Prandlt Number Research Articles

Hall effect on mixed convection MHD flow of nanofluid with clean, copper and Ag water through a vertical rotating channel

AbstractThis article aims to investigate the hall current consequences on the transfer of clean water containing silver in addition to copper nanoparticles suspended nanofluids through vertical rotating channel. Boussinesq approximation is taken into consideration. Exact analytical solutions of governing equations subject to physical boundary conditions are obtained using the techniques of non‐dimensionalization. The impacts of Grashof number, radiation parameter, Prandlt number, frequency parameter, magnetic parameter, Hall parameter, parameter of rotation and its effects on primary as well as secondary velocity and temperature profiles are presented graphically. We analyze the Hall effects and mixed convection in magnetohydrodynamic (MHD) flow of nanofluids through a vertical rotating channel, based on extensive research and applications. An analytical approach is used to derive the exact solution to this problem. A comparative study for silver and copper nanoparticles is made. A tabular solution for impacts of said parameters on convection rate and skin friction coefficient is also presented. This study is applicable for geophysical dynamics and engineering applications.

Designing a Fully‐Tunable and Versatile TKE‐l Turbulence Parameterization for the Simulation of Stable Boundary Layers

AbstractThis study presents the development of a so‐called Turbulent Kinetic Energy (TKE)‐l, or TKE‐l, parameterization of the diffusion coefficients for the representation of turbulent diffusion in neutral and stable conditions in large‐scale atmospheric models. The parameterization has been carefully designed to be completely tunable in the sense that all adjustable parameters have been clearly identified and the number of parameters has been minimized as much as possible to help the calibration and to thoroughly assess the parametric sensitivity. We choose a mixing length formulation that depends on both static stability and wind shear to cover the different regimes of stable boundary layers. We follow a heuristic approach for expressing the stability functions and turbulent Prandlt number in order to guarantee the versatility of the scheme and its applicability for planetary atmospheres composed of an ideal and perfect gas such as that of Earth and Mars. Particular attention has been paid to the numerical stability and convergence of the TKE equation at large time steps, an essential prerequisite for capturing stable boundary layers in General Circulation Models (GCMs). Tests, parametric sensitivity assessments and preliminary tuning are performed on single‐column idealized simulations of the weakly stable boundary layer. The robustness and versatility of the scheme are assessed through its implementation in the Laboratoire de Météorologie Dynamique Zoom GCM and the Mars Planetary Climate Model and by running simulations of the Antarctic and Martian nocturnal boundary layers.

Oscillatory onset of rotating thermal convection subject to spatially varying magnetic fields and stable stratification

Convective fluid motions in deep planetary cores induce spatially and temporally varying magnetic fields observable as secular variation. Oscillatory instabilities occurring as onset modes in rapidly rotating thermal convection influenced by heterogeneous magnetic fields and stratification, investigated in this study, can be potentially linked to such observations. A plane layer approximation to near-polar and near-equatorial regions of the Earth's outer core is considered. In addition to penetrative convection and flow suppression effects, the simultaneous interaction of convective instabilities with asymmetrical magnetic fields and stable stratification is the focus of this study. The fundamental modes of magnetoconvection onset are preserved even under stable stratification, although the convection threshold is lowered irrespective of the parameter regime. The axial invariance of rapidly rotating columnar convection loses its midplane symmetry with intense localization of kinetic energy in the unstably stratified regions. The parameter regimes supporting the onset of the magnetically modified viscous oscillatory (mVO) modes are further extended toward Earthlike conditions such as high thermal conductivity (low Prandlt number, Pr) and magnetic dissipation (low Roberts number, q). Moreover, compared to fully unstable stratification, partial stable stratification enhances the range of imposed magnetic field length scale (δ) over which the onset of mVO modes is favored. The critical field length scale (δ⋆), above which the onset regime of the mVO modes is bounded by a minimum q, is determined. Irrespective of the rotation rate, below δ=δ⋆, the onset of the mVO mode is supported for asymptotically small q for Pr values close to the Earth's outer core.

Radiative MHD Boundary Layer Flow and Heat Transfer Characteristics of Fe-Casson Base Nanofluid over Stretching/Shrinking Surface

In this paper, there is considered MHD boundary layer flow and heat transfer characteristics of Fe-Casson base nanofluid over an exponentially stretching/shrinking surface along the heat source/sink and Newtonian heating effects. In this regard, to develop the system of the governing equations, the one phase model named as Tiwari and Das model is considered with iron nanoparticles. The non-linear governing PDEs are first changed into the system of ODEs using suitable similarity transformations. Later on, the equations are solved numerically by using bvp4c in Matlab software. Effects of certain physical parameters on skin friction coefficient and the local Nusselt number are illustrated graphically. Moreover, the velocity and temperature profiles are examined to observe the influence of various physical parameters such as, Casson, magnetic, suction, radiation, Newtonian heating, heat source/sink and the nanoparticles volume fractions. It is seen that an increase in Casson, magnetic, suction and the nanoparticle volume fractions decrease the velocity profiles for both shrinking and shrinking cases of surface. The temperature profile recedes due to augmentation of Prandlt number and the suction parameter for both stretching/shrinking case while increases with increase in Magnetic, radiation and nanoparticles volume fractions.

Role of suction/injection on electromagnetohydrodynamics natural convection flow in a porous microchannel with electroosmotic effect

This paper examined the role of suction/injection on time-dependent electromagnetohydrodynamics (EMHD) natural (free) convection flow in a vertical microchannel with electroosmotic effect. With the aid of Laplace transformation method, the governing energy and momentum equations are converted from partial differential equation (PDE) into ordinary differential equation (ODE) to obtain fluid temperature and velocity in Laplace domain. The semi-analytical solutions of the velocity profile and temperature distribution have been derived using the Riemann sum approximation. After which a MATLAB program was written to study the effects of Prandlt number Pr, Hartmann number Ha, electric field strength on x and z directions (Ex and Sz) and Grashof number Gr in fluid velocity, temperature, skin-friction and mass flow rate in terms of line graphs. Result shows the role of suction/injection parameter alters the temperature distribution and velocity profile, so also how effective the governing parameters contribute to the flow formation. .

A stagnation point flow of cross nanofluid flow: Levenberg Marquardt backpropagation computational approach

AbstractIn the present article, the intelligence based numerical investigation by Levenberg–Marquardt algorithm with Back‐propagated Artificial Neural Networks (LMA–BANN) is exploited to analyze the nonlinear radiative stagnation point flow of cross nanofluid system (NRS‐CNFS) past a stretching surface. The designed NRS‐CNFS, initially represented by system of partial differential equations (PDEs), is converted into system of non‐linear ODEs through the applicability of mathematical conversion analysis. The desired reference solution in the form of a dataset for LMA–BANN is achieved from the Adam method for NRS‐CNFS's different scenarios by varying the magnetic parameter (Ha), Prandlt number (Pr), thermal radiation parameter(R), Eckert number (Ec), thermophoresis diffusion coefficients (Nt), and non‐dimensional activation energy (E). The obtained results are construed for the NRS‐CNFS through the performances of the testing, validation, and training. In addition, the comparison is provided using the LMA ‐BANN analysis, which is validated in terms of regression, fitness estimations, and histograms along with MSE performances. Hence, the article provides an impeccable innovative approach to problem‐solving by implementing a soft computation paradigm. It promotes the use of an effective and dependable alternative framework that is based on soft computing environments and emphasizes descriptive analysis to successfully handle the problems brought on by varied physical features.

Heat flux and forces acting on a vaporising droplet in a superheated vapor flow

In this paper, we present Direct Numerical Simulations of the interaction between a vaporising spherical droplet and a high-temperature vapour flow. The numerical simulations investigate on the Nusselt number and drag force when varying the Reynolds, Prandtl and Spalding numbers. Classical correlations based on experimental and numerical approaches, are revisited by performing simulations with interface capturing methods. The parametric study carried out by varying Reynolds and Spalding numbers, shows a quantitative agreement with previous correlations on the Nusselt number for a given Prandtl number. However, new trends are obtained by varying the Prandtl number. In particular, it is observed that the Nusselt number variations depend only on the Péclet number. Regarding the total force, we present an analysis on the three components acting on a vaporising droplet, i.e., the forces related to viscous effects, pressure effects and vaporisation effects. Our numerical investigations enable to clarify the impact of the component related to phase change, and a simple semi-empirical model is proposed to describe this force. It is shown that the three components of the total drag force depend on all the dimensionless numbers, if considered one by one, whereas the sum of their contributions is almost independent on the Prandtl and Spalding numbers, unlike what suggests usual correlations. The importance of a new dimensionless number based on the ratio between the Spalding and the Prandlt numbers is also highlighted to describe the flow around the droplet.

Simulations of dynamo action in slowly rotating M dwarfs: Dependence on dimensionless parameters

Aims. The aim of this study is to explore the magnetic and flow properties of fully convective M dwarfs as a function of rotation period Prot and magnetic Reynolds ReM and Prandlt numbers PrM. Methods. We performed three-dimensional simulations of fully convective stars using a star-in-a-box set-up. This set-up allows global dynamo simulations in a sphere embedded in a Cartesian cube. The equations of non-ideal magnetohydrodynamics were solved with the PENCIL CODE. We used the stellar parameters of an M5 dwarf with 0.21 M⊙ at three rotation rates corresponding to rotation periods (Prot) of 43, 61, and 90 days, and varied the magnetic Prandtl number in the range from 0.1 to 10. Results. We found systematic differences in the behaviour of the large-scale magnetic field as functions of rotation and PrM. For the simulations with Prot = 43 days and PrM ≤ 2, we found cyclic large-scale magnetic fields. For PrM > 2, the cycles vanish and the field shows irregular reversals. In the simulations with Prot = 61 days for PrM ≤ 2, the cycles are less clear and the reversal are less periodic. In the higher PrM cases, the axisymmetric mean field shows irregular variations. For the slowest rotation case with Prot = 90 days, the field has an important dipolar component for PrM ≤ 5. For the highest PrM the large-scale magnetic field is predominantly irregular at mid-latitudes, with quasi-stationary fields near the poles. For the simulations with cycles, the cycle period length slightly increases with increasing ReM.

Comparative dynamics of mixed convection heat transfer under thermal radiation effect with porous medium flow over dual stretched surface

Due to enhanced heat transfer rate, the nanofluid and hybrid nanofluids have significant industrial uses. The principal objective of this exploration is to investigate how thermal radiation influences the velocity and temperature profile. A water-based rotational nanofluid flow with constant angular speed {Omega } is considered for this comparative study. A similarity conversion is applied to change the appearing equations into ODEs. Three different nanoparticles i.e., copper, aluminum, and titanium oxide are used to prepare different nanofluids for comparison. The numerical and graphical outputs are gained by employing the bvp-4c procedure in MATLAB. The results for different constraints are represented through graphs and tables. Higher heat transmission rate and minimized skin friction are noted for triple nanoparticle nanofluid. Skin coefficients in the x-direction and y-direction have reduced by 50% in trihybrid nanofluid by keeping mixed convection levels between the range 3 <upepsilon le 11. The heat transmission coefficient with raising the levels of thermal radiation between 0.5 < {uppi } le 0.9 and Prandlt number 7 < {text{Pr}} le 11 has shown a 60% increase.

The Effect of Concentration on the Radiative Flow of Unsteady Magneto hydro – Dynamics (MHD) in a Vertical Channel

The article discusses the effect of concentration on the radiative flow of unsteady Magneto Hydro – Dynamics (MHD) in a porous vertical channel. The approach used in this article is transversely imposed magnetic field on the combination of porosity and viscosity on unsteady incompressible fluid flow and heat transfer. The resulted sets of non – linear coupled differential equation with the governing equations are solved by perturbation technique. The following parameters: Womersely Parameter (AI), Eckert Number (EC), Reynolds Number (Re), Prandlt Number (Pr), Grashof Number (Gr), Schmidt Number (Sc), Chemical Reaction Parameter (Ki) and the Mass Grashof Number (Gc) are used in investigating the nature of the flow due to velocity, temperature and concentration profile. Graphical results are presented and discussed.

The effect of tilt on turbulent thermal convection for a heated soap bubble

We use direct numerical simulation (DNS) to explore the effect of tilt on two-dimensional turbulent thermal convection on a half-soap bubble that is heated at its equator. In the DNS, the bubble is tilted by an angle δ∈[0°,90°], the Rayleigh number is varied between Ra∈[3×106,3×109], and the Prandlt number is fixed at Pr = 7. The DNS reveals two qualitatively different flow regimes: the dynamic plume regime (DPR) and the stable plume regime (SPR). In the DPR, small dynamic plumes constantly emerge from random locations on the equator and dissipate on the bubble. In the SPR, the flow is dominated by a single large and stable plume rising from the lower edge of the bubble. The scaling behavior of the Nusselt number Nu and Reynolds number Re is different in these two regimes, with Nu∝Ra0.3 for the DPR and Nu∝Ra0.24 for the SPR. Concerning Re, the scaling in the DPR lies between Re∝Ra0.48 and Re∝Ra0.53 depending on Ra and δ, while in the SPR, the scaling lies between Re∝Ra0.44 and Re∝Ra0.45 depending on δ. The turbulent thermal and kinetic energy dissipation rates (εT′ and εu′, respectively) are also very different in the DPR and SPR. The probability density functions (PDF) of the normalized log εT′ and log εu′ are close to a Gaussian PDF for small fluctuations but deviate considerably from a Gaussian at large fluctuations in the DPR. In the SPR, the PDFs of normalized log εT′ and log εu′ deviate considerably from a Gaussian PDF even for small values. The globally averaged thermal energy dissipation rate due to the mean temperature field was shown to exhibit the scaling ⟨ε⟨T⟩⟩B∝Ra−0.23 in the DPR and ⟨ε⟨T⟩⟩B∝Ra−0.28 in the SPR. The globally averaged kinetic energy dissipation rate due to the mean velocity field is shown to exhibit the scaling ⟨ε⟨u⟩⟩B∝Ra−0.47 in the DPR (the exponent reduces from 0.47 to 0.43 as δ is increased up to 30°). In the SPR, the behavior changes considerably to ⟨ε⟨u⟩⟩B∝Ra−0.27. For the turbulent dissipation rates, the results indicate the scaling ⟨εT′⟩B∝Ra−0.18 and ⟨εu′⟩B∝Ra−0.29 in the DPR. However, the dependencies of ⟨εT′⟩B and ⟨εu′⟩B on Ra cannot be described by power-laws in the SPR.

Comparative Evaluation of the Optimal Auxiliary Function Method and Numerical Method to Explore the Heat Transfer between Two Parallel Porous Plates of Steady Nanofluids with Brownian and Thermophoretic Influences

In this study, we used the newly established optimal approach, namely, optimal auxilary function method (OAFM) along with the Adams numerical solver technique in order to investigate the heat transfer between two permeable parallel plates of steady nanofluids (HTBTP-SNFs) through Brownian and thermophoretic consequences. The new scheme model (HTBTP-SNFs) in terms of partial differential equations (PDEs) is changed to nonlinear ordinary differential equations (ODEs) by utilizing similarity transformations. The OAFM and Adams numerical methods are used to solve the resulting ODEs with boundary conditions. The OAFM along with convergence and Adams numerical method are studied in detail. The influences of the physical parameters of HTBTP-SNFs model for instance porosity parameter (m), parameter of magnetic (M), parameter of Brownian (Nb), viscosity parameter (R), Schmidt number (Sc), thermophores parameter (Nt), and Prandlt number (Pr) are discussed with the help of tabular data and graphs. The reliability and effectiveness of the technique are achieved by equating the results available in the literature.

Methodology for estimating the magnetic Prandtl number and application to solar surface small-scale dynamo simulations

Context. A crucial step in the numerical investigation of small-scale dynamos in the solar atmosphere consists of an accurate determination of the magnetic Prandtl number, Prm, stemming from radiative magneto-hydrodynamic (MHD) simulations. Aims. The aims are to provide a reliable methodology for estimating the effective Reynolds and magnetic Reynolds numbers, Re and Rem, and their ratio Prm = Rem/Re (the magnetic Prandlt number), that characterise MHD simulations and to categorise small-scale dynamo simulations in terms of these dimensionless parameters. Methods. The methodology proposed for computing Re and Rem is based on the method of projection on proper elements and it relies on a post-processing step carried out using higher order accurate numerical operators than the ones in the simulation code. A number of radiative MHD simulations with different effective viscosities and plasma resistivities were carried out with the CO5BOLD code, and the resulting growth rate of the magnetic energy and saturated magnetic field strengths were characterised in terms of Re and Rem. Results. Overall, the proposed methodology provides a solid estimate of the dissipation coefficients affecting the momentum and induction equations of MHD simulation codes, and consequently also a reliable evaluation of the magnetic Prandtl number characterising the numerical results. Additionally, it is found that small-scale dynamos are active and can amplify a small seed magnetic field up to significant values in CO5BOLD simulations with a grid spacing smaller than h = 12 km, even at Prm ≃ 0.65. However, it is also evident that it is difficult to categorise dynamo simulations in terms of Prm alone, because it is not only important to estimate the amplitude of the dissipation coefficients, but also at which scales energy dissipation takes place.

Pseudocritical Rapid Energy Dissipation Analysis of Base-Load Electrical Demand Reduction on Nuclear Steam Supply System

Effect of pseudocritical rapid energy dissipation (PRED) from Pressurizer in nuclear steam supply system of Pressurized Water Reactor, where a single event as common cause failure, of considerable reduction of base-load electricity demand causes the temperature of the reactor coolant system (RCS) to increase, and corresponding pressure increases in pressurizer and steam generators above set-points. The study employed the uses of MATLAB/Simulink library tools, to experimentally modelled pressure control as PRED, where the momentum of transport of kinematic viscosity fraction above pseudocritical point dissipated as excess energy, to maintain the safety of the Pressurizer and RCS and keep the water from boiling. The result demonstrated the significance of pressure vector and Prandlt number as heat transfer coefficients that provided detailed activities in 2-D contour and 3-D graphics of specific internal energy and other parameterization of fluid in the pressurizer.

Influence of Hartmann Number on Convective Flow of Maxwell Fluid between Two Hot Parallel Plates through Porous Medium Subject to Arbitrary Shear Stress at the Boundary

Natural convection flow of unsteady Maxwell fluid with the effects of constant magnetic force in the course of a porous media is investigated in this research work. Fluid motion between a channel of parallel plates is tempted by time dependent shear stress applied on one plate. The governing partial differential equations of a model under consideration are transformed into ordinary differential equations by Laplace transform method and then solved for temperature and velocity fields. The obtained results for temperature fields are expressed in terms of complementary error function. The influences of involved parameters likes Hartmann number, Grashf number, Prandlt number and porosity parameter, on temperature and velocity profiles are shown graphically. There is no such result regarding Maxwell fluid in the existing literature.

Thermal visualization of Ostwald-de Waele liquid in wavy trapezoidal cavity: Effect of undulation and amplitude

The present study is concerned with the numerical simulations of Ostwald-de Waele fluid flow in a wavy trapezoidal cavity in the presence of a heated cylinder situated at the center of the cavity. The work consists in characterizing the mixed convection as a function of the intensity of heat flow. The flow behaviour and temperature distribution in a cavity are the main focus of this study. The lower wall of the cavity is fixed and heated while the wavy surface is insulated and moves with a constant speed. The sloping walls are kept cold and are subject to zero no-slip conditions for velocity components. The whole setup is modeled as a set of coupled partial differential equations and is solved by the Finite Element Method. For pressure and velocity approximations, we use the stable finite element pair P2−P1, while for temperature approximation we use the space of linear polynomial as P1. The ranges of the parameters involved in the study are the Ostwald-de Wale index (0.5≤n≤1.5), Prandlt number (1≤Pr≤10), Grashof number (100≤Gr≤1000), the number of undulation (0≤λ≤4), and the non-dimensional amplitude of the wavy surface (0.02≤A≤0.06). The major findings of the study are shown using velocity profile, streamlines, and isotherms. Moreover, the kinetic energy and average (Nuavg) Nusselt number is determined for various values of the parameters involved.

Impact of non-similar modeling for forced convection analysis of nano-fluid flow over stretching sheet with chemical reaction and heat generation

Convection differential equations that include momentum, energy and mass balance equations are used for the simulations of boundary layer (BL) flows. In convection analysis, it is in execution to dimensionless the BL system through similarity variable. However, a wide range of convection differential equations cannot be non-dimensionalized using similarity variable therefore the intent of this manuscript is to develop non-similar model for the forced convective magnetic flow of a viscous fluid above an exponentially expanding surface saturated by nano-fluid. The non-similarity is due to the exponential stretching of the surface and viscous dissipation. Influences of chemical reaction and heat generation are also incorporated. The non-similarity transformation is applied to the nonlinear partial differential system (PDE’s) to transform them into dimensionless PDE’s. Non-similar system is analytically approximated by adapting local non-similarity (LNS) and then it is numerically simulated by finite difference based bvp4c algorithm to explore the influences of significant numbers named as Prandlt number, chemical reaction, magnetic field, Brownian motion, heat generation, Eckert number, thermophoresis on concentration, velocity and temperature distribution. Auxiliary consequences presume that the heat penetration in fluid rises with enhancements in Brownian motion and Prandtl number, while it can be perceived here that for positive estimations of heat generation causes the thicker temperature field. Increasing non-similarity variable and magnetic number become reason for the reduction of temperature profile and chemical reaction is reason of reduced concentration profile.

Heat transfer between two porous parallel plates of steady nano fludis with Brownian and Thermophoretic effects: A new stochastic numerical approach

The design of integrated numerical computing through back-propagated neural networks with Levenberg-Marquard system (BNN-LMS) is presented to explore the fluid mechanics problems governing the system of heat transfer between two porous parallel plates of steady nanofluids (HTPSNF) under the stimulus of thermophoretic and Brownian motion. By introducing the similarity transformations, the original system model HTPSNF in terms of PDEs is converted to nonlinear ODEs. Strength of Homotopy Analysis Method (HAM) is utilized the governing equations of original model HTPSNF to obtain the data set. Reference collection for the suggest BNN-LMS scheme is originated in terms of various scenarios associated HTPSNF such as Porosity parameter, Schmidt number, Brownian parameter, viscosity parameter, Prandlt number and thermophoric parameter. To uphold the trueness of the suggest BNN-LMS, the validation, training and testing process of BNN-LMS are accomplished to govern the estimate solution of HTPSNF for various cases and evaluation with reference results. The comparative studies and performance analyses based on outcomes of MSE, error histograms, correlation and regression intimate the effectiveness and virtue of designed LMBNN technique. Mean Square Errors in the ranges of 10−07 to 10−14 confirm the perfection of the presented methodology for the closed correspondence between suggested and reference results.

Modeling and numerical computation of nonsimilar forced convective flow of viscous material towards an exponential surface

The objective of this paper is to study the mixed convective nonsimilar flow above an exponentially stretching sheet saturated by nanofluid. The leading partial differential equations (PDEs) of the problem have been modified towards dimensionless nonlinear PDEs utilizing newly proposed nonsimilarity transformations. Furthermore, local nonsimilarity procedure up to-second truncation has been operated to change the dimensionless PDEs into ordinary differential equations (ODEs). MATLAB-based algorithm bvp4c is used to observe the consequences of the distinct parameters namely Prandlt number [Formula: see text], magnetic field [Formula: see text], Lewis number [Formula: see text], Brownian motion [Formula: see text], Eckert number [Formula: see text], thermophoresis [Formula: see text] on velocity, concentration and temperature distribution are shown in graphical portray. Additional outcomes presume the heat penetration into the fluid enhances with increase in Biot number and Brownian motion. Increasing values of [Formula: see text] and [Formula: see text] cause decrease of temperature profile.

Mixed convection heat transfer of molten salt outside coiled tube

Mixed convection heat transfer of molten salt outside coiled tube in cylinder tank was experimentally reported with Reynolds number 418–2175 and Prandlt number 5.7–8.8. Nusselt number of salt outside coiled tube was significantly higher than that inside tube, and its tendency had good agreement with available heat transfer correlations for fluid crossing tube. The heat transfer between salt and coiled tube was enhanced by forced convection, and its Nusselt number rose as Reynolds number rising. Based on available heat transfer correlations and experimental data, experimental correlation for forced convection was proposed by considering Prandtl number ratio, and it fitted with experimental data with maximum deviation of 10.1%. Natural convection played important role in heat transfer outside coiled tube for large temperature difference, and its effect increased as Reynolds number dropping. In addition, experimental correlation for mixed heat convection with combination of natural convection and forced convection was proposed, and it fitted with experimental data better than other correlations, and maximum deviation was reduced to 8.5%.