Partial Differential Boundary Value Research Articles

Analysis of Nonlinear Convection-Radiation in Chemically Reactive Oldroyd-B Nanoliquid Configured by a Stretching Surface with Robin Conditions: Applications in Nano-Coating Manufacturing.

Motivated by emerging high-temperature manufacturing processes deploying nano-polymeric coatings, the present study investigates nonlinear thermally radiative Oldroyd-B viscoelastic nanoliquid stagnant-point flow from a heated vertical stretching permeable surface. Robin (mixed derivative) conditions were utilized in order to better represent coating fabrication conditions. The nanoliquid analysis was based on Buongiorno's two-component model, which features Brownian movement and thermophoretic attributes. Nonlinear buoyancy force and thermal radiation formulations are included. Chemical reactions (constructive and destructive) were also considered since coating synthesis often features reactive transport phenomena. An ordinary differential equation model was derived from the primitive partial differential boundary value problem using a similarity approach. The analytical solutions were achieved by employing a homotopy analysis scheme. The influence of the emerging dimensionless quantities on the transport characteristics was comprehensively explained using appropriate data. The obtained analytical outcomes were compared with the literature and good correlation was achieved. The computations show that the velocity profile was diminished with an increasing relaxation parameter, whereas it was enhanced when the retardation parameter was increased. A larger thermophoresis parameter induces an increase in temperature and concentration. The heat and mass transfer rates at the wall were increased with incremental increases in the temperature ratio and first order chemical reaction parameters, whereas contrary effects were observed for larger thermophoresis, fluid relaxation and Brownian motion parameters. The simulations can be applied to the stagnated nano-polymeric coating of micromachines, robotic components and sensors.

Exponential heat source effects on the stagnation‐point heat transport of Williamson nanoliquid with nonlinear Boussinesq approximation

AbstractThe nonlinear two‐point partial differential boundary value problem associated with the nano‐pseudoplastic material flow and heat transport subject to nonlinear Boussinesq approximation is computed and explored statistically. Heat transportation features are analyzed by the consideration of an exponential space‐related heat source and the Buongiorno model of nanofluids. The boundary‐driven expressions of the physical phenomenon are coupled and highly complicated due to the consideration of nonlinear convection terms. Reasonable variables are employed to reform the partial differential equations into a system of ordinary differential expressions and are solved numerically. Furthermore, correlation and regression techniques are employed for the statistical evaluation of the phenomenon. The probable error is implemented to calculate the reliability of the computed correlation factors. The exponential index and Schmidt number are positively correlated with the reduced skin friction coefficient whereas the other parameters are negatively correlated with it. The heat transfer rate is improved predominantly by the nonlinear thermal convection parameter. The temperature is enhanced by the intensification of the exponential‐based heat source factor. The temperature and concentration profiles are boosted by incrementing the Biot number values.

Simulation of unsteady natural convection flow of a Casson viscoplastic fluid in a square enclosure utilizing a MAC algorithm

AbstractNon‐Newtonian fluids are increasingly being deployed in energy systems and materials processing. Motivated by these developments, in the current study, a numerical simulation is performed on two‐dimensional, unsteady buoyancy‐driven flow in a square cavity filled with non‐Newtonian fluid (Casson liquid). The enclosure geometry features vertical isothermal walls (with one at higher temperature than the other) and thermally insulated horizontal walls. The conservation equations for mass, momentum, and energy are normalized via appropriate transformations and the resulting dimensionless partial differential boundary value problem is solved computationally with a marker and cell algorithm, which features a finite difference scheme along with a staggered grid system. The projection method is employed to evaluate the pressure term. Extensive visualizations of the impact of emerging physical parameters (Rayleigh number and Casson viscoplastic parameter) on streamline and isotherm distributions in the cavity are presented for fixed Prandtl number. Nusselt number, that is, heat transfer rate, is increased with rising values of the Casson viscoplastic fluid parameter for any value of Rayleigh number. The density of streamlines increases with increasing values of Casson viscoplastic fluid parameter upto 1. Overall, the Casson fluid parameter plays a vital role in controlling the convective heat transfer within the enclosure. The computations are relevant to hybrid solar collectors, materials fabrication (polymer melts), etc.

Numerical study of slip and radiative effects on magnetic Fe3O4-water-based nanofluid flow from a nonlinear stretching sheet in porous media with Soret and Dufour diffusion

Increasingly sophisticated techniques are being developed for the manufacture of functional nanomaterials. A growing interest is also developing in magnetic nanofluid coatings which contain magnetite nanoparticles suspended in a base fluid and are responsive to external magnetic fields. These nanomaterials are “smart” and their synthesis features high-temperature environments in which radiative heat transfer is present. Diffusion processes in the extruded nanomaterial sheet also feature Soret and Dufour (cross) diffusion effects. Filtration media are also utilized to control the heat, mass and momentum characteristics of extruded nanomaterials and porous media impedance effects arise. Magnetite nanofluids have also been shown to exhibit hydrodynamic wall slip which can arise due to non-adherence of the nanofluid to the boundary. Motivated by the multi-physical nature of magnetic nanomaterial manufacturing transport phenomena, in this paper, we develop a mathematical model to analyze the collective influence of hydrodynamic slip, radiative heat flux and cross-diffusion effects on transport phenomena in ferric oxide ([Formula: see text]-water) magnetic nanofluid flow from a nonlinear stretching porous sheet in porous media. Hydrodynamic slip is included. Porous media drag is simulated with the Darcy model. Viscous magnetohydrodynamic theory is used to simulate Lorentzian magnetic drag effects. The Rosseland diffusion flux model is employed for thermal radiative effects. A set of appropriate similarity transformation variables are deployed to convert the original partial differential boundary value problem into an ordinary differential boundary value problem. The numerical solution of the coupled, multi-degree, nonlinear problem is achieved with an efficient shooting technique in MATLAB symbolic software. The physical influences of Hartmann (magnetic) number, Prandtl number, Richardson number, Soret (thermo-diffusive) number, permeability parameter, concentration buoyancy ratio, radiation parameter, Dufour (diffuso-thermal) parameter, momentum slip parameter and Schmidt number on transport characteristics (e.g. velocity, nanoparticle concentration and temperature profiles) are investigated, visualized and presented graphically. Flow deceleration is induced with increasing Hartmann number and wall slip, whereas flow acceleration is generated with greater Richardson number and buoyancy ratio parameter. Temperatures are elevated with increasing Dufour number and radiative parameter. Concentration magnitudes are enhanced with Soret number, whereas they are depleted with greater Schmidt number. Validation of the MATLAB computations with special cases of the general model is included. Further validation with generalized differential quadrature (GDQ) is also included.

Lie Group Analysis of Nanofluid Slip Flow with Stefan Blowing Effect via Modified Buongiorno’s Model: Entropy Generation Analysis

This article presents a detailed theoretical and computational analysis of alumina and titania-water nanofluid flow from a horizontal stretching sheet. At the boundary of the sheet (wall), velocity slip, thermal slip and Stefan blowing effects are considered. The Pak-Cho viscosity and thermal conductivity model is employed together with the non-homogeneous Buongiorno nanofluid model. The equations for mass, momentum, energy and nanoparticle species conservation are transformed via Lie-group transformations into a dimensionless system. The partial differential boundary value problem is therefore rendered into nonlinear ordinary differential form. With appropriate boundary conditions, the emerging normalized equations are solved with the semi-numerical homotopy analysis method (HAM). To consider entropy generation affects a second law thermodynamic analysis is also carried out. The impact of some physical parameters on the skin friction, Nusselt number, velocity, temperature and entropy generation number (EGM) are represented graphically. This analysis shows that diffusion parameter is a key factor to retards the friction and rate of heat transfer at the surface. Further, temperature of fluid decreases for the higher value of thermal slip parameter. In addition, EGM enhances with nanoparticles ambient concentration and Reynolds number. A numerical validation of HAM results is also included. The computations are relevant to thermodynamic optimization of nano-material processing operations.

Transient 2D Heat Transfer with Convection in an Anisotropic Rectangular Slab

It is not unusual that the properties of a material vary with different crystallographic orientations and this property is said to be anisotropy. Paper presented here analytically studies the transient solution to the anisotropic heat conduction in an anisotropic rectangular thin film subjected to initial arbitrary temperature function throughout the medium. An integral transformation is applied to remove the partial derivatives with respect to two spatial variables and transform the partial differential boundary value problem to ordinary differential equations. The result is the multiple integral with respect to spatial variables where the mathematics and model ling software MAPLE 18 was used to numerically calculate the summation and to plot the temperature and heat flux vector profiles.

Similarity solutions to unsteady pseudoplastic flow near a moving wall

It is shown that similarity solutions to a partial differential boundary value problem for power law fluids may be generated by the application of a multiparameter stretching group. Furthermore, the resulting ordinary differential boundary value problem arising from the initial group analysis is itself transformed to an initial value problem by the application of a further stretching (oneparameter) group transformation. The resultant initial value problem is solved by a standard forward marching method. Mathematics Subject Classification: 34B40, 35C06, 35G30, 76M55, 76M60

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

An Analysis of Collinear Periodic Crack-Tip Stress Field of an Infinite Plate Subjected to Concentrated Force

The Stress field of collinear periodic cracks in an infinite orthotropic composite plate subjected to the concentrated force is studied. The mechanical problem is turned into partial differential boundary value problem. The solution of partial differential equations can be obtained by the complex function methods, which contains unknown coefficients. Using undetermined coefficients method and with the help of boundary conditions, the partial differential boundary value problem is solved. The expressions for stress and displacements near crack tip are derived. Since the distribution of periodic cracks, the stress intensity factor is related to crack shape. Finally, a numerical example shows that the stress changes with the polar angle.

Stress Field Analysis of Mode II Periodic Cracks-Tip

The cracks-tip field on ModeⅡperiodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to the concentrated force was studied. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. Owing to the distribution of periodic cracks, stress intensity factor(SIF) depends on the shape factor, which is greatly influenced by the crack spacing and the crack length. The results show that interaction happens between the periodic cracks, and that scale effect of the stress intensity factor in cracks-tip is obvious.

Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates

The local averaging technique has become a popular tool in adaptive finite element methods for solving partial differential boundary value problems since it provides efficient a posteriori error estimates by a simple postprocessing. In this paper, the technique is introduced to solve a class of symmetric eigenvalue problems. Its efficiency and reliability are proved by both the theory and numerical experiments structured meshes as well as irregular meshes.

A New Variation of Point Collocation: The Meshless Finite Mixture (MFM) Method

In this paper, a new variation of point collocation with finite mixture approximation, the meshless finite mixture (MFM) method, is developed. Based on the finite mixture theorem, the method can consist of two or more existing meshless techniques so as to exploit their respective merits for the numerical solution of partial differential boundary value (PDBV) problems. In this instance, the classical reproducing kernel particle and differential quadrature techniques are mixed in a point collocation framework. For higher numerical accuracy and stability, the least-square approach is then employed to optimize the weighting coefficients in the construction of the approximation. To validate the presently developed MFM method, several 2-D numerical studies are carried out. The numerical results clearly indicate that the optimal weighting coefficients deployed in the MFM method enhance the numerical accuracy and stability, for various PDBV problems.

Development of a meshless finite mixture (MFM) method

A meshless method with novel variation of point collocation by finite mixture approximation is developed in this paper, termed the meshless finite mixture (MFM) method. It is based on the finite mixture theorem and consists of two or more existing meshless techniques for exploitation of their respective merits for the numerical solution of partial differential boundary value (PDBV) problems. In this representation, the classical reproducing kernel particle and differential quadrature techniques are mixed in a point collocation framework. The least-square method is used to optimize the value of the weight coefficient to construct the final finite mixture approximation with higher accuracy and numerical stability. In order to validate the developed MFM method, several one- and two-dimensional PDBV problems are studied with different mixed boundary conditions. From the numerical results, it is observed that the optimized MFM weight coefficient can improve significantly the numerical stability and accuracy of the newly developed MFM method for the various PDBV problems.

Linear stability analysis for the differentially heated rotating annulus

We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier–Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations. A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis. A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.

The Sturm–Liouville group

In several recent publications Everitt and Markus have developed a theory of ordinary and partial differential boundary value problems, through their novel methods of complex symplectic algebra. For instance, in the case of the regular Sturm–Liouville problem, the corresponding boundary complex symplectic space has four dimensions, and can be used to classify all the self-adjoint boundary conditions by means of two-dimensional Lagrangian subspaces. In this investigation the groups Auto( S), of all symplectic automorphisms, are analyzed for all finite-dimensional complex symplectic spaces S, say of dimension n⩾2; and these are demonstrated to be noncommutative, connected Lie groups of (real) dimension n 2. This global topological analysis provides information on continuous families of canonical coordinates in S, and their deformations.

Conjugate gradient algorithms in the solution of optimization problems for nonlinear elliptic partial differential equations

Several variants of the conjugate gradient algorithm are discussed with emphasis on determining the parameters without performing line searches and on using splitting techniques to accelerate convergence. The splittings used here are related to the nonlinear SSOR algorithm. The behavior of the methods is illustrated on a discretization of a nonlinear elliptic partial differential boundary value problem, the minimal surface equation. A conjugate gradient algorithm with splittings is also developed for constrained minimization with upper and lower bounds on the variables, and the method is applied to the obstacle problem for the minimal surface equation.

Numerical Solutions to Free-Surface Axisymmetric Flows

Methods are developed for obtaining solutions by finite differences to free streamline axisymmetric, potential fluid flows. The formulation of the boundary value problem considers the velocity potential and Stokes stream function as the independent variables and the radial and axial coordinates as the dependent variables. The nonlinear partial differential equations are derived which describe axisymmetric flow in terms of the radial and axial coordinates. Because of the nonlinear nature of the equations, the solution is obtained by using a Newton-Raphson inner iteration, as well as the usual outer iteration customarily used for a finite difference solution of a partial differential boundary value problem of the elliptic type. Two axisymmetric problems have been investigated. The first is of a jet of inviscid incompressible fluid issuing from a nozzle into the free atmosphere. The second is the cavity flow resulting from a jet flowing past a body of revolution.