenerative cooling with endothermic chemical reactions

The regenerative cooling technique with endothermic reactions is prospective for scramjet engines, in which the convective heat transfer characteristics are necessary for the thermal system design. In this work, the turbulent heat transfer characteristics of hydrocarbon fuels with endothermic chemical reactions were experimentally and numerically investigated. Three kinds of surrogates and two different pressures were employed to obtain various chemical endotherms, and the distributions of physical properties and dimensionless parameters were analyzed in the chemical region. Results showed that endothermic chemical reactions improved the convective heat transfer performance, even though the variations of fluid Reynolds number and Prandtl number brought by chemistry were small. Then, a dimensionless parameter was proposed to describe the proportion of chemical endotherm capacity. The numerical results revealed that the natural logarithm of Nusselt number enhancement changed linearly with the proposed dimensionless parameter, and the slopes were irrelevant to reactant types or chemical conditions. Last, by modifying the Dittus-Boelter correlation, a novel Nusselt number correlation was put forward for the endothermic reacting flow. Based on the analysis of the existing data and the formula form, continuously increasing the fuel chemical endotherm might bring a smaller benefit of Nusselt number augmentation.

A group theoretic analysis on heat transfer in MHD thermally slip Carreau fluid subject to multiple flow regimes (MFRs)

It is well believed that computational modelling is the best remedy to study heat transfer in complicated manmade or naturally occurring phenomena claimed in general sciences. Therefore, owning such an important belief, we proposed mathematical modelling and corresponding solution of heat transfer in Carreau fluid flow with physical effects namely, heat absorption and generation, thermal slip, magnetic field, porous medium, velocity slip, and chemical reaction. The Carreau flow field is controlled by constructing mathematical equations. Lie symmetry is used to obtain the particular scaling group of transformations. These transformations are carried to step down the higher dimensional equations in terms of ordinary differential equations. Carreau fluid velocity, Carreau temperature, Carreau concentration, Nusselt number, Sherwood number, and skin friction dependency are examined towards involved flow variables. The surface physical quantities are investigated in various frames namely, porous medium, non-porous medium, magnetized flow field, non-magnetized flow field, thermal slip, and non-thermal slip fields. For Carreau fluid flow field we have noticed that the Nusselt number shows direct relation with Prandtl number variation in both non-porous and porous mediums.

Falkner–Skan Equation with Heat Transfer: A New Stochastic Numerical Approach

In this study, a new computing model is developed using the strength of feedforward neural networks with the Levenberg–Marquardt method- (NN-BLMM-) based backpropagation technique. It is used to find a solution for the nonlinear system obtained from the governing equations of Falkner–Skan with heat transfer (FSE-HT). Moreover, the partial differential equations (PDEs) for the unsteady squeezing flow of heat and mass transfer of the viscous fluid are converted into ordinary differential equations (ODEs) with the help of similarity transformation. A dataset for the proposed NN-BLMM-based model is generated in different scenarios by a variation of various embedding parameters, Deborah number ( β ) and Prandtl number (Pr). The training (TR), testing (TS), and validation (VD) of the NN-BLMM model are evaluated in the generated scenarios to compare the obtained results with the reference results. For the fluidic system convergence analysis, a number of metrics such as the mean square error (MSE), error histogram (EH), and regression (RG) plots are utilized for measuring the effectiveness and performance of the NN-BLMM infrastructure model. The experiments showed that comparisons between the results of the proposed model and the reference results match in terms of convergence up to E-05 to E-10. This proves the validity of the NN-BLMM model. Furthermore, the results demonstrated that there is an increase in the velocity profile and a decrease in the thickness of the thermal boundary layer by increasing the Deborah number. Also, the thickness of the thermal boundary layer is decreased by increasing the Prandtl number.

Solar energy aspects of gyrotactic mixed bioconvection flow of nanofluid past a vertical thin moving needle influenced by variable Prandtl number

The current analysis deals with the nanofluid flow over a vertical thin needle having gyrotactic microorganisms. Further, the incompressible liquid is electrically conducted in the existence of magnetic field. The important effect of thermophoresis and Brownian motion has been involved in the model of nanofluid. The system of dimensionless form of ordinary differential equations (ODE's) are obtained by applying appropriate transformations to the formulated problem defined by the system of partial differential equations (PDE's). Later, these reduced ODE's are tackled numerically by using Runge–Kutta-based shooting process. Finally, keeping in view of physical significance of flow parameters, the graphical analysis for prominent parameters is portrayed and discussed in detail. The outcome exposed that the velocity gradient reduces for rising values of magnetic and buoyancy ratio parameter. The upsurge in Biot number and radiation parameter enhances the heat passage. The motile microorganisms density number improves for growing values of Lewis number and declines for improved values of Peclet number.

Magnetohydrodynamics Williamson Fluid Comprising Gyrotactic Microorganisms Flows Through a Permeable Stretching Layer with Variable Fluid Properties

The present study shows the impacts of Williamson fluid with magnetohydrodynamics flow containing gyrotactic microorganisms under the variable fluid property past permeable stretching sheet. Variable Prandtl number, mass Schmidt number, and gyrotactic microorganisms Schmidt number were all considered. The momentum, energy, mass, and microorganism equations’ governing PDEs are converted into nonlinear coupled ODEs and numerically solved with the bvp4c solver using suitable transformations. The main outcome of this study is that Williamson fluid parameter constantly decreases in velocity profile, however reverse effects can be shown in temperature profile. Also, M parameter and Kp parameter enhance the heat transfer rate, concentration rate and microorganisms boundary layer thickness but declines in momentum boundary layer thickness and velocity profile. The aim of this research is to see how velocity slide, temperature jump, concentration slip, and microorganism slip affect MHD Williamson fluid flow with gyrotactic microorganisms over a leaky surface embedded in spongy medium, with non-linear radiation and non-linear chemical reaction.

Shock wave structure in non-ideal dilute gases under variable Prandtl number

This paper investigates the structure of normal shock waves for a planar steady flow of non-ideal dilute gases under variable viscosity and thermal conductivity using the Navier–Stokes–Fourier approach to the continuum model. The gas is assumed to follow the simplified van der Waals equation of state along with the power-law temperature-dependent coefficients of shear viscosity, bulk viscosity, and thermal conductivity. A closed system of nonlinear differential equations having a variable Prandtl number ( $$\Pr $$ ) is formulated. Exact analytical solutions of the shock wave structure in non-ideal gases are derived for $$\Pr \rightarrow \infty $$ and $$\Pr \rightarrow 0$$ limits, and the corresponding profiles for velocity and temperature are obtained. For $$\Pr \rightarrow 0$$ , an isothermal shock is encountered for high Mach numbers. It appears sooner in non-ideal gases. The solution profiles for $$\Pr =2/3$$ are obtained numerically and compared with the corresponding profiles for $$\Pr \rightarrow 0$$ , 3/4, and $$\infty $$ under the same initial conditions. Qualitative agreement is obtained with the theoretical and experimental results for the shock wave structure. The inverse shock thickness is computed for different values of $$\Pr $$ , and it is found that the inverse shock thickness increases with an increase in the Prandtl number. The bulk viscosity, the non-idealness parameter, the specific heat ratio, the power-law index, and the pre-shock Mach number have a significant effect on the shock wave structure.

Simultaneous effects of bioconvection and velocity slip in three-dimensional flow of Eyring-Powell nanofluid with Arrhenius activation energy and binary chemical reaction

Recent investigations relate the enhancement of extrusion processes and transportation of heat systems via solar radiation and utilization of nanoparticles. The prime aim of the current exploration is to analyze the rheology of three-dimensional magnetized Eyring-Powell nanofluid in the presence of velocity slip, thermal radiation and Arrhenius activation energy. The suggested flow model attains both nanoparticles and gyrotactic microorganisms. The dimensionless system for flow problem is introduced with utilization of convenient transformation. The self similar solution of converted equations is integrated numerically with the help of well-known shooting technique via MATLAB software. The aspects of pertinent parameters like combine parameter Ha, slip parameter α, stretching parameter β, Biot number Bi, Lewis number Le, Peclet number Pe, bio-convection Lewis number Lb are graphically influenced for velocities, temperature, nanoparticles concentration and density of motile microorganism. The nanoparticles significance for each physical quantity is comprehensively pointed out. Study revealed that an interaction of buoyancy forces resists the fluid particles movement near the surface effectively. It is further observed that slip factor enhanced the both heat and mass transportation phenomenon. An increment in local Nusselt and Sherwood numbers is resulted with variation of mixed convection parameter and Prandtl number. The reported results are valuable for the energy transportation enhancement and applications of bio-nanofuels cells.

Large eddy simulations of wall jets with coflow for the study of turbulent Prandtl number variations and data-driven modeling

A machine-learning technique is used to develop data-driven models for turbulent heat flux prediction in wall jets with co-flow The training data are obtained by performing highly-resolved large-eddy simulations of nine cases covering various flow and geometry conditions. Robust Reynolds-averaged Navier-Stokes based heat-transfer closures are obtained, and a significant error reduction in predicting adiabatic wall effectiveness is achieved using the machine-learnt models.

Experimental and semi-analytical investigation of heat transfer in nucleate pool boiling by considering surface structuring methods

ABSTRACT Pool boiling is the process in which the heating surface is submerged in a large body of stagnant liquid. In the present work, heat transfer in nucleate pool boiling is modeled both experimentally and semi-analytically with the consideration of surface structuring methods including wire electrical discharge machining (WEDM) and etching. The developed models consider variations of nucleation site density (NSD), contact angle and Prandtl number taking into account both pure and impure fluids. According to the results, by increasing wall superheat and contact angle, heat flux enhances. Also, rising Prandtl number results in heat flux enhancement. Comparing Sample 2 (structured by etching method), Sample 3 (structured by WEDM method), and Sample 4 (the surface with best performance) with Sample 1 (plain surface) indicates that the HTC is approximately 37%, 121%, and 177% higher than the plain surface, respectively. Comparison of the obtained results based on the semi-analytical model with existing experimental data and previous analytical models shows a good agreement.