System Of Dimensionless Equations Research Articles

Entropy optimization of lid-driven micropolar hybrid nanofluid flow in a partially porous hexagonal-shaped cavity with relevance to energy efficient storage processes

Hydromagnetically associated heat convection can greatly enhance the performance of high-efficiency thermal appliances and renewable energy sources through an optimized design. This investigation examines the production of thermodynamic irreversibility and heat convection for a double lid-driven flow within a partially porous stratified hexagonal enclosure. The top and bottom-wall are moving in the opposite direction with an equal velocity U0. The top-wall and the bottom-wall are kept at temperature Tc and Th (Th >\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$>$$\\end{document} Tc) while the slanted walls are assumed to be thermally insulated. A constant magnetic field is employed in the horizontal x-direction. The hexagonal cavity was filled with a micropolar hybrid nanofluid Ag-MgO/water. The system of dimensionless equations was solved by the finite difference method (FDM) associated with successive over-relaxation (SOR), successive under-relaxation (SUR), and Gauss–Seidel iteration tactics and required results are computed with problem specific program in MATLAB code. The results indicate that the Ra and the thickness of the porous layer (Xp) significantly influences heat convection and thermal irreversibility processes. The Nuavg and STotal rises 6.299% and 3.373% as ‘ϕhnf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\upphi }_{{\ ext{hnf}}}$$\\end{document}’ enhances from 0 to 4%, respectively. Furthermore, as the values of Ra, Ha, K0, and ϕhnf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\upphi }_{{\ ext{hnf}}}$$\\end{document} increase, Beavg experiences a decline of 53.73%, 11.04%, 38.36%, and 0.09% respectively. Also, movement of wall has a significant impact on heat transfer rates and entropy production. The present study may be extended in numerous areas to mimic the problems like—(1) onset of thermo-mechanical process for solid–fluid interaction in a conduit. (2) Thermos-chemical process with extraction of ions in two-phase fluid for double layer plating on a continuously moving sheet, as region of porous stratum saturated with a class of fluid and region without porous medium occupied with other fluid.

Models based analysis of radiative induced MHD hybrid nanofluid flow over an exponentially stretching sheet

The study focuses on the flow of hybrid nanofluid, induced by magnetic and radiation effects, across an exponentially stretched sheet. The research examines the impact of temperature-dependent properties of the hybrid nanofluid on the sheet. Water is used as the base fluid, and SWCNT and MWCNT are employed as nanoparticles. The study includes a discussion of the Yamada–Ota, Xue and Tiwari–Das models of hybrid nanofluids. The governing system of flow is presented mathematically, and boundary layer approximations are used to reduce differential equations. The differential equations are transformed into dimensionless ordinary differential equations (ODEs) by using transformations. The dimensionless system of equations is then solved numerically. The results of the flow model are offered in tabular and graphical forms. We observed that Tiwari–Das model of hybrid nanofluid achieved more heat transfer and friction factor values when compared to other models of Xue and Yamada–Ota models of hybrid nanofluid. Temperature curves are noted to be enhanced by enlargement in the nano-concentration factor. If the nano-concentration increased in the fluid which boosted the thermal conductivity of the liquid, then as a result, the temperature of fluid enhanced at surface.

Caputo–Fabrizio fractional model of MHD second grade fluid with Newtonian heating and heat generation

In this research article the heat transfer of generalized second grade fluid is investigated with heat generation. The fluid flow is analyzed under the effects of Magneto hydrodynamics over an infinite vertical flat plate. The Newtonian heating phenomenon has been adopted at the boundary. For this purpose the problem is divided into two compartments i.e. momentum equation and energy equations. Some specific dimensionless parameters are defined to convert the model equations into dimensionless system of equations. The solutions for dimensionless energy and momentum equations are obtained by using the Laplace transform technique. From obtained results by neglecting magneto hydrodynamic effects and heat source some special case are achieved which are already published in literature. The case for which the fractional parameter approaches to the classical order is also discussed and it has been observed that it is convergent. Finally, the influences of different physical parameters are sketched graphically. It has been observed that for increasing values of Prandtl number the velocity and temperature decreases, for increasing values of Grashof number the velocity of the fluid increases. Also it has been investigated that for increasing values of fractional parameter the velocity and temperature of the fluid increases.

Significance of geothermal viscosity and heat generation/absorption on magnetic nanofluid flow and heat transfer

This model includes the study of incompressible magnetic nanofluid flow due to a convectively heated rotating disk under the influence of geothermal viscosity. An equation of energy has been constructed by taking the heat generation/absorption effect into consideration. With the help of suitable transformation, the obtained fundamental equations are simplified into dimensionless system of equations. The non-dimensional system of equations is solved numerically by MATLAB bvp4c solver. The tabular and graphical expressions for velocity and temperature are shown for some values of involved parameters. The outcome shows that heat energy is generated in the flow due to the higher value of the heat generation parameter.

Natural Convection Modeling in a Solar Tower

Objectives: We present in this study a modeling of thermal laminar convection airflow in a solar tower. Methods: To formulate with precision, the boundary conditions of the solar chimney model chosen, the Cartesian equations are transformed into hyperbolic coordinates. An orthogonal grid is elaborated. It then makes it possible to draw up the diagrams of physique and calculation fields. The computer code uses the heat equation, the vorticity, and the stream function formalism as the boundary conditions for pressure are difficult to set. We use the Boussinesq approximation, which consists in considering that the density (r) of the fluid varies only in the term of the gravity forces, whose variations with temperature, assumed to be linear, generate natural convection. These variations are then translated into an equation of state which relates density to temperature. The system of dimensionless equations is solved by using an intégro-interpolation method referring to finite differences scheme. Findings: The solutions obtained from the dimensionless equations enabled us to determine the space evolution parameters (temperatures and velocities) in the tower according to the Rayleigh number. The fluid temperature and velocity evolution in the collector increase when one moves in the direction of radius decrease. The fluid temperature evolution in the chimney showed that the highest temperature is located at the chimney base while we obtained a parabolic profile of the transverse temperature distribution within the chimney. Finally, the evolution of the fluid velocity in the chimney showed that there was a preferred zone for turbine installation. Novelty : The use of dimensionless geometric parameters is unique and in general, the approach adopted in this paper differs from that encountered in the literature.

Shape effect of Cu, Al2O3 and TiO2 nanoparticles on stagnation point nanofluid flow in a microgravity environment

The unsteady viscous nanofluid flow near a three-dimensional stagnation point was studied numerically under microgravity environment. g-Jitter is one of the effects occurs under microgravity environment that producing a fluctuating gravitational field. Three different types of nanoparticles were induced in the study that is copper (Cu), alumina (Al2O3), and titania (TiO2) which then produce a water-based typed of nanofluid. In addition, different shape of nanoparticle was applied on the study in analyzing the performance of each types of nanoparticle. The fluid system was then mathematically formulated into a system of partial differential equation based on physical law and principle such as conservation of mass, Newton’s second law and conservation of energy. The system of equation then undergoes semi-similar transformation technique in reducing the complexity of the problem into non dimensionless form. Keller box method was applied into the dimensionless system of equations in solving the problem numerically. The problem was analyzed in term of velocity and temperature profiles together with skin friction coefficient and Nusselt number. The results shown that temperature profile, skin friction coefficient and Nusselt number were increase while velocity profile decreased as nanoparticle volume fraction decreased. The results indicated that, the needle-shaped nanoparticles give the highest enhancement on the heat transfer of the nanofluid compared to sphere and disk-shaped nanoparticles with more than 14% significant different. In addition,  alumina hold the highest velocity profile while copper hold the lowest velocity profile.

Boundary layer flow with forced convective heat transfer and viscous dissipation past a porous rotating disk

The Coriolis effect with forced convective heat transfer on steady ferrohydrodynamic flow past a rotating porous disk in the presence of viscous dissipation has been investigated. The basic idea of the Neuringer-Rosensweig model has been used for the equation of motion of the nanofluid flow. With help of suitable transformations, the governing non-linear system of coupled partial differential equations is simplified into the dimensionless system of ordinary differential equations. Further, the dimensionless system of equations is solved numerically by the MATLAB routine bvp4c solver package. The findings for the motivating parameters of physical interest are expressed by the table and discussed with graphs. The outcomes show that heat transfer rate and thermal boundary layer thickness increase due to the higher value of the dissipation parameter.

СТРУКТУРА ПОТОКА И КИНЕМАТИКА ТЕЧЕНИЯ НЕНЬЮТОНОВСКОЙ ЖИДКОСТИ В ТРУБЕ С ВНЕЗАПНЫМ РАСШИРЕНИЕМ

This paper deals with a problem of a laminar steady-state flow of a non-Newtonian incompressible fluid in a pipe with a sudden expansion. The flow is described by a system of dimensionless equations in terms of stream function and vorticity in a cylindrical coordinate system: an equation of vorticity transfer and Poisson's equation for stream function. Rheological properties of the medium are defined by the Ostwald-de Waele model. The problem is solved numerically. The false transient method is applied to obtain a steady-state solution to the problem. The equations are discretized in accordance with the finite-difference method based on the alternating direction scheme. The final system of equations is solved by the tridiagonal matrix algorithm. Flow structures of Newtonian, pseudoplastic, and dilatant fluids are found to include twodimensional flow zones before and after expansion plane. A recirculation region occurs in the inner corner. To assess the effect of the Reynolds number, expansion ratio, and power-law index on the lengths of the two-dimensional flow zones and recirculation region, the graphs are plotted over a wide range of variation in the parameters. Local pressure losses are presented as functions of the governing parameters of the problem.

Rotating flow of carbon nanotubes subject to prescribed heat flux condition

This article intends to describe the three-dimensional flow of nanofluid consisting of carbon nanotubes (CNTs) subject to rotating frame. Disturbance in flow is generated by an exponentially stretching sheet. Prescribed heat flux condition is considered. Darcy-Forchheimer relation is employed to characterize the flow in porous space. System of dimensionless equations is obtained by utilizing adequate transformations. Optimal homotopy analysis algorithm is employed for solution development. Physical interpretation of emerging variables on flow fields and physical quantities is discussed. Skin friction coefficient enhances for local porosity parameter, nanoparticle volume fraction and Forchheimer number. Augmentation for heat transfer rate is noticed through solid volume fraction of nanoparticles.

Effects of g-jitter and radiation on three-dimensional double diffusion stagnation point nanofluid flow

The unsteady double diffusion of the boundary layer with the nanofluid flow near a three-dimensional (3D) stagnation point body is studied under a microgravity environment. The effects of g-jitter and thermal radiation exist under the microgravity environment, where there is a gravitational field with fluctuations. The flow problem is mathematically formulated into a system of equations derived from the physical laws and principles under the no-slip boundary condition. With the semi-similar transformation technique, the dimensional system of equations is reduced into a dimensionless system of equations, where the dependent variables of the problem are lessened. A numerical solution for the flow problem derived from the system of dimensionless partial differential equations is obtained with the Keller box method, which is an implicit finite difference approach. The effects studied are analyzed in terms of the physical quantities of principle interest with the fluid behavior characteristics, the heat transfer properties, and the concentration distributions. The results show that the value of the curvature ratio parameter represents the geometrical shape of the boundary body, where the stagnation point is located. The increased modulation amplitude parameter produces a fluctuating behavior on all physical quantities studied, where the fluctuating range becomes smaller when the oscillation frequency increases. Moreover, the addition of Cu nanoparticles enhances the thermal conductivity of the heat flux, and the thermal radiation could increase the heat transfer properties.

Transportation of water-based trapped bolus of SWCNTs and MWCNTs with entropy optimization in a non-uniform channel

Here, we communicate the peristalsis of single- and multi-walled CNTs through nonlinear porous, non-uniform propagating channel boundaries. Non-uniform channel boundaries are of asymmetric characteristics. Flow equations are modeled by taking variable viscosity, linear and nonlinear porous medium (i.e., Darcy and Darcy–Forchheimer), nonlinear thermal radiation, mixed convection, heat generation and absorption aspects. Convective heat transfer aspects are at the convectively heated surface. The entropy generation rate is modeled via the second law of thermodynamics. Modeled equations are simplified with the help of long wavelength assumption. Dimensionless system of equations with respective boundary conditions is solved numerically via built-in shooting algorithm in Mathematica 8 software. Further, these numerical results are directly received in the form of curves. Such curves are made for velocity, temperature, pressure gradient, trapping, entropy rate and Bejan number. Heat transfer rate at lower and upper walls is achieved through bar charts.

Effectiveness of Cattaneo–Christov double diffusion in Sisko fluid flow with variable properties: Dual solutions

The nature and properties of generalized Newtonian fluid flows are of the most significant phenomena applicable to engineering applications. But this particular portion of research work incorporates the importance of flow, heat and mass transfer analysis of non-homogeneous Sisko fluid transport model. The flow source in this study is assumed by the stretching and shrinking velocities of the sheet. Therefore, the influence of these two velocities creates a phenomenon of multiple solutions. The effect of magnetic field is another significant physical parameter in the flow analysis and has been considered in this study. Moreover, the impacts of variable thermal conductivity, mass diffusivity and suction/injection are also incorporated. The system of conservative governing partial differential equations are converted into a dimensionless system of equations by using the suitable transformations. The new system of equations along with the corresponding transformed boundary conditions are then solved numerically with the help of collocation method in Matlab. This method is a built-in approach for the solution of nonlinear boundary value problem. In comparison with other user-defined numerical approaches, this method is little bit fast and works accurately because this method uses finite difference method for modifying weak initial guess and the CPU timing is very small as compared to other built-in approaches. The present results are shown for the existence of multiple (upper branch and lower branch) solutions for a specific range of involved physical parameters. The critical values of shrinking parameter corresponding to suction parameter and Sisko fluid parameter are computed in the certain range of ( $$\chi _{{\rm c}}<\chi <0$$ ). The behaviors of various dimensionless parameters on different profiles are discussed graphically. The increase in material parameter causes a reduction in skin friction for both cases, i.e., shear-thinning as well as shear-thickening fluids. The first and second solutions of temperature and concentration profiles, respectively, show increasing and decreasing trends for an increase in temperature and concentration time relaxation parameters.

Magnetoconvection and Entropy Analysis in T-Shaped Porous Enclosure Using Finite Element Method

In this paper, a computational work has been investigated on mixed convection and entropy generation in a T-shaped porous cavity filled with a copper–water nanofluid in the presence of a sinusoidal heated wall. The dimensionless system of equations is tackled with the help of Galerkin finite element method, and the obtained nonlinear discretized system of equations is solved via Newton’s method. The computed results have an excellent agreement with the already-published literature in a related area for special cases of governing parameters. The results are investigated for various parameters, including particular ranges as given: Richardson number (), Darcy number (), orientation of magnetic field (), aspect ratio (), and Hartmann number (). It was inferred that heat transfer, entropy generation and Bejan number are decreased by increasing the Hartmann number. At and as the Darcy number increases from to , the average Nusselt number has a growth of 30% or more. The effects of the aforementioned parameters on heat transfer and entropy generation are studied and analyzed in detail.

Numerical study of Hall effects on the peristaltically induced motion of a viscous fluid through a non-uniform regime: An application to the medical science

Impacts of Hall current (potential) on the peristaltically induced motion of a magneto-hydrodynamics (MHD) viscous incompressible fluid is analysed in a curved geometry. This study is novel in term of integrating numerically the Hall effects with peristaltic propulsive phenomena bounded within the curved regime. The usage of these electro-kinetically controlled devices in the modern era of bio-medical industries makes this study relatively new and interesting. Firstly, the governing equations are modelled in a curvilinear coordinates system. Secondly, these equations are transformed into a dimensionless system of equations by using dimensionless variables under long-wavelength and low-Reynold-number assumptions. The numerical solution of these governing equations is obtained with the appropriate boundary conditions (BCs) by using the BVP4C technique. The significant influences of several embedded physical parameters such as the curvature parameter, Hartmann number, Hall parameter in the velocity profile, pumping and trapping phenomena's are argued expansively through graphs. It is visible that the effects of the Hall current are dominant over the boundary layer (BL) phenomena for large values of the Hall parameter. Moreover, comparison among the straight channel and the curvedchannel is also highlighted. Furthermore, the validation of the numerical code is given at some particular values of the curvature parameter through numeric tables.

Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes

In this paper, we analyze the effects of Dufour number and fractional-order derivative on unsteady natural convection flow of a viscous and incompressible fluid over an infinite vertical plate with constant heat and mass fluxes. The fractional constitutive model is obtained using fractional calculus approach. The Caputo fractional derivative operator is used in this problem. The dimensionless system of equations has been solved by employing Laplace transformation technique. Closed form solutions for concentration, temperature and velocity are presented in the form of Wright function and complementary error function. Effects of fractional and physical parameters on temperature and velocity profiles are illustrated graphically.

Sprinkler Irrigation Droplet Dynamics. II: Numerical Solution and Model Evaluation

AbstractA system of equations, with strong physical basis, was derived for sprinkler irrigation droplet dynamics in the companion paper. Numerical solution of these equations and model evaluation is discussed in this paper. With the aim of enhancing computational efficiency and robustness, the droplet dynamics equations were scaled using four characteristic variables: characteristic time, length, velocity, and density. The characteristic time, length, and velocity are derived based on consideration of the motion of a droplet falling freely, starting from rest, through a quiescent ambient air, to an eventual steady-state condition. The characteristic density was set to the density of water at standard conditions. The dimensionless system of equations was then solved numerically with a fourth-fifth order pair Runge-Kutta method capable of local error estimation and time-step size control. The numerical model was first evaluated through successful comparisons with simplified solutions derived based on more l...

On soft stability loss in rotating turbulent MHD flows

The problem of the stability of turbulent flows of liquid metal in a cylindrical cavity against small velocity disturbances under the action of a rotating magnetic field (RMF) has been studied. The flow is considered in the induction-free approximation using the ‘external’ friction model. A system of dimensionless equations is examined in cylindrical coordinates. The results of computations performed on the basis of this mathematical model using the exchange of stabilities principle have shown a good consistency between the critical values of computed and experimental Reynolds numbers.

Influence of Thermal Radiation on Unsteady Free Convection MHD Flow of Brinkman Type Fluid in a Porous Medium with Newtonian Heating

The focus of this paper is to analyze the influence of thermal radiation on some unsteady magnetohydrodynamic (MHD) free convection flows of an incompressible Brinkman type fluid past a vertical flat plate embedded in a porous medium with the Newtonian heating boundary condition. The fluid is considered as a gray absorbing-emitting but nonscattering medium and the Rosseland approximation in the energy equations is used to describe the radiative heat flux for optically thick fluid. For a detailed analysis of the problem, four important situations of flow due to (i) impulsive motion of the plate (ii) uniform acceleration of the plate (iii) nonuniform acceleration of the plate, and (iv) highly nonuniform acceleration of the plate are considered. The governing equations are first transformed into a system of dimensionless equations and then solved analytically using the Laplace transform technique. Numerical results for temperature and velocity are shown graphically, while skin friction and Nusselt number are computed in tables. The results show that temperature and velocity increase on increasing radiation and Newtonian heating parameters. However, the results of magnetic and porosity parameters on velocity are found quite opposite.

Cooling Intensification of a Continuously Moving Stretching Surface Using Different Types of Nanofluids

The effect of different types of nanoparticles on the heat transfer from a continuously moving stretching surface in a concurrent, parallel free stream has been studied. The stretching surface is assumed to have power-law velocity and temperature. The governing equations are converted into a dimensionless system of equations using nonsimilarity variables. Resulting equations are solved numerically for various values of flow parameters. The effect of physical quantities on the temperature profiles is discussed in detail.

Modeling frost heave in freezing soils

A generalized model for secondary frost heave in freezing fine-grained soils is presented. The cryostatic suction effect, which causes an increase in upward water permeation, ice-lens growth during freezing, and, as a consequence, the increase of soil heave, is considered to be the main mechanism of moisture transfer. Although the model in this paper has a number of approaches in common with the model of Fowler and Krantz (1994) it differs in at least several important respects. We recognize the need to determine the distribution of the moisture within the frozen fringe by approximation of the experimental data for the equilibrium unfrozen water content. This distribution is the result of the complicated interaction between water, ice and the mineral skeleton during the freezing process. The generalization of the Clapeyron relation, which is used in the work cited above, only estimates the drop in initial freezing temperature and does not define the connection with the external temperature gradient ∇T, which is responsible for the frost heave process. This very important aspect is discussed in detail in the Introduction to our paper. Another difference is the fact that our solution is based on a dimensionless system of equations. We take into account the ratio Pe/Ste ≠ 1 (where Pe << 1). This approach allows us to obtain both a more general solution as well as analyze frost heave and propagation of the freezing front as they depend upon the convective (Pe) and phase transition (Ste) characteristics (criteria) of the process. The theoretical results derived from our solution of the analysis for fine-grained soils are compared, in good agreement, with experimental investigations and numerical models. A singularity of the solution at the initial point in time is discussed. In this respect the asymptotic solution for short and long times is obtained. The results are compared with both solutions (modeling and asymptotic). The model presented predicts the frost heave and freezing processes in porous media with reasonable accuracy and satisfactorily reflects observed phenomena, and thus can be suitable for engineering practice.