Nonlinear Convection Research Articles

Numerical implementation and error analysis of nonlinear coupled fractional viscoelastic fluid model with variable heat flux

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

Nonsimilar solution of a boundary layer flow of a Reiner–Philippoff fluid with nonlinear thermal convection

AbstractThe thermodynamics modeling of a Reiner–Philippoff‐type fluid is essential because it is a complex fluid with three distinct probable modifications. This fluid model can be modified to describe a shear‐thinning, Newtonian, or shear‐thickening fluid under varied viscoelastic conditions. This study constructs a mathematical model that describes a boundary layer flow of a Reiner–Philippoff fluid with nonlinear radiative heat flux and temperature‐ and concentration‐induced buoyancy force. The dynamical model follows the usual conservation laws and is reduced through a nonsimilar group of transformations. The resulting equations are solved using a spectral‐based local linearization method, and the accuracy of the numerical results is validated through the grid dependence and convergence tests. Detailed analyses of the effects of specific thermophysical parameters are presented through tables and graphs. The study reveals, among other results, that the buoyancy force, solute and thermal expansion coefficients, and thermal radiation increase the overall wall drag, heat, and mass fluxes. Furthermore, the study shows that amplifying the space and temperature‐dependent heat source parameters allows fluid particles to lose their cohesive force and, consequently, maximize flow and heat transfer.

Electroosmosis-Optimized Thermal Model for Peristaltic Transportation of Thermally Radiative Magnetized Liquid with Nonlinear Convection.

The present study addresses the heat transfer efficiency and entropy production of electrically conducting kerosene-based liquid led by the combined impact of electroosmosis and peristalsis mechanisms. Effects of nonlinear mixed convection heat transfer, temperature-dependent viscosity, radiative heat flux, electric and magnetic fields, porous medium, heat sink/source, viscous dissipation, and Joule heating are presented. The Debye–Huckel linearization approximation is employed in the electrohydrodynamic problem. Mathematical modeling is conducted within the limitations of δ << 1 and Re → 0. Coupled differential equations after implementing a lubrication approach are numerically solved. The essential characteristics of the production of entropy, the factors influencing it, and the characteristics of heat and fluid in relation to various physical parameters are graphically evaluated by assigning them a growing list of numeric values. This analysis reveals that heat transfer enhances by enhancing nonlinear convection and Joule heating parameters. The irreversibility analysis ensures that the minimization of entropy generation is observed when the parameters of viscosity and radiation are held under control. Fluid velocity can be regulated by adjusting the Helmholtz–Smoluchowski velocity and magnetic field strength.

Unsteady nonlinear convective flow of a nanofluid over a vertical plate due to impulsive motion: Optimization and sensitivity analysis

The classical Boussinesq approximation (linear density-temperature variation) is valid for small temperature differences in the system. However, density varies with temperature nonlinearly in various applications, such as heat exchangers, solar collectors, and nuclear reactors. Therefore, unsteady nonlinear mixed convection (quadratic density-temperature variation) in a nanofluid on a vertical plate due to impulsive motion is investigated. Modified Nield and Kuznetsov nanofluid model is incorporated that considers physical realistic boundary conditions. Unsteady flow is governed by the conservation of mass, momentum, energy, and the continuity equation of nanoparticles. Williams-Rhyne transformations are used to parameterize and reduce the infinite time domain to a finite time domain. Subsequent nonlinear differential equations system is solved by using the Finite Difference Method (FDM) based routine. Rayleigh and Hiemenz type equations are obtained respectively in an initial-unsteady and final-steady state. The Response Surface Method (RSM) and Central Composite Design (CCD) are used for the simultaneous optimization of the friction factor and the rate of heat transport. It is found that the quadratic density-temperature variation significantly affects the flow fields. The friction factor has the highest and positive sensitivity toward the thermal buoyancy number. The heat transport rate has the highest and negative sensitivity toward the thermo-migration of nanoparticles.

Quadratic Convective Nanofluid Flow at a Three-Dimensional Stagnation Point with the g-Jitter Effect

The nonlinear density variation with temperature happens in many thermal applications like solar collectors, energy production, heat exchangers, and combustions, and it gives a significant impact on heat transfer and fluid flows. Thus, a nonlinear convective of an unsteady stagnation point flow under the influence of gravity modulation in the presence of water-based nanoparticles alumina (Al2O3) is studied here. Suitable variables are utilized to reduce the highly coupled nonlinear governing equations into a system of dimensionless simple partial differential equations. The Keller-box method is then applied to solve the consequent governing equations. Velocity and temperature profiles for various values of pertinent parameters are displayed graphically and discussed. The results indicate that the quadratic convection has enhanced the fluid flow and heat transport. Furthermore, the nonlinear convection parameter and the nanoparticles volume fractions have delivered a positive effect on the skin friction and the rate of heat transfer.

Unsteady nonlinear thermal convection flow of MWCNT-MgO/EG hybrid nanofluid in the stagnation-point region of a rotating sphere with quadratic thermal radiation: RSM for optimization

The time-dependent enhanced heat transport and nonlinear thermal buoyancy-driven flow of the MWCNT-MgO/EG hybrid nanofluid at the stagnation-point of the rotating sphere subjected to the thermal jump boundary condition are studied theoretically. The quadratic density-temperature variation is also examined together with the quadratic Rosseland thermal radiation. For realistic modelling, experimental data of MWCNT-MgO/EG viscosity and thermal conductivity are used. The monophasic model and the concept of the boundary layer are used to derive the governing partial differential equations, and then they are handled for self-similar solutions using the Finite Element Method (FEM). The effects of driving parameters are examined using 3D contour and surface plots. The further goal of the study is to optimize the heat transport and shear stress of the flow system by applying the central composite-response surface method (RSM) based on the desirability approach. The hybrid nanoparticles volume fraction (NVF) improves friction factors and heat transport. Thermal buoyancy and Coriolis forces condense the thickness of the thermal layer. Heat transport is greater for quadratic thermal convection/radiation than for linear thermal radiation/convection. By RSM, the maximum heat transport (19.0894), the minimum friction factor in the x-direction (2.2564), and the minimum friction factor in the z direction (0.7333) are simultaneously conquered at the low level of NVF, and mixed convection factor and at a high level of QTR factor.

Chemically reactive Maxwell nanoliquid flow by a stretching surface in the frames of Newtonian heating, nonlinear convection and radiative flux: Nanopolymer flow processing simulation

AbstractThe effects of a chemical reaction and radiative heat flux in a nonlinear mixed thermo-solutal convection flow of a viscoelastic nanoliquid from a stretchable surface are investigated theoretically. Newtonian heating is also considered. The upper-convected Maxwell (UCM) model is deployed to represent the non-Newtonian characteristics. The model also includes the influence of thermal radiation that is simulatedviaan algebraic flux model. Buongiorno’s two-component nanofluid model is implemented for thermophoretic and Brownian motion effects. Convective thermal and solutal boundary conditions are utilized to provide a more comprehensive evaluation of temperature and concentration distributions. Dimensionless equations are used to create the flow model by utilizing the appropriate parameters. The computed models are presented through a convergent homotopic analysis method (HAM) approach with the help of Mathematica-12 symbolic software. Authentication of HAM solutions with special cases from the literature is presented. The impact of various thermophysical, nanoscale and rheological parameters on transport characteristics is visualized graphically and interpreted in detail. Temperatures are strongly enhanced with Brownian motion and thermophoresis parameters. Velocity is boosted with the increment in the Deborah viscoelastic number and mixed convection parameter, and the hydrodynamic boundary layer thickness is reduced. A stronger generative chemical reaction enhances concentration magnitudes, whereas an increment in the destructive chemical reaction reduces them and also depletes the concentration boundary layer thickness. Temperature and concentration are also strongly modified by the conjugate thermal and solutal parameters. Greater radiative flux also enhances the thermal boundary layer thickness. Increasing the Schmidt number and the Brownian motion parameter diminish the concentration values, whereas they elevate the Sherwood number magnitudes,i.e.enhance the nanoparticle mass transfer rate to the wall.

Gravitational Modulation Effect on Double-Diffusive Oscillatory Convection in a Viscoelastic Fluid Layer

The complex Ginzburg-Landau model has been employed to derive the temporal amplitude of convection in a viscoelastic fluid layer under gravity modulation. The finite amplitude is based on a weakly nonlinear thermal convection. The perturbation technique is applied to simplify the nonlinear system of partial differential equations into a non-autonomous amplitude equation. Heat and mass transfer obtained in terms of the Nusselt and Sherwood numbers, and the corresponding results are presented for the small amplitude of modulation. The moderate values of amplitude and frequencies of gravity modulation are considered, their effect is to enhance the heat and mass transfer in the layer. Thus, the modulation plays dual role in heat and mass transfer. The variations of Nusselt and Sherwood numbers with slow time becomes rapid on either increasing the parameters Rs, Pr, λ, δ or decreasing the parameters Γ, ε, Ω. Each parameter effect has been presented graphically with their effect on Nu and Sh. Further, it is found that an oscillatory mode of convection enhances the heat and mass transfer than a stationary mode.

Double Phase Implicit Obstacle Problems with Convection and Multivalued Mixed Boundary Value Conditions

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.

New stabilized [formula omitted] × [formula omitted] finite element methods for nearly inviscid and incompressible flows

This work proposes a new stabilized P1×P0 finite element method for solving the incompressible Navier–Stokes equations. The numerical scheme is based on a reduced Bernardi–Raugel element with statically condensed face bubbles and is pressure-robust in the small viscosity regime. For the Stokes problem, an error estimate uniform with respect to the kinematic viscosity is shown. For the Navier–Stokes equation, the nonlinear convection term is discretized using an edge-averaged finite element method. In comparison with classical schemes, the proposed method does not require tuning of parameters and is validated for competitiveness on several benchmark problems in 2 and 3 dimensional space.

The parametric computation of nonlinear convection magnetohydrodynamic nanofluid flow with internal heating across a fixed and spinning disk

The heat and mass transition characteristics with nonlinear convection flow through impermeable stationary and porous spinning disks are evaluated in the present analysis. The gyrotactic microorganism and magnetic nanoparticulate have been used in the carrier fluid ethylene glycol and water to synthesized nanofluid. The heat and mass propagation are examined by considering heat absorption/generation, thermal radiation and rate of chemical reaction, respectively. The governing equations are transformed to the system of nonlinear ordinary differential equations all through a similarity context. The basic modeled equations are numerically computed using the parametric continuation method. Physical interest variables are illustrated through tables and figures versus velocity, energy, mass and gyrotactic microorganism profiles. It has been observed that the enhancement in nonlinear convection owing to heat generation and temperature coefficient elevated the wall friction along the radial direction of the gyrating disk. The mass transition rate lessens with the consequences of Schmidt number, while enhancing with the increment of the chemical reaction. The application, such as biomedical imaging, cancer therapy, information storage, wastewater treatment and thermal decomposition of ferric oxide nanoparticles made the present analysis more plausible.

Arbitrary multi-resolution multi-wavelet-based polynomial chaos expansion for data-driven uncertainty quantification

Various real world problems deal with data-driven uncertainty. In particular, in geophysical applications the amount of available data is often limited, posing a challenge in the construction of an appropriate stochastic discretization. Arbitrary polynomial chaos is an alternative to tackle this challenge. Approximating the dependence of model output on the uncertain model parameters by expansion in an orthogonal polynomial basis using data-driven principles. This type of global polynomial representation suffers often from Gibbs’ phenomena, especially if applied in non-linear convection dominated problems that require to deal with discontinuities. The multi-resolution or multi-element framework has been successfully used for reducing Gibbs’ phenomena in intrusive stochastic discretizations. In the present work, we introduce a multi-resolution extension of the arbitrary polynomial chaos expansion which is based on the construction of piecewise polynomial. Gaussian quadrature nodes and weights that are computed using only stochastic (localized) moments provided by the underlying raw data. We enhance our approach by a multi-wavelet based stochastic adaptivity that assures a significant reduction of the computational costs. Numerical experiments of increasing complexity demonstrate the performance of the non-intrusive implementation of the introduced methods in relevant scenarios. The use of a carbon dioxide storage benchmark scenario allows one to compare the presented methodology with other stochastic discretization techniques applied to this benchmark.

Influence of Liquid Hydrogen Diffusion on Nonlinear Mixed Convective Circulation around a Yawed Cylinder

A yawed cylinder is a cylinder inclined in the plane of a flowing liquid. The liquid flow past the yawed cylinder is important for practice, namely, for bubble suppression and control of the boundary layer transition in undersea applications. It should be noted that an inclined cylinder characterizes an asymmetrical behavior of fluid flow and heat transfer. Energy and mass transference characteristics of a steady nonlinear convective flow over the yawed cylinder by accounting for chemically reactive species and viscous dissipation are analyzed in this investigation. The differential equations defining the boundary layer parameters are then transformed into a dimensionless view, taking into account the non-similar transformation. It should be noted that the governing equations have been written using the conservation laws of mass, momentum, energy, and concentration. These considered equations allow the simulation of the analyzed phenomenon using numerical techniques. Further, quasilinearization and implicit finite difference approximation are used to work out the non-dimensional governing equations. A parametric investigation of all the pertinent characteristics accompanies this. A descriptive system of computation outcomes for the velocity, temperature, and concentration patterns, the drag coefficients, Nu and Sh, is demonstrated by graphs. Enhancing the magnitudes of the Eckert number raises the temperature pattern while energy transport strength is reduced. As the species concentration profile diminishes, the mass transfer characteristics are enhanced for raising magnitudes of the nonlinear chemical reaction parameter. Further, a velocity profile along the chordwise direction rises with enhancing magnitudes of nonlinear convection characteristics and yaw angle. Furthermore, the velocity pattern along the spanwise direction enhances with the growing magnitudes of yaw angle. For assisting buoyancy flow, the friction parameter at the border in the spanwise direction enhances with rising values of yaw angle.

Entropy generation analysis of radiative heat transfer in Williamson fluid flowing in a microchannel with nonlinear mixed convection and Joule heating

In this article, the spectral quasi-linearization (SQLM) method is implemented to solve the complicated differential equations governing the nonlinear mixed convective heat transfer of a Williamson fluid through a vertical microchannel. Unlike the conventional Boussinesq approximation, the quadratic Boussinesq approximation is taken into account in the formulation. The effects of Rosseland thermal radiation, Joule heating, and viscous dissipation are described in the thermal analysis subjected to the boundary conditions of convective thermal heating. The analysis of entropy production is also performed. The importance of various parameters governing velocity, Bejan number, temperature, and entropy generation was explored using graphic illustrations. It was found that the nonlinear density change with a temperature significantly affects the heat transport in the microchannel and thus increases the magnitude of the Bejan number and the production of entropy. Entropy production occurs maximum due to the boundary conditions of convection heating at the walls of the microchannel. Furthermore, due to a stronger viscous heating mechanism, the magnitude of the Bejan number is reduced, while the production of entropy increases significantly. As a limiting case of the problem, a comparison was made with results previously published in the literature and excellent agreement was established. The calculations provide a solid reference point for future CFD models and are relevant to the dynamics of polymers in microfluidic devices and the polymer industries.

Insight into the dynamics of micropolar fluid about a vertical cone when nonlinear thermal radiation is significant: The case of triple mixed convection

AbstractThis novel research investigates the nonlinear triple diffusive combined convective micropolar liquid flow past a vertical cone in the presence of nonlinear thermal radiation, cross‐diffusion, and a convective boundary condition. We aim to analyze this present problem using nonsimilar transformations. This report presents the significance of nonlinear mixed convection, energy flux due to the concentration gradient, and mass flux due to the temperature gradient and nonlinear thermal radiation in the dynamics of the fluid subject with micropolar fluid is presented. The differential equations defining the boundary‐layer parameters are then transformed into dimensionless view, taking into account the nonsimilar transformation. Furthermore, the method of quasilinearization and implicit finite difference approximation is used to work out the nondimensional governing equations for the solution. The velocity pattern diminishes, while dimensionless temperature and concentration distributions enhance with growing values of microrotation parameter. Furthermore, species concentrations of the fluid increase with increasing Soret effect values, while opposite results appear for mass transfer rates. Also, drag coefficient enhances for assisting buoyancy flow whilst diminishes for opposing buoyancy flow with increasing values of the microrotation parameter. The microrotation pattern reduces with growing values of the nonsimilarity characteristics. Furthermore, the Prandtl number is displayed on a comparison graph, and the results are very similar.

A convection–diffusion model on a star-shaped graph

In this paper we study a convection–diffusion equation on a star-shaped graph composed by n incoming edges and m outgoing edges with a nonlinearity $$f\in C^1({\mathbb {R}})$$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $$\partial _x(f(u(t, x))$$ with $$f(s)=-a|s|^{q-1}s$$ with $$q\ge 2$$ and $$a\in {\mathbb {R}}$$ verifying $$(n-m)a\ge 0$$ , we analyze the long time behavior of the solutions. For $$q> 2$$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case $$q=2$$ , the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers’ equation. Explicit representations of the limit profiles are obtained.

An implicit difference scheme for the fourth-order nonlinear partial integro-differential equations

In this paper, an implicit difference scheme is constructed and analyzed for the fourth-order nonlinear partial integro-differential equations. The backward Euler scheme and convolution quadrature are combined for time derivative and Riemann–Liouville (R–L) fractional integral terms. A fully discrete difference scheme is established with the space discretization by the standard central difference approximation. For nonlinear convection term , by the Galerkin method based on piecewise linear test functions, we can handle nonlinear term implicitly and attain a system of nonlinear algebraic equations. The stability and convergence are derived by the energy method. Especially, the existence and uniqueness of the numerical solutions are strictly proved for the nonlinear system. Also, we introduce three methods for solving the nonlinear systems, where two existing methods and a new linearized iterative algorithm are given to show the efficiency of our scheme. Numerical results are consistent with the theoretical analysis.

Contributions of nonlinear mixed convection for enhancing the thermal efficiency of Eyring-Powell nanoparticles for periodically accelerated bidirectional flow

Due to enhanced thermal aspects, the nanofluids present novel applications in the many engineering phenomenon like thermal extrusion, heat transmitting devices, energy production, nuclear processes, cooling procedure, thermal systems, solar energy, bio-medical processes, etc. This mathematical model proposed the more impressive thermal features of Eyring-Powell nano-material with new impact of nonlinear mixed convection. Additionally, the heat transfer enhancement is claimed by utilizing thermal radiation and magnetic force features. The accelerated flow pattern is observed via three-dimensional periodically moving configuration. The nanofluidic outcomes of Brownian and thermophoretic properties are focused out via Buongiorno’s nanofluid model. Apposite conversions are engaged to acquire dimensionless form of formulated nonlinear problem and then such transmuted equations elucidated analytically via homotopic technique. A complete graphical investigation for diverse influential parameters on velocity change, diffusion phenomenon, and wall shear surfaces is observed. Furthermore, the tables for numerical calculations of local Nusselt and Sherwood numbers are structured and discussed. It is perceived that the effects of material fluid parameter and magnetic parameter on both components of velocity are quite conflicting. The highly improved change in heat transfer is visualized for radiation parameter. Moreover, both temperature and concentration distributions show decreasing tendency for enhancement in fluid parameter.

Nonlinear mode analysis of periodic roll waves and wave splitting

ABSTRACT This paper describes the characteristics of the formation of periodic roll waves and the splitting of waves based on nonlinear mode analysis. A novel system is developed to investigate the formation of roll waves through linear and nonlinear modal analyses. The nonlinear analysis shows the feasibility of the periodic roll waves and their splitting when the wavenumber is extremely small. It is observed that there are phase shifts in each term of the momentum equation for the periodic roll waves. The phase of the nonlinear convection term in the fundamental mode is shifted forward relative to the phase of the linear convective term, and it acts as the viscosity term. In contrast, the phase of the nonlinear convection term in the higher mode is delayed, and it acts as the friction term. At a smaller wavenumber, the phase shift in the nonlinear interaction between the fundamental and second modes leads to the suppression of the fundamental mode development, resulting in the wave splitting.

Non-linear periodic long waves based on Boussinesq equation for shallow water waves: A coupled FEM modeling

A coupled mathematical model is developed based on the non-linear Boussinesq equation (BE) for shallow water waves to investigate the influence of the periodic non – linear long waves inside the irregular domain. A coupled numerical model based on the solution of the BE in a bounded region with the incorporation of the effects of viscous dissipation, dispersion, convective non-linearities, and variable bathymetry is obtained by using Galerkin Finite Element Method (GFEM), and an analytical approach based on Fourier-Bessel series solution for the scattered waves in unbounded region. The current coupled numerical model is validated by comparing the simulation results with existing studies. Moreover, the accuracy of the current scheme is also examined using convergence analysis for the rectangular domain. The present numerical model is applied on T and TT–shaped harbors to investigate the role of geometry to induce the amplification. In addition, the current approach implemented on the realistic Paradip port, Odisha, India, and Pohang New Harbor (PNH), South Korea at six record stations for practical implication. Therefore, the present numerical model can be utilized to construct/redesign an artificial port in the coastal region including variable bathymetry considering the effect of partial reflection, dissipation, and convective non – linearities.