Dimensionless Systems Research Articles

Cattaneo-Christov fluxes for nonlinear mixed convective entropy generated Reiner-Rivlin material flow

Thermal and solutal transportation processes involve in tremendous useful applications related to heat exchanger, scientific, micro-electronic device technologies and industry sectors like nuclear reactors cooling, marine engineering, nanotechnology, thin-film technology, biomedical applications and thermal conduction in soft tissue. In view of such useful applications the Cattaneo-Christov analysis for nonlinear mixed convective Reiner-Rivlin material flow in presence of constant applied magnetic field is considered. Heat source, magnetohydrodynamics and radiation are incorporated in energy equation. Cattaneo-Christov flux is employed to discuss the thermal and mass transport characteristics. Entropy calculation in radiating flow is discussed. The proposed non-linear systems are transformed into dimensionless ordinary systems through appropriate transformation. Non-linear ordinary expressions are computed for convergent series solutions through Optimal homotopy analysis technique (OHAM). Graphical interpretations for quantities under interest against pertinent variables are explored. Large magnetic parameter has opposite impact for entropy and fluid flow. An increase of temperature through thermal relaxation time parameter is noticed. Velocity has same trend for both mixed convection and Reiner-Rivlin fluid variables. Reduction in concentration is seen for solutal relaxation time parameter. Temperature enhancement is witnessed in presence of heat generation. Entropy rate enhancement for radiation is observed. Decrease in concentration is noticed for reaction. Entropy rate increases against diffusion variable is enhanced.

Entropy generated nonlinear mixed convective beyond constant characteristics nanomaterial wedge flow

Background and objectiveNanomaterial flows at present have received much attention of the researchers. Such motivation stems for their utilization in pharmaceutical, industrial, petroleum, manufacturing and engineering applications including regenerative medicine, treatment of cancer, electronic device cooling, solar collector, mineral oil, antimicrobial agents, thermal storage system, transformer cooling and X-imaging etc. Higher thermal energy requirement in several electrical and mechanical systems is desired in recent time due to high advancement of science and technology. For important application here the stagnation point flow due to wedge with nonlinear convection is explored. For porous media, the Darcy-Forchheimer relation is considered. Characteristics of nanofluid is added by utilizing Buongiorno's model. Variable fluid characteristics are under consideration. Applied magnetic field is accounted. Energy expression comprises thermal radiation, Brownian motion, Ohmic heating, thermophoresis and heat generation. First order reaction and entropy rate are considered. MethodologyNonlinear differential expressions are changed into dimensionless ordinary systems through adequate transformations. The resultant nonlinear ordinary systems are solved through optimal homotopy analysis technique (OHAM). ResultsOutcomes for influential variables about temperature, velocity, concentration and entropy rate have been arranged. Here results show that for magnetic field the liquid flow and entropy rate have opposite scenarios. Thermal distribution and concentration have reverse effects for random motion variable while similar effect holds for thermophoresis parameter. Magnetic field and diffusion parameters contribute to augment entropy generation. Higher values of temperature dependent electrical conductivity variable lead to increase entropy rate. Thermal distribution for radiation and Eckert number is similar.

Entropy optimized radiative flow conveying hybrid nanomaterials [formula omitted] with melting heat characteristics and Cattaneo-Christov theory: OHAM analysis

Here flow for radiative hybrid-nanomaterial (MgO+MoS2) satisfying Darcy-Forchheimer relation is addressed. Ethylene glycol C2H6O2 is utilized as a base fluid. Magnesium oxide (MgO) and Molybdenum disulfide (MoS2) are considered as distinct nanoparticles. Thermal expression is examined through heat generation and viscous dissipation. Analysis for Cattaneo-Christov theory is carried out. Condition for melting heat is deliberated. Entropy generation is examined. Variable thermal conductivity of hybrid nanomaterial is taken. Suitable transformations are implemented to obtain nonlinear dimensionless systems. Optimal homotopy analysis method (OHAM) is implemented for computations. Temperature, velocity and entropy generation are scrutinized physically. Nusselt number and skin friction are discussed. Conclusion synthesis salient points. Present work is relevant in metallurgical engineering and polymer processing. The findings conclude that Forchheimer number correspond to decline in velocity. Entropy rate and thermal field have similar trend for heat generation variable. A decrease in velocity against melting variable is noted. Temperature increases for variable thermal conductivity and thermal relaxation parameters. Entropy optimization improved for Brinkman number. Skin friction decays for porosity parameter and Forchheimer number. Skin friction improves for melting variable and Forchheimer number. Entropy augments against higher thermal relaxation time. Higher melting variable lead to an enhancement for Nusselt number. There is an improvement for radiation through Nusselt number and temperature.

Neural network design for cubic autocatalysis chemical processes, the flow of a Darcy‐Forchheimer viscous fluid is optimized for entropy

AbstractIn this article, neural network backpropagation (NNBP) is used to present two‐dimensional Darcy‐Forchheimer viscous fluid (DFVF) of viscous liquid against a stretching sheet under the influence of heat sink/source and viscous dissipation. Mathematical modelling and numerical simulation are also discussed. Thermal polymer processing, engineering processes, and industrial processes all benefit from the dynamic entropy generation phenomenon. Scientists are primarily interested in finding ways to decrease entropy formation to polish up the thermal performance of industrial systems. In this continuation, the optimal frame for the Darcy‐Forchheimer flow with a curved surface has been developed. Non‐linear partial systems are converted to dimensionless differential systems through the employment of the appropriate variables. Utilizing tools from the bvp4c method, the problem's highly nonlinear equations are numerically solved. These ODEs are solved by applying the bvp4c solution approach to produce the reference dataset for NNBP. Using this reference dataset in MATLAB, the graphs for the solution and error analysis for changing various parameters are analyzed. The performance of NNBP is validated using mean squared error data, regression analysis, and error histogram. The DFVF solution is examined using the testing, validation, and training procedures. Examined through influential variables the effects of fluid flow, thermal field, Bejan number, and concentration.

Artificial neural network (ANN) analysis of non-similar solution of MHD nanofluid flow past a curved stretching surface

The objective of this study is to predict the non-similar solution of Magnetohydrodynamics (MHD) mixed convection transports of viscous nanofluids over a curved stretching surface by using numerical simulation with an artificial neural network (ANN) algorithm, namely the back-propagation Levenberg–Marquardt scheme. In this article, aluminum oxide and iron oxide are used as nanoparticles to enhance the thermal conductivity of the base fluid. Additionally, aluminum oxide and iron oxide nanoparticles are used in certain biomedical applications, including composite materials, photocatalysis, wastewater treatment, and nanofiltration membranes. The radiation, Joule heating, and viscous dissipation effects are studied along with the convective boundary condition. The flow problem is attained in the form of a system of Partial differential equations (PDEs) and converted into dimensionless PDEs via non-similarity transformation. These dimensionless systems of PDEs are converted into a system of Ordinary differential equations (ODEs) via local, non-similar techniques up to second-order truncations. The Bvp4c technique is used to simulate the numerical results. The data sets are collected using the Bv-p4c technique. These data sets are utilized for the validation, training, and testing phases of the developed Back-propagated Levenberg-Marquardt Scheme- Artificial Neural Network (BLMS-ANN) model to determine the solution of the problem for a variety of physical scenarios. The ANN model is utilized to choose data, construct and train a network, and evaluate the performance of the network through the use of mean square error and regression analysis. The ANN models are used to predict the performance of the Nusselt number for both nanofluids and the results demonstrated that ANN is capable of making accurate predictions of the local Nusselt number for both nanofluids. In the entire study, the ANN model predicts the result of the local Nusselt number of nanofluids with higher accuracy as compared to nanofluids.

Dynamics of Tangent Hyperbolic Fluid Past a Semi-infinite Plate with the Significance of Joule Heating, Thermal Radiation and Soret-Dufour Mechanisms

The present investigation concentrates on the unsteady flow of tangent hyperbolic liquid past a vertical plate under the influence of Lorentz force, Joule heating, and viscous dissipation. The mathematical modelling leads to nonlinear coupled partial differential equations (PDEs). Suitable non-dimensional quantities are applied to the governing PDEs to obtain dimensionless systems of equations. The transformed boundary layer PDEs are solved with the aid of the spectral relaxation method (SRM). The SRM employs the Gauss-Seidel techniques to linearize and decouple the system of nonlinear PDEs. The applied magnetic field acts as an opposition to the flow by producing the Lorentz force. The Weissenberg parameter, alongside the magnetic parameter, is observed to decline the velocity profile. An increment in thermal radiation parameter is observed to enhance the thickness of the hydrodynamic and thermal boundary layer. Therefore, the thermal condition and convective flow are improved with heat generation and thermal radiation in the flow phenomenon. This investigation is unique because it investigates the combined influence of Soret-Dufour and MHD, viscous dissipation, and Joule heating. This study plays a significant role in astrophysics, heat exchanger devices, MHD power generation, and geothermal energy extraction. When this study is compared to studies that have already been done, it agrees with those studies.

Selection of cross sections for solitary wave transmission efficient for submeged vertical porous breakwater using Flow-3D model

Climate change is causing progressively severe coastal erosion, especially in beaches and tourist areas. Vertical porous breakwaters were chosen to protect the beaches for their effectiveness in reducing waves and preventing erosion. However, submerged vertical porous breakwaters must be applied due to the characteristics of beaches that do not allow shielding. Thus, the width of the porous breakwaters should be increased, incurring construction costs. This study proposes cross-sections of submerged vertical porous breakwaters that have the same width and determines the selected cross-section in those sections that effectively meets the wave height reduction. The Flow-3D numerical model with porous structures and solitary wave boundary conditions was used to simulate 72 scenarios of the interactions between solitary waves and submerged vertical porous breakwaters. The selected cross-section is a parallel configuration, where the distance between the two breakwaters is twice the width of the first breakwater. Dimensionless graph systems defining parallel breakwater parameters according to incoming solitary waves were established.

Thermal radiation impact on chemical reactive flow of micropolar nanomaterial subject to Brownian diffusion and thermophoresis phenomenon

Background and objectiveEntropy generation (a novel prospective) in different thermodynamical processes has usefulness in polymer process. Importance of entropy optimization is noticed in nuclear reactions, heat exchangers, biological system, electronic cooling, combustion, porous media, thermal systems, turbo machinery, turbine systems and many others. Entropy minimization has been utilized for assessment of thermal energy through thermodynamical processes. Entropy minimization rate is employed to boosts up efficiency of thermodynamical system. Therefore, present investigation is aimed to investigate optimized radiative flow of micropolar nanofluid due to exponential stretched curved surface. Heat expression with heat generation and radiation is explored. Moreover, interesting aspects of homogenous and heterogeneous chemical reactions are also presented. For nanofluid characteristics the Buongiorno's model is incorporated. Velocity and thermal slip conditions are explored. Mathematical modeling of flow expressions are presented through curvilinear coordinates. Entropy rate is modeled employing second law of thermodynamics. MethodologySuitable relations have been employed to get the dimensionless systems. Resultant dimensionless systems are computed through ND-solve technique. ResultsGraphical analysis for sundry variables on liquid flow, entropy generation, temperature, microrotation velocity and concentration are addressed. An opposite scenario for microrotation and velocity holds against material parameter. Heat transport rate augments in presence of slip constraint. Change in radiation constant and thermophoresis variable correspond to higher temperature. Entropy generation has enhancing trend for radiation parameter. Concentration has opposite scenario for random and thermophoresis diffusion variables. Larger thermal ratio parameter enhances entropy generation.

Entropy optimization in radiative flow of Reiner-Rivlin material with heat source and modified Cattaneo-Christov heat and mass fluxes

The present investigation deals with magnetized convective flow of Reiner-Rivlin liquid by stretched cylinder. Solutal and thermal transport analyses are discussed through Cattaneo-Christov heat and mass fluxes. Heat source and radiation effects are considered in thermal equation. Physical descriptions of chemical reaction and entropy generation are examined. By utilizing appropriate transformations, the model is transformed into dimensionless ordinary differential systems (ODEs). The obtained non-dimensional expressions are solved for convergent solutions by using optimal homotopy analysis method (OHAM). Influences for prominent variables on flow, concentration, temperature and entropy rate are explored. It is noticed that liquid flow enhances for mixed convection variable while opposite trend observed against magnetic field. Liquid flow is enhanced for buoyancy ratio parameter. Higher thermal relaxation time parameter results in thermal field enhancement. Entropy generation enhances against higher radiation variable. An enhancement in entropy rate is found for higher Brinkman number. Larger approximation of Schmidt number results in concentration reduction. Concentration decays for higher solutal relaxation time variable. Higher approximation of reaction variable decrease concentration.

Unsteady MHD Flow of Casson Fluid Past Vertical Surface Using Laplace Transform Solution

The study explores the velocity of Casson fluid that is time-independent over an exponentially infinite isotherm vertical permeable sheet. The impact of magnetohydrodynamic (MHD) with Casson flow over the permeability sheet is examined. In contrast, thermal radiation and heat sink parameters have been incorporated. This study’s primary goal is to determine the significance of thermal radiation on Casson flow with MHD using an analytical solution over a permeable sheet. The flow of the fluid occurs above the sheet when [Formula: see text] is greater than zero, and the sheet extends far away in the [Formula: see text]-direction. The model of governing equations is reduced by applying a suitable set of dimensionless parameters. These dimensionless systems of equations are solved through the Laplace transformation method. The impacts of various variables over velocity, temperature, concentration, skin friction, and Nuseelt number are scrutinized. These variables contain magnetic field [Formula: see text], Casson fluid parameter [Formula: see text], Ghroshof number [Formula: see text], modified Grashof number [Formula: see text], Prandtl number [Formula: see text], thermal radiation [Formula: see text] and Scimdth number [Formula: see text]. These plots are sketched for the considerable magnitude of these variables through the Mathematica Software, and these plots are discussed in detail. Results show that the increasing value of [Formula: see text] reduces the fluid velocity but velocity of fluid is enhanced with larger values of [Formula: see text], [Formula: see text] and [Formula: see text]. The impact of Skin friction and Nusselt number is elaborated by tabular outlined.

Modeling and computational framework of radiative hybrid nanofluid configured by a stretching surface subject to entropy generation: Using Keller box scheme

This study examines the characteristics of the velocity, thermal field and entropy profiles for hybrid nanofluid flow passing through a starching sheet with thermal radiation. The carbon nanotube (SWCNT and MWCNT) are used as a nanoparticles with Cattaneo-Christov (CC) heat flux. Ethylene glycol is utilized as a base fluid in this case. To achieve an improved solution, the fluid flow over the geometric properties is designed using highly non-linear PDEs, and the governing equations must be converted into dimensionless non-similar equation systems using the highly efficient well-known Keller-box scheme in computational software MATLAB. The practical feasibility of these solutions is determined by the range of the controlling parameters. The velocity distribution reduces as the magnetic parameter estimate increases, however, the temperature field and entropy production increase as the magnetic parameter fluctuation esclates. As the slip parameter is increased, the velocity field diminish. The thermal field is enhanced for rising the radiation parameter, and the entropy profile is boosted for increasing Brinkman parameter values. The findings of this research might have a significant impact on industries where local cooling and heating via impingement jets are needed in electronic devices, heat sinks, drying technologies, and so on. To the best of the authors' knowledge, this is the first effort to employ a hybrid nanofluid to analyze entropy formation due to magnetohydrodynamics flow over a starching sheet.

Modeling and simulation for Cattaneo-Christov heat analysis of entropy optimized hybrid nanomaterial flow.

Here, the hydromagnetic entropy optimized flow of a hybrid (Pb + Fe2O3/C2H6O2) nanoliquid by a curved stretchable surface is addressed. The Darcy-Forchheimer model is utilized for porous space. Lead (Pb) and ferric oxide (Fe2O3) are considered the nanoparticles and ethylene glycol (C2H6O2) as the base liquid. Thermal expression consists of dissipation and ohmic heating. Entropy generation is under consideration. The Cattaneo-Christov heat flux impact is discussed. Non-dimensional partial expressions by adequate transformations have been reduced to ordinary differential systems. The ND-solve technique is implemented for numerical solutions of dimensionless systems. Graphical illustrations of velocity, thermal field and entropy against influential variables for both nanoliquid (Pb/C2H6O2) and hybrid nanoliquid (Pb + Fe2O3/C2H6O2) are presented. Graphical illustrations of velocity, thermal field and entropy against sundry variables for both nanoliquid (Pb/C2H6O2) and hybrid nanoliquid (Pb + Fe2O3/C2H6O2) are presented. Influences of sundry variables on the Nusselt number and drag force for both nanoliquid (Pb/C2H6O2) and hybrid nanoliquid (Pb + Fe2O3/C2H6O2) are examined. A higher thermal relaxation time tends to intensify the heat transport rate and temperature. An increment in the magnetic variable leads to an enhancement of the entropy and thermal field. An improvement in liquid flow is seen for volume fraction variables. Velocity against the porosity variable and Forchheimer number is reduced. The Brinkman number leads to maximization of entropy generation.

Darcy–Forchheimer flow of nanofluid invoking irreversibility

Magnetohydrodynamic Darcy–Forchheimer nanoliquid flow with entropy generation subject to stretched sheet is explored. Thermal expression with ohmic heating, dissipation and radiation is taken. In addition, random and thermophoresis motion impacts are considered. Soret and Dufour effects are addressed. Physical features of the entropy rate are analyzed. Entropy generation rate is used to amplify thermodynamical system performances. Nonlinear dimensionless systems are developed through the implementation of non-similarity transformations. The nonlinear resultant systems are numerically solved by local non-similarity via numerical scheme (ND-solve method). Physical description of various flow variables on velocity field, temperature distribution, entropy rate, concentration and Bejan number is scrutinized. Behaviors of different sundry parameters for Nusselt number and solutal transport have been discussed. For larger magnetic variable the velocity decays. A reduction is seen in temperature for Prandtl number. Higher porosity variable declines the velocity. An augmentation in temperature is improved for Dufour number. Similar trend of temperature through random and thermophoresis parameters is seen. Reverse effect of velocity is seen in concentration for Soret and Schmidt numbers. An improvement in concentration is noticed for the thermophoresis parameter. A reverse trend is noted for entropy rate versus radiation and Brinkman number.

Non‐similar modelling of Williamson Fluid over a non‐linear stretching sheet with temperature dependent thermal conductivity and Chemical Reactions

AbstractIn this paper we investigate the mixed convection flow of a Williamson fluid under the effects of temperature dependent thermal conductivity and chemical reaction. The mathematical model for this phenomenon is described in a set of partial differential equations (PDEs) and further this set is reduced to a dimensionless system of nonlinear partial differential equations (PDEs) using a non‐similarity method. The non‐similarity method has an advantage over similarity methods for handling non‐similar problems. The method first uses transformations to reduce governing equations into dimensionless systems of PDEs. The effects of Grashof number of heat and mass transfer Weissenberg number, Prandtl number, Schmidt number and reaction rate parameters on velocity, temperature, and concentration profiles are given in forms of graphs.

Entropy optimization for nanofluid flow with radiation subject to a porous medium

Here hydrodynamic Darcy-Forchheimer nanofluid flow with entropy rate by a porous surface is examined. Mathematical background of current flow is developed through implementing single-phase approach. Here silver (Ag) and gold (Au) are used as nanoparticles, while sodium alginate ((C6H7NaO6)n) is considered as base fluid. Moreover, the impact of radiation, heat generation, Ohmic heating and dissipation are considered in energy equation. Feature of irreversibility (entropy rate) analysis is scrutinized. A set of suitable non-similar variables are employed for conversion of nonlinear partial system into dimensionless ordinary system. The obtained dimensionless systems are tackled through numerical scheme (ND-solve technique). Influence of physical variables on velocity field, Bejan number, thermal field and entropy generation are examined against both silver and gold nanoparticles. Reverse effect for entropy rate and velocity is noted through magnetic variable. Higher approximation of volume fraction has reverse effect on velocity and temperature. Bejan number is diminished for magnetic variable. An intensification in radiation effect increases thermal field. A reverse trend is seen for entropy and Bejan numbers for Brinkman number.

Entropy optimized chemical reactive Darcy–Forchheimer flow of unsteady viscous fluid with magnetic field

Entropy generated in magnetohydrodynamic flow by a porous surface is an examined porous medium through the Darcy–Forchheimer relation is discussed. A thermal expression consists of dissipation, heat generation, Joule heating, and radiation. A first-order chemical reaction is studied. By using appropriate variables, the nonlinear systems are transformed into dimensionless differential systems. The finite-difference technique is employed for the development of computational results. Significant performance of thermal field, velocity, entropy rate, and concentration versus physical parameters are presented graphically. A decrease in velocity is noted for the Forchheimer number and porosity parameter. The behavior of concentration and thermal field is similar to the suction variable. The opposite trend holds for velocity and thermal field for the Hartman number. A larger estimation of radiation parameter and Brinkman number correspond to entropy generation enhancement.

Entropy optimized radiative flow of viscous nanomaterial subject to induced magnetic field

Here we examine the time-dependent viscous nanofluid flow with induced magnetic field. Energy expression comprising dissipation and radiation is accounted. Additionally Brownian movement and thermophoresis have been addressed. Thermal expression examines the thermodynamical system performance. Flow subject to first order reaction is chemically reactive. Dimensionless system is obtained through adequate variables. Dimensionless differential systems are solved employing finite difference scheme. Variations in velocity, induced magnetic field, thermal field and concentration against influential parameters are explored. Entropy generation is designed. Computational results of drag force, thermal transport rate and Sherwood number have been organized. Decay in velocity is seen for suction variable. Larger magnetic Prandtl number rises the thermal field. An increment in induced magnetic field is noticed through magnetic Prandtl number. Reverse results of entropy and thermal field for radiation are observed. Larger approximation of thermophoresis variable has reverse impact on concentration and thermal field. Higher estimation of Brinkman number correspond to an increase of entropy rate.

Numerical analysis of bioconvection-MHD flow of Williamson nanofluid with gyrotactic microbes and thermal radiation: New iterative method

Abstract The aim of this study is to investigate the numerical analysis of an innovative model containing, bioconvection phenomena with a gyrotactic motile microorganism of magnetohydrodynamics Williamson nanofluids flow along with heat and mass transfer past a stretched surface. The effect of thickness variation and thermal conductivity feature is employed in the model. Bioconvection in nanofluid helps in bioscience such as in blood flow, drug delivery, micro-enzyme, biosensors, nanomedicine, for content detection, etc. For simulation procedure, the mathematical partial differential equations are converted into dimensionless systems owing to dimensionless variations such as magnetic field, power index velocity, Williamson parameter, wall thickness parameter, thermal conductivity variation, Prandtl number, thermal radiation, Brownian motion, Lewis number, Peclet number, and different concentration parameter, etc. For numerical simulation, New Iterative Technique (NIM) numerical algorithm is adopted and employed for the linear regression planned for the proposed model. For comparison purposes, the homotopy technique is employed on the flow model. Close agreement is seen between both methods revealing the accuracy and consistency of NIM numerical technique. Many features of no-scale constraints are evaluated through graphical data for a key profile of the flow model. Results show that microorganism concentration is heavy due to the magnetic effect and Hall current.

Eliminating Radiative Losses in Long-Range Exciton Transport

We demonstrate that it is possible to effectively eliminate radiative losses during excitonic energy transport in systems with an intrinsic energy gradient. By considering chain-like systems of repeating `unit' cells which can each consist of multiple sites, we show that tuning a single system parameter (the intra-unit-cell coupling) leads to efficient and highly robust transport over relatively long distances. This remarkable transport performance is shown to originate from a partitioning of the system's eigenstates into energetically-separated bright and dark subspaces, allowing long range transport to proceed efficiently through a `dark chain' of eigenstates. Finally, we discuss the effects of intrinsic dipole moments, which are of particular relevance to molecular architectures, and demonstrate that appropriately-aligned dipoles can lead to additional protection against other (non-radiative) loss processes. Our dimensionless open quantum systems model is designed to be broadly applicable to a range of experimental platforms.

Thermal aspects of radiation in Casson fluid with nonlinear stretching surface: non-similar solutions

A mathematical model is established to peruse the effects of viscous dissipation and thermal radiation on non-Newtonian, incompressible, steady, mixed convective flow next to a nonlinear vertical stretching surface. The Rosseland approximation is used in the energy equation to describe the radiative heat flux. Non-similarity transformation has been used to convert coupled partial differential equations (PDEs) into dimensionless PDEs. For dealing with non-similar problems, the non-similarity method is more effective as compared to similarity transformation. First, this method utilizes transformations to reduce governing equations into dimensionless systems of PDEs, and further, it utilizes different truncation levels to get a detailed set of PDEs. MATLAB built-in solver bvp4c is used to solve obtained dimensionless PDEs. Parameters describing the nature of flow like local Sherwood number, local Nusselt number and local skin friction coefficient are explored in detail after finding the numerical solution. The effects of Casson number, Radiation number, Eckert number, Schmidt number, Prandtl number and Grashof number for heat and mass transfer on concentration, temperature and velocity profiles are specified with the aid of graphs.