Self-similar Equations Research Articles

Magnetohydrodynamic Stagnation Point Flow and Heat Transfer of Casson Fluids Over a Stretching Sheet

This research investigates the effects of heat transfer on the stagnation-point flow of a non-Newtonian Casson fluid in a two-dimensional magnetohydrodynamic (MHD) boundary layer over a stretched sheet, considering thermal radiation impacts. By employing similarity transformations, the governing partial differential equations are transformed into nonlinear ordinary differential equations. The obtained self-similar equations are numerically solved using the Optimal Homotopy Analysis Method (OHAM). The numerical results are graphically represented, showcasing the influence of various parameters on fluid flow and heat transfer characteristics. The study uncovers important dynamics in transport phenomena. Examining and illustrating the effects of dimensionless parameters on velocity, temperature, and concentration profiles reveal significant insights. Moreover, skin friction and Nusselt number results for Casson fluids are analyzed and presented. The findings indicate that the Casson parameter and Hartman number act in opposition to fluid momentum, while the thermal conductivity parameter enhances fluid temperature. Thus, this research provides valuable insights into MHD boundary layer flows of non-Newtonian Casson fluids with thermal radiation effects, and the OHAM solution method proves effective in predicting flow transport properties.

Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect

Abstract A numerical framework is established for a two-dimensional steady flow of the magnetized Jeffery fluid model over elongated/shrinking sheets, with potential applications such as coating sheets, food products, fiber optics, drilling fluids, and the manufacturing processes of thermoplastic polymers. The model also demonstrates the influence of chemical reaction, magnetic field, and stability analysis which provide a novel contribution to this study. To ensure the ease and effectiveness of this analysis, we transform the set of difference equations governing the system into ordinary equations using the similarity transformation. The reliability of the solution is examined by using stability analysis. The Navier–Stokes equations have been transformed into self-similar equations by adopting appropriate similarity transformations and subsequently solved numerically using the bvp4c (three-stage Labatto-three-A formula) approach. The comparison between the derived asymptotic solutions and previously documented numerical results is quite remarkable. The self-similar equations display a duality of solutions within a limited range of the shrinking parameter, as observed from the data. For each stretching scenario, there is a unique solution. Hence, an examination of temporal stability has been conducted through linear analysis to establish the most fundamentally viable solution. The determination of stability in the analysis is based on the sign of the smallest eigenvalue, which indicates whether a solution is unstable or stable. The analysis of stability reveals that the first solution, which describes the primary flow, remains stable. Through the utilization of graphs, we thoroughly examine and discuss the influence of emerging factors. The numerical results obtained from this analysis demonstrate multiple solutions within a certain range of M 1 ≥ M ci {M}_{1}\ge {M}_{{ci}} , i = 1 , 2 , 3 i=1,2,3 , and no solution in the range M 1 < M ci {M}_{1}\lt {M}_{{ci}} . M ci {M}_{{ci}} denotes the critical values, which increase as the quantities of Sc {\rm{Sc}} increase from 0.3 to 0.9. Similarly, multiple solutions exist for λ ≥ λ ci \lambda \ge {\lambda }_{{ci}} , i = 1 , 2 , 3 i=1,2,3 , and no solution in the range λ < λ ci \lambda \lt {\lambda }_{{ci}} is observed.

Study of nanofluid flow in a stationary cone–disk system with temperature-dependent viscosity and thermal conductivity

The substantial temperature gradient experienced by systems operating at relatively high temperatures significantly impacts the transport characteristics of fluids. Hence, considering temperature-dependent fluid properties is critical for obtaining realistic prediction of fluid behavior and optimizing system performance. The current study focuses on the flow of nanofluids in a stationary cone–disk system (SCDS), taking into account temperature-dependent thermal conductivity and viscosity. The influence of Brownian motion, thermophoresis, and Rosseland radiative flux on the heat transport features are also examined. The Reynolds model for viscosity and Chiam's model for thermal conductivity are employed. The Navier–Stokes equation, the energy equation, the incompressibility condition, and the continuity equation for nanoparticles constitute the governing system. The Lie-group transformations lead the self-similar ordinary differential equations, which are then solved numerically. Multi-variate non-linear regression models for the rate of heat and mass transfers on the disk surface were developed. Our study reveals a notable decrease in the rate of heat and mass transfer when pre-swirl exists in the flow. The significant influence of nanofluid slip mechanisms on the effective temperature and nanofluid volume fraction (NVF) within the system is highlighted. Furthermore, the variable viscosity property enhances the temperature and NVF of the SCDS.

Magnetized hybrid nanofluid flow across a wedge: A computational three-dimensional study

The boundary layer fluid flow past a wedge surface occurs in various industrial processes, including wire drawing, metal foaming, polymer extraction, fiber processing, and in determining the dynamic properties (shear stress, drag) of working fluids. Theoretically, the boundary layer three-dimensional flow of a magnetized hybrid nanoliquid over a wedge surface is studied with viscous dissipation and radiation effects. The flow is due to moving wedge surface and power-law free-stream velocities. The current study unifies three different classical flow problems, namely, the Blasius flow, the Hiemenz flow, and the Falkner-Skan flow. Single-phase description of hybrid nanofluid is considered, and the problem is kept realistic by considering the experimental thermophysical properties of the nanomaterial. The governing system consists of conservation of mass, momentum, and the energy equations for hybrid nanofluids. Prandtl’s boundary layer approximations are applied to study the viscous dominant fluid regimes. The entropy equation and the Bejan number are also derived. Numerical computations of self-similar equations are obtained to explore the rheological and the heat transport features against the physical parameters. The Nusselt number and the drag on the wedge surface are computed and analyzed. Of the three classical flow problems, the maximum thickness of the momentum layer occurs for the Blasius flow problem. A minimum drag force on the wedge surface occurs for lower Hartmann number values. Furthermore, an increase in the volume fraction of the hybrid nanoparticles leads to an enhanced heat transport. The use of carbon nanotubes which has exceptional characteristics are taken into consideration in this study. This study also demonstrates the comparison on three-different classical flow problems (Blasius flow, Hiemenz flow, and Falkner-Skan flow).

Frictional heating effects on entropy generation with nonlinear Rosseland radiation: Implementation of generalized differential quadrature method

This article examines heat transfer and entropy generation analysis for boundary layer flow, considering the impacts of frictional heating and nonlinear Rosseland thermal radiation. The equations governing momentum and energy in a two-dimensional boundary layer are transformed into self-similar equations via similarity operations. The generalized differential quadrature method (GDQM) is then used to numerically solve the no-dimensional governing equations. Entropy generation and Bejan numbers are computed using the acquired solutions and are then plotted against the emerging flow parameters. An in-depth analysis of temperature distribution, entropy generation, velocity profile, and Bejan number. A high level of agreement was seen when the results from the literature were replicated using GDQM to verify the accuracy of the numerical code. It is found that a self-similar solution exists in the presence of viscous dissipation. The primary contributor to entropy generation at the interface is fluid friction, in contrast to heat transfer. Further, the findings indicate that entropy generation positively correlates with the Prandtl number, heating parameter, radiation parameter, and Eckert number. Notably, the results show that a decrease in operational temperature causes a decrease in entropy production.

Study of hybrid nanofluid flow in a stationary cone-disk system with temperature-dependent fluid properties

Cone-disk systems find frequent use such as conical diffusers, medical devices, various rheometric, and viscosimetry applications. In this study, we investigate the three-dimensional flow of a water-based Ag-MgO hybrid nanofluid in a static cone-disk system while considering temperature-dependent fluid properties. How the variable fluid properties affect the dynamics and heat transfer features is studied by Reynolds’s linearized model for variable viscosity and Chiam’s model for variable thermal conductivity. The single-phase nanofluid model is utilized to describe convective heat transfer in hybrid nanofluids, incorporating the experimental data. This model is developed as a coupled system of convective-diffusion equations, encompassing the conservation of momentum and the conservation of thermal energy, in conjunction with an incompressibility condition. A self-similar model is developed by the Lie-group scaling transformations, and the subsequent self-similar equations are then solved numerically. The influence of variable fluid parameters on both swirling and non-swirling flow cases is analyzed. Additionally, the Nusselt number for the disk surface is calculated. It is found that an increase in the temperature-dependent viscosity parameter enhances heat transfer characteristics in the static cone-disk system, while the thermal conductivity parameter has the opposite effect.

Lie group and stability analysis of tangent hyperbolic fluid flow driven by a shrinking sheet

This study elaborates on the mathematical modeling of tangent hyperbolic fluid flow driven by a shrinking sheet in a porous medium. The roles of Soret-Dufour and thermal radiation are considered to model the energy and concentration equations. The primary objective is to find an appropriate similarity transformation through Lie group analysis, which helps to produce the self-similar equations from the governing nonlinear partial systems. The similarity solutions are calculated numerically using the “bvp4c” solver in Matlab. Additionally, adequate suction is required to obtain the similarity solutions. As expected, dual solutions are found within a certain range of the suction parameter. The skin friction coefficient, heat, and mass transfer rates at the sheet are observed for pertinent flow constraints of physical interest. It reveals that the dual solutions bifurcate from a critical point, and beyond this point, no solution exists. Furthermore, the branches of dual solutions on the nondimensional velocity, temperature, and concentration profiles depict different characteristics. Thus, a linear temporal stability analysis has been employed to test the flow stability, revealing that the upper branch is physically reliable. The heat transfer rate increases by 7.69% and 9.55% in the upper and lower branches, respectively, for Soret parameter changes from 0.6 to 0.9. Meanwhile, the mass transfer rate decreases by 85.30% and 324.60% in the upper and lower branches, respectively. The consequences of Weissenberg and suction parameters show an opposite character on the velocity, temperature, and concentration profiles in the upper branch.

Time-dependent stagnation point flow of nano Casson fluid with Joule heating over an elongated surface subjected to viscous heating and exponential space-based heat source/sink: Boungiorno model

The unsteady two-directional flow of a non-Newtonian magneto-Casson nanoliquid flow over an elongated flat surface with the porous matrix is investigated. The flow is subjected to space-based exponential heat generation/absorption (ESHS), thermophoresis, Brownian motion of nanoparticles, and transverse magnetic field. Within the base fluid, the diffusion of chemically reactive nanoparticles is assumed to be highly significant; hence considered. The governing equations of the flow model admit self-similar equations and are numerically solved by employing the Runge-Kutta-based shooting technique (RKSM). The significance of key parameters on the temperature, velocity, friction factor at the surface, heat transfer rate, and mass transfer rate distributions is analyzed. The use of high-Prandtl number base fluid and nanoparticles of high thermal conductivity could be of practical use to increase the heat transfer rate and avoid nanoparticle accumulation. The occurrence of nanoparticles in the operating liquids reduces the shearing stress at the plate surface to avoid backflow.

Blasius–Rayleigh–Stokes nanofluid dusty flow influenced by Cattaneo–Christov double diffusion and melting heat transfer — application of response surface methodology

This paper focuses on the study of Blasius–Rayleigh–Stokes nanofluid dusty flow with generalized Fourier and Fick laws under the influence of a transitive magnetic field. The Blasius–Rayleigh–Stokes nanofluid dusty flow has significant applications in various fields, including the design of heat exchangers, microfluidic devices and industrial processes involving the transportation of suspended particles in fluids. This paper considers the impact of melting heat transfer at the surface boundary. By employing relevant transformations, the mathematical model is transformed into a system of self-similar equations. The solution to this set of highly nonlinear equations is obtained using the bvp4c numerical method in combination with the response surface methodology (RSM) statistical approach. The results are presented through graphical illustrations and numerically calculated tabulated values. It is observed that the fitted model for the skin friction coefficient [Formula: see text] optimal. Moreover, the model parameters [Formula: see text], Q, [Formula: see text], M and L are linearly and quadratically significant in explaining the variation presented for skin friction coefficient [Formula: see text]. This indicates the response skin friction coefficient [Formula: see text] is minimized to 0.005 at [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] and is maximized to 1.387 at [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

IMPACT OF POROUS AND MAGNETIC DISSIPATION ON DISSIPATIVE FLUID FLOW AND HEAT TRANSFER IN THE PRESENCE OF DARCY-BRINKMAN POROUS MEDIUM

In this article, the boundary layer flow of an electrically-conducting fluid through a porous medium attached with a radiative permeable stretching sheet is analyzed. Following the Brinkman theory, an extended Darcy model (Darcy-Brinkman model) is utilized for the model momentum equation. Heat transfer analysis is also performed in the presence of viscous and Joule dissipation. Moreover, in the modeling of the energy equation, the effects of internal heating resulting from the mechanical effort required to squeeze out the fluid through the porous medium are also included in porous dissipation. Suitable dimensionless variables are introduced to convert the governing boundary layer equations into a dimensionless form, which are then converted into self-similar, nonlinear ordinary differential equations by utilizing similarity transformations. The exact solution of the nonlinear self-similar momentum equation is obtained in the form of the exponential function. In contrast, the solution of the energy equation is computed through the Laplace transform technique in the form of Kummer confluent hypergeometric functions. Effects of involved physical parameters on the momentum boundary layer (MBL), thermal boundary layer (TBL), wall shear stress, and local Nusselt number are explored through graphs and tables. Moreover, the slope linear regression (SLR) technique is used to calculate the rate of decrease/increase in shear stress and the rate of heat transfer at the boundary. The velocity and momentum boundary layer decreases for large values of porosity parameter and increases by increasing the viscosity ratio. The shear stress increases by increasing the porosity parameter, Hartman number, and suction parameter, while the opposite effect is examined with increasing values of viscosity ratio parameter. Heat transfer rate also enhances by increasing the Brinkman viscosity ratio parameter and wall suction velocity.

On the Analytical Determination of the Contour of Well - Styeamlined Bodies

The problem is solved with the help of a modified Prandtl equation applied to the case under study. This is a two-dimensional problem of flowing around a flat body when the essential factor is to take into account the limitation of its dimensions in the longitudinal and transverse directions. Thanks to the above Prandtl equation it was possible to reduce the problem to a self-similar equation. An analytical solution has been found. Thanks to this solution the shape of the body is analytically determined when the resistance is at its lowest. An analysis of the solution of the problem for different Reynolds numbers is carried out. The resulting equation is solved numerically for different values of its included parameters. With the help of a graphic illustration the different shapes of such contours are shown.

Stability analysis of MHD stagnation point flow of Casson fluid past a shrinking sheet in porous medium considering heat sink or source, thermal radiation and suction effects

This study investigates the magnetohydrodynamic stagnation point flow of Casson fluid and heat transfer phenomena past a shrinking sheet in porous medium with thermal radiation, heat sink/source and suction effects. The Lie group analysis approach transforms the governing equations into self-similar equations, which are then solved through the shooting methodology. Asymptotic solutions for large stretching and suction parameters are reported. Dual solutions are found for a certain range of shrinking parameter. Stability analysis is performed in order to identify the stable and physically realizable solution. The stability is examined by calculating the lowest eigenvalues of the linearized boundary value problem associated with the self-similar equations. The analysis indicates that the first solution is stable, while the second solution is unstable. It is observed that as the suction parameter (s), magnetic field parameter (M) and Casson fluid parameter (β) increase, the range of dual similarity solutions increases.

Insight of boundary layer structure with heat transfer through a diverging porous channel in Darcy-Forchheimer porous material with suction/injection: A study of separation control

Separation control and formation of boundary layer Newtonian flow in a diverging perme-able channel in Darcy-Forchheimer porous material having suction/injection are discussed. Self-similar equations from governing equations are acquired and existence conditions for boundary layer structure are derived using nature of velocity gradient inside boundary layer. It reveals that if sum of Darcy permeability parameter and twice of Forchheimer parameter exceeds 2, then the boundary layer flow always exists with all type of mass suction/injection and even without suction/injection. Also, if mass suction parameter goes beyond 22, then there is no matter what are the values of Darcy permeability parameter and Forchheimer pa-rameter, a boundary layer exists inside the divergent channel. In addition, obtained numerical solutions are exhibited graphically. It reveals that thicknesses of velocity and thermal bound-ary layers reduce with Darcy and non-Darcy resistances of porous medium and fluid tempera-ture also diminishes. The velocity and temperature reduce with increment of mass suction and contrary results are found for mass injection.

Study of nanofluid flow and heat transfer in a stationary cone-disk system

Rheometric, viscosimetric, bio-medical, and several other pharmaceutical machineries utilize the structural advantages provided by the geometry of a stationary conical diffuser. The problem of the Buongiorno nanofluid flow in the conical gap of a stationary cone-disk system for isothermal boundaries is studied. The governing system, comprising the incompressibility condition, Navier–Stokes equation, energy conservation equation, and conservation of Nanoparticle Volume Fraction (NVF) equation, is analyzed. The Lie-group theory has been used to derive a self-similar model. Solutions of the self-similar equations were computed numerically, and the expressions for the Nusselt number and Sherwood number are obtained. The parametric investigation reveals that the heat and mass transfer rate subside significantly when pre-swirl is introduced to the flow. Furthermore, the nanofluid slip mechanisms enhance the effective temperature of the system.

Removing rotated discretely self-similar singularity for the Euler equations

In this paper we exclude the scenario of rotated discretely self-similar singularities of the 3D incompressible Euler equations in the case when the singularity point is isolated and the vorticity satisfies appropriate decay condition at spatial infinity. For the proof we use the maximum principle for the transformed self-similar equations.

Computation of stagnation coating flow of electro-conductive ternary Williamson hybrid mathrm{GO}-mathrm{AU}-{mathrm{Co}}_{3}{mathrm{O}}_{4}/mathrm{EO} nanofluid with a Cattaneo–Christov heat flux model and magnetic induction

Modern smart coating systems are increasingly exploiting functional materials which combine multiple features including rheology, electromagnetic properties and nanotechnological capabilities and provide a range of advantages in diverse operations including medical, energy and transport designs (aerospace, marine, automotive). The simulation of the industrial synthesis of these multi-faceted coatings (including stagnation flow deposition processes) requires advanced mathematical models which can address multiple effects simultaneously. Inspired by these requests, this study investigates the interconnected magnetohydrodynamic non-Newtonian movement and thermal transfer in the Hiemenz plane's stagnation flow. Additionally, it explores the application of a transverse static magnetic field to a ternary hybrid nanofluid coating through theoretical and numerical analysis. The base fluid (polymeric) considered is engine-oil (EO) doped with graphene left(GOright), gold left(Auright) and Cobalt oxide left(C{o}_{3}{O}_{4}right) nanoparticles. The model includes the integration of non-linear radiation, heat source, convective wall heating, and magnetic induction effects. For non-Newtonian characteristics, the Williamson model is utilized, while the Rosseland diffusion flux model is used for radiative transfer. Additionally, a non-Fourier Cattaneo–Christov heat flux model is utilized to include thermal relaxation effects. The governing partial differential conservation equations for mass, momentum, energy and magnetic induction are rendered into a system of coupled self-similar and non-linear ordinary differential equations (ODEs) with boundary restrictions using appropriate scaling transformations. The dimensionless boundary value problem that arises is solved using the bvp4c built-in function in MATLAB software, which employs the fourth-order Runge–Kutta (RK-4) method. An extensive examination is conducted to evaluate the impact of essential control parameters on the velocity f{^{prime}}left(zeta right), induced magnetic field stream function gradient g{^{prime}}left(zeta right) and temperature theta left(zeta right) is conducted. The relative performance of ternary, hybrid binary and unitary nanofluids for all transport characteristics is evaluated. The inclusion of verification of the MATLAB solutions with prior studies is incorporated. Fluid velocity is observed to be minimized for the ternary mathrm{GO}–mathrm{Au}–{mathrm{Co}}_{3}{mathrm{O}}_{4} nanofluid whereas the velocity is maximized for the unitary cobalt oxide left({mathrm{Co}}_{3}{mathrm{O}}_{4}right) nanofluid with increasing magnetic parameter (beta ). Temperatures are elevated with increment in thermal radiation parameter (Rd). Streamlines are strongly modified in local regions with greater viscoelasticity i.e. higher Weissenberg number (We). Dimensionless skin friction is significantly greater for the ternary hybrid GO–Au–C{o}_{3}{O}_{4}/EO nanofluid compared with binary hybrid or unitary nanofluid cases.

Hybrid nanofluid flow over an unsteady stretching/shrinking disk with thermal radiation

The aim of this paper is to examine the combined effects of unsteadiness and thermal radiation on the flow of a mixture nanoliquid due to a shrinking disk. Investigation of flow near a stagnation-point naturally arises in several industrial sectors and is useful owing to its significance in thermal enhancement. The motion is persuaded by a radially stretched/shrinked exterior of the disk in adding to the flow near a stagnation point. As a result, this research examines the thermophysical belongings of the unsteady flow near a stagnation-point past a stretching/shrinking disk by using twin-kind nanoelements, viz. a mixture nanoliquid. Cu and Al2O3 nanoparticles have been considered in water to produce hybrid nanofluid. The numerous uses like electronic cooling system, heat exchangers and radiators are available. Using similarity transformations, self-similar equations have been obtained which are then solved numerically using MATHEMATICA software. Comparisons have been made with the available data in the literature and found a complete agreement for some special case of the problem. Double solutions subsist for equally stretched and shrinked disks, whereas a unique solution exists for the static disk. For flow past a stretching or a shrinking disk, a 27.79% of increase in warmth transport speed is noted for mixture nanofluid, contrasted to the nanoliquid. Hence, this study will be of interest to several scientists and engineers, owing its capability to augment the speed of heat transport in contrast to the regular nanofluids.

Semi-Numerical Investigation of Boundary Layer Flow and Heat Transfer of Magnetohydrodynamics Nano-Fluid Flow in Presence of Chemical Reaction Over a Non-Isothermal Porous Medium

Abstract The technical brief presents, analysis of boundary layer flow and heat transfer in nanofluids under the influence of magnetic field, thermal radiation and chemical reaction over non-isothermal stretching surface through permeable porous medium. The self-similarity equations obtained from governing equations are solved using shifted Chebyshev and Haar wavelet collocation methods. The prescribed surface temperature, prescribed heat flux cases and impact of various flow governing parameters are discussed in detail. The established results are compared with earlier results and are comparable.

Simultaneous features of ternary hybrid nanoparticles on thermal radiative flow configured by Darcy–Forchheimer porous surface

Due to its remarkable thermodynamic development in various engineering sciences, the dispersion of nanostructures in classical base liquids is receiving greater attention from researchers and scientists. In light of the aforementioned inspirations, the suggested research is associated with using a water-based ternary-hybrid nanofluid that contains three unique nanostructures, silicon dioxide, titanium dioxide and aluminum oxide to optimize the heating process in science and engineering. In this situation, the influence of a chemically reactive magneto-hydrodynamics (MHD) Darcy–Forchheimer ternary hybrid composites transport involving nonlinear radiation across an exponentially permeable stretched surface is highlighted with the model’s physical initial and boundary constraints. The simulation of the heat expression also includes the impacts of nonlinear thermal radiations, energy supply, energy dissipation and magnetic force. Furthermore, the interpretation of Buongiorno’s theory involves spontaneous and thermophoresis diffusions. Entropy analysis is explained using the second law of thermodynamics. We converted the model framework for fluid movement, energy and mass transfer (together with ternary hybrid nanofluid characteristics) into self-similar nondimensional differential equations, that were then further analytically evaluated by using the homotopy asymptotic method (HAM) technique via Mathematica software. The acquired results are illustrated numerically and graphically to study the behavior of fluid flow, heat, concentration distribution, drag force, rate of heat and mass transfer for various emergent components to gain scientific comprehension. With the growing values of magnetic, porosity and Forchheimer number, the nature of the velocity profile goes down.

Unsteady flow with heat transfer over a shrinking surface and linear temporal stability analysis

The present study accentuates an unsteady flow and heat transfer over a shrinking surface in the presence of magnetic field and suction effects. Similarity solutions of the self-similar equations are generated via Matlab solver ‘bvp4c’. The solutions corresponding to different values of the velocity ratio parameter (c) reveal some interesting results, which are discussed in detail. It has been observed that adequate suction is required to obtain the similarity solutions. The shear stresses and heat transfer rate are prominently affected by the intensity of suction, velocity ratio parameter, and magnetic field. Unlike an unsteady stretching sheet problem, dual solutions of the self-similar equations are found in a certain range of unsteadiness parameter. An emphasis has been given to performing a temporal stability analysis which reveals that the upper branch is a stable solution branch.