Structure Of Partial Differential Equations Research Articles

Phase change heat transfer in a vertical metal foam-phase change material thermal energy storage heat dissipator

A metallic foam heat dissipator for cooling electronic components was addressed. A heat dissipator is a partitioned aluminum container loaded along with aluminum metallic foam and saturated with paraffin wax. A heat flux at a surface contains a basic uniform flux and the step transient raise, which should be managed by a heat dissipator and a Phase Change Material (PCM). The regulating equations for a melting/solidification transfer of heat & momentum transport in a heat dissipator were instituted into a structure of partial differential equations. Then, the vital monitoring equations were converted into a general dimensionless type and solved by the Finite Element Method. A mesh adjustment technique & automated time-step control was employed to control the accuracy & convergence of the result automatically. An adaptation technique controls the mesh resolution at the melting/solidification interface. The dimensionless temperature of fusion is a vital factor in the control of the surface temperature and heat dissipator efficiency. Considering a fixed amount of material for walls, a heat dissipator with thick sidewalls and thin top and bottom walls results in slightly better thermal performance. Using a PCM heat sink could reduce the heated surface temperature by >175 % during the pulse load.

Numerical analysis on the heat and mass transfer MHD flow characteristics of nanofluid on an inclined spinning disk with heat absorption and chemical reaction

AbstractThis work examines the heat transfer properties of magnetohydrodynamic nanofluid flow. Through a similarity conversion, the leading structure of partial differential equations is changed to that of ordinary differential equations. A rigorous mathematical bvp4c methodology is used to generate numerical results. The purpose of this study is to characterize the different temperature, concentration, and velocity limitations on a nanofluid with a magnetic effect that is spinning. The findings for rotating nanofluid flow and heat transfer characteristics of nanoparticles are shown using graphs and tables. The influence of physical factors such as heat transfer rates and skin friction coefficients is studied. When the magnetic parameter M is raised, the velocity of the nanoliquid decreases. A rise in thermal radiation (Rd) causes the temperature graphs to grow substantially, although the concentration profiles exhibit the opposite tendency. The effect of the convective heat transfer factor Bi on temperature is shown to increase as Bi increases, but the concentration distribution decreases as Biot increases.

Study of MHD SWCNT-Blood Nanofluid Flow in Presence of Viscous Dissipation and Radiation Effects through Porous Medium

In this analysis, a computational study is conducted to examine the two-dimensional flow of an incompressible MHD SWCNT-blood nanofluid, saturated mass and porous medium .In addition to viscous dissipation, thermal radiation is taken into consideration. We developed the mathematical model and useful boundary intensity approximations to diminish the structure of partial differential equations based on the fluid for blood-based SWCNT underflow assumptions. Converted the partial differential equations by applying corresponding transformations to arrive at ODE’s. The above results are solved numerically by the Runge-Kutta 4th order technique. Noticed that there is desirable conformity when interpolated with the numerical one. The effects exhibited the velocity of SWCNT-blood nanofluid enhanced for defined standards of the viscosity parameter. Rise in temperature when various parameters like Prandtl number, Eckert number, and slip parameter are applied on SWCNT-blood. The impact of fluid flow on blood-based SWCNT is discussed graphically, and our results are tabulated along with illustrations. The design concepts, such as the Nusselt quantity and the local skin friction, conform to the analytical approach. Velocity reductions with an increase in CNT’s volume fraction, whereas enhancement in the blood temperature, is noted, which is directed to the rise in the heat mass transfer rates.

Marangoni transport of Jeffrey nanofluid due to circular horizontal cylinder with motile microorganisms

The concept of macroscopic convection mobility of a fluid induced by a density differential caused by directional collective swimming of microorganisms is known as bioconvection. These motile microorganisms are characterized as oxytactic, gravitactic, gyrotactic, and chemo-tactic on their movements. The horizontal circular cylinder is exploited to transport Jeffrey nanofluids with the repercussions of viscous dissipation and motile microorganisms. Moreover, the Cattaneo-Christov theory and thermal radiation are also assumed. The regulating structure of partial differential equations is renovated into ordinary differential equations (ODEs) incorporating the shooting method with appropriate similarity transformation. For numerical simulation, the attained set of ODEs is cracked through bvp4c built-in mathematical tool MATLAB. The numerical and illustrated fallouts of prominent parameters like bioconvection Lewis number, Lewis number, mixed convective number, Prandtl parameter, Peclet number, thermophoresis parameter and Brownian motion parameter against velocity distribution, temperature gradient, volumetric and motile microorganism concentration. The escalating valuation of velocity slip parameter, bioconvection Rayleigh number and solutal Biot number exaggerated the concentration profile. The intensifying values of bioconvective Lewis, Peclet and Marangoni parameters and fluid parameter the microorganisms profile of nanoparticles. The paper presents a model as well as numerical statistics on the upshot of bioconvection on small solid particle mixing.

A thermal conductivity model for hybrid heat and mass transfer investigation of single and multi-wall carbon nano-tubes flow induced by a spinning body

The enhancement of thermal heat transfer utilizing the hybrid nanofluids is explored widely by numerous researchers over the years. Because of these exciting characteristics features of hybrid nanofluids, the prime focus of this article is the investigation of engine oil and mineral oil-based hybrid nanofluids consisting of carbon nanotubes (SWCNT-MWCNT) over a rotating cone present in a convectively heated permeable environment. In the flow supposition, a mathematical model is formulated, Boundary layer approximations are used to minimize the structure of partial differential equations. The dimensionless system of highly non-linear partial differential equations is obtained through boundary layer approximation. The findings obtained showed a rise in temperature for both CNT's, the thickness of the thermal boundary layer has increased under the impact of solid volume fraction of nanotubes. SWCNTs provide a reduction in skin friction and enhancement in Nusselt number. The hybrid nanofluid with EO (engine oil) base has demonstrated a greater heat transfer coefficient and reduced fluid motion's skin friction than MO (mineral oil) based. The model hybrid flow can be used in industries like turbine heat production, wind energy generation, and pharmaceutical processes.

Three-Dimensional Water-Based Magneto-Hydrodynamic Rotating Nanofluid Flow over a Linear Extending Sheet and Heat Transport Analysis: A Numerical Approach

This comparative study inspects the heat transfer characteristics of magnetohydrodynamic (MHD) nanofluid flow. The model employed is a two-phase fluid flow model. Water is utilized as the base fluid, and zinc and titanium oxide (Zn and TiO2) are used as two different types of nanoparticles. The rotation of nanofluid is considered along the z-axis, with velocity ω*. A similarity transformation is used to transform the leading structure of partial differential equations to ordinary differential equations. By using a powerful mathematical BVP-4C technique, numerical results are obtained. This study aims to describe the possessions of different constraints on temperature and velocity for rotating nanofluid with a magnetic effect. The outcomes for the rotating nanofluid flow and heat transference properties for both types of nanoparticles are highlighted with the help of graphs and tables. The impact of physical concentrations such as heat transference rates and coefficients of skin friction are examined. It is noted that rotation increases the heat flux and decreases skin friction. In this comparative study, Zn-water nanofluid was demonstrated to be a worthy heat transporter as compared to TiO2-water nanofluid.

Elegant scheme for one-way wave propagation in Kerr media

In this article, an intelligent computational scheme is presented for the dynamical solution of fractional-order linear and nonlinear (Kerr nonlinearity) Helmholtz equations with boundary conditions, which have a significant impact on electromagnetics, hydrodynamics and acoustic phenomena. The propagation of electromagnetic waves in Kerr media describes the linearly polarized electric field in an isotropic material, for a range of important phenomena in nonlinear optics and other areas. Here, the generalized Helmholtz equation of $$ n $$ dimensions is considered with fractional-order derivative. The fractional Laplace transform is utilized, and parameter $$ s^{\alpha } $$ is linearized by series expansion. Ultimately, the inverse fractional Laplace transform of this expansion reduces the fractional-order derivative to integer-order derivative. The new attained structure of partial differential equation becomes equivalent to the so-called fractional-order Helmholtz equation. The contribution also provides an artificial neural network-based technique, designed with a metaheuristic algorithm, to analyze the governing model numerically and graphically. Taking sigmoidal function as an activation function, the unsupervised error function is formulated, which is then optimized with the aid of the firefly algorithm (FFA). By using FFA, the minimum search path of the error function is tracked with the convergent values of the network weights. Besides, the validity and accuracy of the adapted technique are ascertained by calculating the error norms that ensure the convergence of the approximation. Accordingly, the achieved facts and figures accentuate comprehensively the novel implication of this attempt.

Magnetohydrodynamic flow of molybdenum disulfide nanofluid in a channel with shape effects

PurposeThe purpose of this paper is to examine the combined effects of thermal radiation and magnetic field of molybdenum disulfide nanofluid in a channel with changing walls. Water is considered as a Newtonian fluid and treated as a base fluid and MoS2 as nanoparticles with different shapes (spherical, cylindrical and laminar). The main structures of partial differential equations are taken in the form of continuity, momentum and energy equations.Design/methodology/approachThe governing partial differential equations are converted into a set of nonlinear ordinary differential equations by applying a suitable similarity transformation and then solved numerically via a three-stage Lobatto III-A formula.FindingsAll obtained unknown functions are discussed in detail after plotting the numerical results against different arising physical parameters. The validations of numerical results have been taken into account with other works reported in literature and are found to be in an excellent agreement. The study reveals that the Nusselt number increases by increasing the solid volume fraction for different shapes of nanoparticles, and an increase in the values of wall expansion ratio α increases the velocity profile f′(η) from lower wall to the center of the channel and decreases afterwards.Originality/valueIn this paper, a numerical method was utilized to investigate the influence of molybdenum disulfide (MoS2) nanoparticles shapes on MHD flow of nanofluid in a channel. The validity of the literature review cited above ensures that the current study has never been reported before and it is quite new; therefore, in case of validity of the results, a three-stage Lobattoo III-A formula is implemented in Matlab 15 by built in routine “bvp4c,” and it is found to be in an excellent agreement with the literature published before.

Thermal radiation and slip effects on magnetohydrodynamic (MHD) stagnation point flow of Casson fluid over a convective stretching sheet

Current study examines the combined effect of thermal radiation and velocity slip along a convective heated stretching sheet. magnetohydrodynamic (MHD) effect near the stagnation point is also under consideration. The main structure of partial differential equations attained in the form of momentum, energy and concentration equations. For a self-similar solution, the system of governing PDEs assimilated into the set of nonlinear ordinary differential equations by applying suitable similarity transformation. Resulting nonlinear ODEs are successfully solved via Runge-Kutta-Fehlberg method. All obtained unknown functions are discussed in detail after plotting the numerical results against different arising physical parameters. The validations of numerical results have been taken into account with two different numerical approaches and it is found to be in excellent agreement with the numerical results. The study reveals that concentration boundary layer thickness decreases by increasing the values of Schmidt number and increases by increasing the chemical reaction parameter.

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

The theory of multidimensional Poisson vertex algebras provides a completely algebraic formalism for studying the Hamiltonian structure of partial differential equations for any number of dependent and independent variables. We compute the cohomology of the Poisson vertex algebras associated with twodimensional, two-component Poisson brackets of hydrodynamic type at the third differential degree. This allows obtaining their corresponding Poisson–Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of nonequivalent infinitesimal deformations, nor in the third group, corresponding to the obstructions to extending such deformations.

Convex hull property and maximum principle for finite element minimisers of general convex functionals

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for $$\mathbb{P }_1$$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $$p$$ -Laplacian and the mean curvature problem. In the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.

Differential geometric computations and computer algebra

The use of computer algebra in the field of differential geometry and its applications to geometric structures of partial differential equations is discussed. The differential geometric setting is shortly described; a number of programs are slightly touched, some examples given, and an application to the construction of supersymmetric extensions of the Korteweg-de Vries equation is demonstrated.