Partial Differential Form Research Articles

Tangent hyperbolic nanofluid: Significance of Lorentz and buoyancy forces on dynamics of bioconvection flow of rotating sphere via finite element simulation

Enhanced thermal transportation for bio convection and mixed convection of Tangent hyperbolic nano fluid flow in stagnation region of rotating sphere is scrutinized in the existence of magnetic field. The nano-sized tiny particles are incorporated due to their unusual qualities like enhanced the thermal transport in the base fluid, which play significance role in modern nanotechnology. Further, improve the stability of nano-sized particles for long time period and avoid possible sedimentation, also incorporated auto-motile and gyrotactic microorganisms. The bioconvection of nano fluid transportation across spherical surface is considered important for heat storage and thermal balancing in technological processes. In this communication, time dependent partial differential formulation is transmuted to a set of dimensionless partial differential equation via similarity transformation. A numerical solution is sought through Galerkin based finite element discretization while it is coded and run on Matlab platform. The role of variables for the dependent quantities is monitored through computational procedure. The parameters of rotating sphere and Casson fluid speeded up the fluid flow in x-direction, but it recedes the flow in y-direction. The larger inputs of Brownian motion and thermophoresis parameters raised the nano fluid temperature.

An Amplitude-Frequency Curve-Based Parameter Sensitivity Analysis for the Conveying Fluid Pipe

Sensitivity analysis is a common way to research the parameters’ impact on complex systems, for instance national economy, environment, ecosystem, etc. The parametric analysis of a conveying fluid pipe is a challenge because it is a complex high-order partial differential system. In this brief, a mechanism-based method is proposed to analyze the parameter sensitivity of the conveying fluid pipe. Firstly, the mathematical model in a high-order partial differential form of the conveying fluid pipe under a harmonic load is deduced and transformed into a nonlinear ordinary differential equation form through the Galerkin truncation. The amplitude-frequency curve of the target system is established by harmonic balance method and matrix analysis technique. Then, the parameter sensitivity is discussed by the analytical results of the amplitude-frequency curve through Newton’s method. Finally, some examples are applied to intuitively illustrate the conveying fluid pipe’s parameter sensitivity.

Similarity Solutions of a non-Newtonian Fluid’s Momentum and Thermal Boundary Layers: Cross Fluid Model

The steady, incompressible and laminer flow of a non-Newtonian fluid that fits the Cross-fluid model over a flat plate is investigated. Dimensionless momentum and energy equations in partial differential form are derived to examine the variation of fluid velocity and temperature. The equations are simplified by the boundary layer theory based on the assumption that the change occurs in a narrow region, then scaling symmetries are calculated. By means of symmetries, equations in a partial form are reduced to an ordinary form by computing similarity variables and functions. The sbvp2.0 package developed for the Matlab environment based on collocation methods was used for the numerical solutions of the equations. In the light of analytical approach and solutions, the heat transfer is investigated by the Nusselt number. The study reveals that increases in Weissenberg number and power-law index, as non-Newtonian properties, are in charge of the thinner boundary layers, thus causing less friction and effective convection. As a result of numerical parts of the study, sbvp2.0 package is recomended for stiff equations with high nonlinearity, especially arising from boundary layer flows.

Analytical Solution for Vibrations of a Modified Timoshenko Beam on Visco-Pasternak Foundation Under Arbitrary Excitations

An analytical solution is derived for dynamic response of a modified Timoshenko beam with an infinite length resting on visco-Pasternak foundation subjected to arbitrary excitations. The modified Timoshenko beam model is employed to further consider the rotary inertia caused by the shear deformation of a beam, which is usually neglected by the traditional Timoshenko beam model. By using Fourier and Laplace transforms, the governing equations of motion are transformed from partial differential forms into algebraic forms in the Laplace domain. The analytical solution is then converted into the time domain by applying inverse transforms and convolution theorem. Some widely used loading cases, including moving line loads for nondestructive testing, travelling loads for seismic wave passage, and impulsive load for impact vibration, are also discussed in this paper. The proposed generic solutions are verified by comparing their degraded results to the known solutions in other literature. Several examples are performed to further investigate the differences of the beam responses obtained from the modified and the traditional Timoshenko beam models. Results show that the modified Timoshenko beam simulates the beam responses more accurately than the traditional model, especially under the dynamic loads with a high frequency. The analytical solutions proposed in this paper can be conveniently used for design and applied as an effective tool for practitioners.

Thermal analysis of nanofluidic flow through multi-walled carbon nanotubes subjected to perfectly and imperfectly bonded wall conditions

This research work scrutinizes thermal analysis of nanofluidic flow through MWCNTs subjected to perfectly and imperfectly bond wall conditions. Employing continuity of flow, Navier stokes equation (NS), energy equation for fluid and energy equation for MWCNTs, the thermal-fluidic models in partial differential forms are developed. Thereafter, the established unique models are converted into ordinary differential equations via appropriate similarity transform and solved analytically using the classical differential transform method. With the obtained solutions, thermal-fluidic profile visualizations as well as parametric studies were executed. The range of parameters considered are from 0 to 0.05 for Bi number and 0.0 to 8.0 for the magnetic field term. The results point out that an augmentation in Bi generally reduces thermal profile under slip and non-slip conditions. The solutions as enumerated in this work match with existing results from literatures, hence the verification and validation of this study. It is foreseen that the results obtained will provide better physical insights into the thermal analysis of internal flow through nanotubes and will enhance the designs of nanomechanical carbon nanotubes-based devices that convey fluid under different wall conditions.

Bioconvection: Significance of mixed convection and mhd on dynamics of Casson nanofluid in the stagnation point of rotating sphere via finite element simulation

Enhanced thermal transportation for bio convection and mixed convection of Casson nano fluid flow in stagnation region of rotating sphere is scrutinized in the existence of magnetic field. The bioconvection of nano fluid transportation across spherical surface is considered important for heat storage and thermal balancing in technological processes. To enhanced heat transportation, the gyrotactic microorganisms are used with nano sized particles to improve stability. In this communication, time dependent partial differential formulation is transmuted to a set of dimensionless partial differential equation via similarity transformation. A numerical solution is sought through Galerkin based finite element discretization while it is coded and run on Matlab platform. The role of variables for the dependent quantities is monitored through computational procedure. The parameters of rotating sphere and Casson fluid speeded up the fluid flow in x-direction, but it recedes the flow in y-direction. The larger inputs of Brownian motion and thermophoresis parameters raised the nano fluid temperature. The absolute value of Nusselt number is up-surged with enhanced buoyancy and rotation but diminishes against parameters of Brownian motion, bio convection, and Rayleigh number.

An improved lumped model for freezing of a freely suspended supercooled water droplet in air stream

This work deals with the mathematical modeling of the transient freezing process of a supercooled water droplet in a cold air stream. The aim is to develop a simple yet accurate lumped-differential model for the energy balance for a freely suspended water droplet undergoing solidification, that allows for cost effective computations of the temperatures and freezing front evolution along the whole process. The complete freezing process was described by four distinct stages, namely, supercooling, recalescence, solidification, and cooling. At each stage, the Coupled Integral Equations Approach (CIEA) is employed, which reduces the partial differential equation for the temperature distribution within the spherical droplet into coupled ordinary differential equations for dimensionless boundary temperatures and the moving interface position. The resulting lumped-differential model is expected to offer improved accuracy with respect to the classical lumped system analysis, since boundary conditions are accounted for in the averaging process through Hermite approximations for integrals. The results of the CIEA were verified using a recently advanced accurate hybrid numerical-analytical solution through the Generalized Integral Transform Technique (GITT), for the full partial differential formulation, and comparisons with numerical and experimental results from the literature. After verification and validation of the proposed model, a parametric analysis is implemented, for different conditions of airflow velocity and droplet radius, which lead to variations in the Biot numbers that allow to inspect for their influence on the accuracy of the improved lumped-differential formulation.

Applications of temperature dependent viscosity for Cattaneo–Christov bioconvection flow of couple stress nanofluid over oscillatory stretching surface: A generalized thermal model

This continuation reports the thermal consequences of couple stress nanofluid containing gyrotactic microorganisms model in presence of novel features like nonlinear thermal, temperature dependent viscosity and uniform magnetic force. The generalized expressions of diffusion theory are entertained for the modeling of energy and concentration equations. The thermal radiative flow is addressed by using nonlinear thermal radiation instead of traditional linear thermal expressions. The effective Brownian motion and thermophoresis mechanisms are focused in view of Buongiorno's nano-material model. The current nanofluid model with couple stress nonlinear model is the most improved thermal model which preserves many results as a limiting case. The flow is persuaded by periodically accelerated and stretching surface. The resulted equations based on various elementary laws are formulated in partial differential form. The most convergent technique homotopy analysis method is proposed to discuss the solution procedure. The convergence analysis is performed and later on a comprehensive graphical observation with interesting physical justifications is presented. Various physical quantities are graphically entertained. The results show that the velocity profile oscillates periodically and magnitude of oscillation increases with couple stress parameter. The temperature of nanoparticles boosted with the viscosity parameter while a declining pattern of temperature is noted for the thermal Biot number. The skin friction coefficient periodically increases with the viscosity vector and couple stress parameter.

Analytical Solution for Longitudinal Dynamic Responses of Long Tunnels under Arbitrary Excitations

In this paper, an analytical solution is proposed for longitudinal dynamic responses of long tunnels under arbitrary excitations. For the derivation, the tunnel is assumed as a Timoshenko beam resting on a visco-Pasternak foundation. The Timoshenko beam theory is employed to consider both effects of the shear distortion as well as the rotary inertia of the tunnel, which are neglected by the Euler–Bernoulli beam. The visco-Pasternak foundation is applied to represent the viscoelastic compressive and shear resistance of soil. The governing equations of motion are transformed from partial differential forms into algebraic forms through integral transformations, and thus the solutions are conveniently obtained. The analytical solutions of the tunnel under several specific dynamic loads, including impulsive loads, moving line loads as well as traveling loads, are presented in detail and verified by comparing to the known degraded solution in literature and finite element results. Several examples are also conducted to investigate the influence of the relative stiffness ratio between the soil and the tunnel structure on the tunnel responses.

Hybrid integral transform analysis of supercooled droplets solidification.

The freezing phenomena in supercooled liquid droplets are important for many engineering applications. For instance, a theoretical model of this phenomenon can offer insights for tailoring surface coatings and for achieving icephobicity to reduce ice adhesion and accretion. In this work, a mathematical model and hybrid numerical–analytical solutions are developed for the freezing of a supercooled droplet immersed in a cold air stream, subjected to the three main transport phenomena at the interface between the droplet and the surroundings: convective heat transfer, convective mass transfer and thermal radiation. Error-controlled hybrid solutions are obtained through the extension of the generalized integral transform technique to the transient partial differential formulation of this moving boundary heat transfer problem. The nonlinear boundary condition for the interface temperature is directly accounted for by the choice of a nonlinear eigenfunction expansion base. Also, the nonlinear equation of motion for the freezing front is solved together with the ordinary differential system for the integral transformed temperatures. After comparisons of the solution with previously reported numerical and experimental results, the influence of the related physical parameters on the droplet temperatures and freezing time is critically analysed.

Convective heat and zero-mass flux conditions in the time-dependent second-grade nanofluid flow by unsteady bidirectional surface movement

In this paper, a theoretical model is developed for the deliberation of time-dependent 3D magnetohydrodynamics second-grade (SG) nanofluid with radiation effects past an unsteady stretched surface. The work is carried out by involving the physical effects of thermophoresis and Brownian movement. To find the desired solutions, more practical boundary conditions are introduced familiar as the Robin and zero-mass diffusion conditions. The derived model of partial differential form is reframed into the structure of ordinary differential expressions by implementing the dimensionless variables which are reported numerically through the Runge-Kutta-Fehlberg (RKF) scheme with shooting procedure. Numerically determined skin-friction data and energy transportation rate are also addressed with appropriate analysis. Solutions are performed for nanofluid velocity, concentration of nanoparticles and nanofluid temperature. The analysis of dissimilar parametric values on liquid velocity, concentration of nanoparticles and temperature is reported using plots and tabulated data. The presence of Hartman and Biot numbers demonstrated the higher temperature curves. The nanoparticle concentration and temperature distributions reduced for larger unsteady parameter values. A comparison is also provided for limiting cases to authenticate our obtained results.

Finite Element Study for Magnetohydrodynamic (MHD) Tangent Hyperbolic Nanofluid Flow over a Faster/Slower Stretching Wedge with Activation Energy

The below work comprises the unsteady flow and enhanced thermal transportation for Carreau nanofluids across a stretching wedge. In addition, heat source, magnetic field, thermal radiation, activation energy, and convective boundary conditions are considered. Suitable similarity functions use to transmuted partial differential formulation into the ordinary differential form, which is solved numerically by the finite element method and coded in Matlab script. Parametric computations are made for faster stretch and slowly stretch to the surface of the wedge. The progressing value of parameter A (unsteadiness), material law index ϵ, and wedge angle reduce the flow velocity. The temperature in the boundary layer region rises directly with exceeding values of thermophoresis parameter Nt, Hartman number, Brownian motion parameter Nb, ϵ, Biot number Bi and radiation parameter Rd. The volume fraction of nanoparticles rises with activation energy parameter EE, but it receded against chemical reaction parameter Ω, and Lewis number Le. The reliability and validity of the current numerical solution are ascertained by establishing convergence criteria and agreement with existing specific solutions.

Stagnation point Walter’s-B nanofluid flow over power-law lubricating surface with slip conditions: Hybrid HAM solutions

Modern bearings used the lubricants having polymer components of larger molecular weight for higher speed and high load operations. These lubricants show rheological activity that is non-Newtonian. Power-law lubricants are commonly used for industrial and bearing lubrication operations. This article is modeled through the slip as interfacial condition to demonstrate the role of power-law lubrication on the stagnation point flow of Walters-B nanofluid. The thermophoretic force and Brownian movement factors are accounted in energy and mass species expressions. The expressions in partial differential forms are constructed for both the base and lubricant fluids. The system is re-structured into ordinary differential expressions by the implication of dimensional analysis. An approximate and numerical scheme namely the hybrid homotopy analysis method is reported to demonstrate the solutions of nonlinear expressions. The behavior of involved constraints on practical quantities is shown and analyzed. The nature of friction factor, thermal and solutal rates at the surface is reported through numeric data. Match of the present solution with previously published data shown better agreement. It can be seen that the slip phenomenon dominates over the free-stream phenomenon. The large values of viscoelastic parameter decay the thermal and solutal fields.

The Relations between the Various Critical Temperatures of Thin ‎FGM Plates

This work investigates the relations between the critical temperature of the thin FGM plates under various temperature distributions through the thickness resting on the Pasternak elastic foundation. Both rectangular and skew plates are investigated. The uniform, linear, and nonlinear temperature distributions through the plate’s thickness are considered. Formulations are derived based on the classical plate theory (CPT) considering the von Karman geometrical nonlinearity taking the physical neutral plane as the reference plane. The partial differential formulation is separated into two sets of ordinary differential equations using the extended Kantorovich method (EKM). The stability equations and boundary conditions terms are derived according to Trefftz criteria using the variational calculus expressed in an oblique coordinate system. Novel multi-scale plots are presented to show the linear relations between the critical temperatures under various temperature distributions. The critical temperature of plates with different materials are also found linearly related. Resulting relations should be a huge time saver in the analysis process, as by knowing one critical temperature of the one FGM plate under one temperature distribution many other critical temperatures of many other FGM plates under any temperature distributions can be obtained instantly.

Relation of Integro-Partial Differential Equations With Delay Effect Based on the Maxwell Equations to the Heaviside and Pocklington Equations

We derive the integro-partial differential equations with delay effect based on the Maxwell equations for a two-line system under the thin-wire approximation and the approximation of neglecting the interwire distance in the delay term. We introduce the normal- and common-mode and write the integro-partial differential equations in terms of these two modes. For the case of the same radius, we show that the equation of the normal-mode reduces to the Heaviside telegraph equation in the partial differential form and the equation of the common-mode becomes the Pocklington equation in the integro-partial differential form. We perform numerical calculations of the coupled integral and partial differential equations in the time domain and compare the results with those of various approximations. We show that the Heaviside telegraph equation is valid for the normal-mode, while the full treatment of the delay term in the integro-partial differential equations is necessary for the common-mode.

Impact of Stefan blowing on thermal radiation and Cattaneo–Christov characteristics for nanofluid flow containing microorganisms with ablation/accretion of leading edge: FEM approach

The impacts of Stefan blowing on Cattaneo–Christov characteristics and bioconvection of self-motive microorganisms mixed in water-based nanofluids with a ablation/accretion of leading edge are examined in the present investigation. Governing partial differential formulation is transmuted into ordinary differential form via similarity functions. The finite element method is harnessed to yield solution of numerical for the resulting set of nonlinear coupled equations with coding implementation in MATLAB. It is noteworthy that the reliability and validity of the current numerical solution are an excellent agreement with existing specific solutions in the literature. The interest in computational effort centered about the formation of boundary layer patterns for microorganism distribution, fluid temperature, volume fraction of nanoinclusions and fluid velocity when influential parameters are varied. The most important results of the current examination are that upgrade in Stefan blowing parameter undermines the fluid velocity while an increment in ablation/accretion impact at leading edge shows in a deceleration in flow velocity. Another significant result is that an increment in ablation/accretion at leading edge upsurges the fluid temperature and concentrations.

Brownian motion and thermophoretic diffusion influence on thermophysical aspects of electrically conducting viscoinelastic nanofluid flow over a stretched surface

Current investigation elaborates the impacts of Brownian motion and thermophoretic force on electrically conducting Prandtl-Eyring nanofluid flow yielded by stretched surface. Buongiorno nano-model is used to trace the heat and mass transfer characteristics in the flow regime. The mathematical formulation concerning to the adopted physical parameters is modeled in the form of complex partial differential structure. Boundary layer theory is obliged to reduce non-linearity of subsequent equations by truncating higher order terms. To facilitate the computation process, the governing problem in partial differential form is converted into dimensionless ordinary differential expressions. Numerical solution for attained boundary value problem is procured by R-K-Fehlberg methodology. The consequences of flow governing parameters on interested physical quantities (momentum, heat, concentration) are depicted in graphical manner while tabular representation is used to demonstrate the variations in wall drag coefficient, wall thermal flux and particles concentration flux. The computed results show that the presence of magnetic field is not favorable for fluid momentum, albeit, both Brownian motion and thermophoresis phenomenon surges the thermal energy of fluid. Besides these, concentration profile increases versus Brownian motion while thermophoresis phenomenon has reverse impacts on it. Surface drag force enriched by magnifying the magnetic field intensity, furthermore, the surface heat flux shows reduction versus Brownian motion and thermophoresis parameters. In addition, the surface mass flux shows increasing trend versus all governing parameters.

Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which higher dimensional algebras this characterization can exactly be extended and under which circumstances. It turns out to be crucial that this characterization requires a domain of definition that lies in a subalgebra that has the norm composition property and that is either associative (Clifford algebra case) or at least alternative (octonionic case). The orthonormal frames are elements of the spin group Spin(n+1). We round off by relating the nature of the orthonormal frames to the associated Möbius transformation which are related to SO(9, 1) in the octonionic case and to the Ahlfors–Vahlen group in the case of a Clifford algebra.

Study of thermal and hydrodynamic characteristics of water-nano-encapsulated phase change particles suspension in an annulus of a porous eccentric horizontal cylinder

In this paper, the thermal and hydrodynamic characteristics of a suspension with water-Nano-Encapsulated Phase Change Material (NEPCM) in an annulus of a porous eccentric horizontal cylinder are investigated. The NEPCM particles have a core-shell structure and stability suspended in water. Hence, the particles, along with the liquid, could freely circulate inside the annuli of the horizontal cylinder due to the buoyancy forces. The cores of these particles are made from a Phase Change Material (PCM). Moreover, such cores are in a continuous exchange of heat transfer between the solid and liquid phases. The heat transfer is acting in a combination of absorption, storage, and release mechanisms. The governing equations for the fluid motions and conservation of energy could be written in partial differential forms and by using the appropriate non-dimensional variables converted into non-dimensional ones. Then, the numerical approach is applied by implementing the finite element method (FEM) to solve such equations iteratively. The impact of various non-dimensional parameters including the fusion temperature, Stefan number, Rayleigh number, Darcy number, the volume fraction of nanoparticles, and eccentricity of the inner cylinder is addressed on the flow and heat transfer. It is observed that the most favourable fusion temperature ranges for the maximum heat transfer rate vary as a function of the Rayleigh number. In addition, the heat transfer rate can be enhanced by applying the phase change core of nanoparticles.

Partial differential Chow forms and a type of partial differential Chow varieties

We first present an intersection theory of irreducible partial differential varieties with quasi-generic differential hypersurfaces. Then, we define partial differential Chow forms for irreducible partial differential varieties whose Kolchin polynomials are of the form And we establish for partial differential Chow forms most of the basic properties of their ordinary differential counterparts. Furthermore, we prove that a certain type of partial differential Chow varieties exist.Communicated by Jason Bell