Transformation Of Partial Differential Equations Research Articles

Soret-Dufour Effects on Heat and Mass Transfer of Newtonian Fluid Flow over the Inclined Sheet and Magnetic Field

Newtonian fluid is ideal for lubrication purposes because the viscosity of this fluid remains as a function of the shear. Besides, the heat and mass transfer are an important study area in fluid dynamics due to its vast applications in industrial processes. The heat-mass transfer can be defined as the Soret and Dufour effect, which implemented in many industrial applications such as in chemical engineering and geosciences field. In addition, the fluid flow over an extending/compressing sheet has significant industrial applications such as the cooling of continuous strips, glass fibre production, the extrusion of plastic sheets from a die, etc. As a response, this study aims to investigate the impacts of Soret and Dufour parameters on the Newtonian fluid flow over an inclined stretching/shrinking sheet. The methodology of this mathematical model are stated as follow: 1) the transformation of partial differential equations (PDEs) to the ordinary differential equations (ODEs), and 2) The ODEs are solved using bvp4c solver in MATLAB software. The bvp4c solver is a MATLAB program directory that solves general form and multi-point boundary layer problems. The main sections of bvp4c are: 1) The solution of the ODEs, 2) The related boundary conditions that can produce the expected results, and 3) An initial guess to run the bvp4c solver. As a result, the numerical and graphical results show that the Soret effect increases the concentration profile whereas decreases the temperature profile. The vice versa occurrence is true for Dufour effect. The convective mass transfer caused by a temperature gradient is known as thermal-diffusion (Soret) effect. The convective heat transfer produced by concentration differences is known as diffusion-thermo (Dufour) effect. However, since the process of heat and mass transfers are related to each other, the Soret and Dufour effects are able to influence both of this process simultaneously. In conclusion, the convective heat transfer is enhanced by increasing Soret and Dufour number while the convective mass transfer is declined by increasing the two numbers.

Numerical Computation of Stagnation Point Flow and Heat Transfer over a Nonlinear Stretching/Shrinking Sheet in Hybrid Nanofluid with Suction/Injection Effects

The steady, laminar, stagnation point flow of hybrid nanofluid past a nonlinearly stretching and shrinking sheet is studied. Hybrid nanofluid is regarded by disseminated two distinct nano-sized particles, silver (Ag) and copper oxide (CuO) in pure water. Similarity technique was used for the transformation of partial differential equations (PDEs) into an ordinary differential equation (ODEs). Obtained ODEs were solved using Matlab’s built-in function (bvp4c). The results of important governing parameters which are nonlinear parameter, suction/injection parameter, stretching/shrinking parameter and nanoparticle volume fraction are evaluated and discussed in graphical and tabular form for the velocity and temperature profiles, along with local skin friction, local Nusselt number. Nonunique solutions (first and second branch) are visible for some limit of shrinking parameter. It is noticed that suction parameter delays the boundary layer hastens flow separations.

Influence of induced magnetic field and surface roughness of Casson nanofluid flow over an exponentially stretching sheet

In the present analysis, we have discussed the mixed convective of flow over an exponential sheet in presence of induced magnetic field and surface roughness. The effects of hall current and chemical reaction on flow are also considered. The non-Newtonian model behavior is characterized by Casson nanofluid. The similarity transformations are employed for the transformation of partial differential equations into non-dimensional form and obtained equations are solved by employing the Differential transformation method (DTM). Further, the importance of various parameters on various profiles and gradients are explored graphically with some suitable discussion. At the wall, the small (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}) and frequency (n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n$$\\end{document}) parameters influence on gradients are illustrated.

Thermal Radiation and Chemical Reaction Effects of Unsteady Magnetohydrodynamic Dissipative Squeezing Flow of Casson Nanofluid Over Horizontal Channel

This work investigates the effects of chemical reaction and thermal radiation on the magnetohydrodynamic (MHD) squeezing Casson fluid flow through a porous media under the influence of heat absorption, which is motivated by the application of flow in industrial issues. The flow is generated by the compression of two plates together. The transformation of partial differential equations (PDEs) to ordinary differential equations (ODEs) is accomplished via similarity variables. The numerical analysis was carried out using the shooting method, which involved solving the competent controlling equations with dominant parameters for a thin liquid film. It is necessary to compare the numerical findings with previous research to verify the present solutions. The findings reveal that the velocity and wall shear stress increase as the distance between the two plates becomes closer. The higher values of squeezing number diminish the squeezing force on the fluid flow, which in turn reduces the thermal field. The thinner temperature boundary layer corresponds to the lower values of thermal diffusivity and it shows the higher values of temperature gradient for the increasing values of β. Due to the presence of stronger Lorentz forces the temperature and velocity fields behave like decreasing functions of Hartmann number.

A numerical study of the micropolar nanofluid flow containing aluminum alloy nanoparticles over a variable thickened stretching sheet

Aluminum alloys are used to make wheels that are suitable for aeroplanes and automobiles, as well as all types of ground vehicles and watercraft. Aluminum alloys are made through melting, sintering (assembly of formed parts utilizing metal particles that melt together at intense temperatures), or mechanical braiding. Aluminum alloys have had a major impact on aeroplane manufacturing. Aluminum alloys like AA7075 and AA7072 are especially useful in transportation applications including maritime, aviation, and automotive, and are also utilized in the construction of bicycles, glider rock climbing equipment, and planes. This attempt sheds light on the magnetically influenced methanol-based micropolar nanofluid containing aluminum alloy nanoparticles (AA7075) over a variable thickened stretching sheet. A variable magnetic field is applied normal to the flow direction. The flow is considered at a stagnation point. Also, the Joule heating impact is considered in this analysis. The similarity transformations are used for the transformation of partial differential equations into ordinary differential equation. The authors have chosen to solve the proposed model with the help of NDSolve technique which can handle a wide range of ordinary and partial differential equations (ODEs and PDEs). The results showed that, as the volume fraction of AA7075 nanoparticles grows the velocity profile of the AA7075–methanol nanofluid decreases, while the microrotation and temperature profiles of the AA7075–methanol nanofluid increases. The velocity profile of the AA7075–methanol nanofluid reduces, while the microrotation and temperature profiles of the AA7075–methanol nanofluid increase with the increasing magnetic parameter. The growing micropolar parameter augments the velocity and temperature profiles of the AA7075–methanol nanofluid, whereas a dual impact of the micropolar parameter is found against the microrotation profile of the AA7075–methanol nanofluid. The growing variable wall thickness factor augments the velocity, microrotation and temperature profiles of the AA7075–methanol nanofluid. It is found that the embedded factors highly affected the AA7075–methanol nanofluid as compared to methanol fluid.

A numerical study of the micropolar nanofluid flow containing aluminum alloy nanoparticles over a variable thickened stretching sheet

Aluminum alloys are used to make wheels that are suitable for aeroplanes and automobiles, as well as all types of ground vehicles and watercraft. Aluminum alloys are made through melting, sintering (assembly of formed parts utilizing metal particles that melt together at intense temperatures), or mechanical braiding. Aluminum alloys have had a major impact on aeroplane manufacturing. Aluminum alloys like AA7075 and AA7072 are especially useful in transportation applications including maritime, aviation, and automotive, and are also utilized in the construction of bicycles, glider rock climbing equipment, and planes. This attempt sheds light on the magnetically influenced methanol-based micropolar nanofluid containing aluminum alloy nanoparticles (AA7075) over a variable thickened stretching sheet. A variable magnetic field is applied normal to the flow direction. The flow is considered at a stagnation point. Also, the Joule heating impact is considered in this analysis. The similarity transformations are used for the transformation of partial differential equations into ordinary differential equation. The authors have chosen to solve the proposed model with the help of NDSolve technique which can handle a wide range of ordinary and partial differential equations (ODEs and PDEs). The results showed that, as the volume fraction of AA7075 nanoparticles grows the velocity profile of the AA7075–methanol nanofluid decreases, while the microrotation and temperature profiles of the AA7075–methanol nanofluid increases. The velocity profile of the AA7075–methanol nanofluid reduces, while the microrotation and temperature profiles of the AA7075–methanol nanofluid increase with the increasing magnetic parameter. The growing micropolar parameter augments the velocity and temperature profiles of the AA7075–methanol nanofluid, whereas a dual impact of the micropolar parameter is found against the microrotation profile of the AA7075–methanol nanofluid. The growing variable wall thickness factor augments the velocity, microrotation and temperature profiles of the AA7075–methanol nanofluid. It is found that the embedded factors highly affected the AA7075–methanol nanofluid as compared to methanol fluid.

Soret and Dufour influences on forced convection of Cross radiative nanofluid flowing via a thin movable needle

The main feature of the current investigation is to analyze the magnetohydrodynamic mixed convection flow of Cross fluid. Flow is due to a movable thin needle with Soret and Dufour effect. Heat generation/absorption and nonlinear heat radiation are used in the energy equation. Characteristics of the chemical reaction and thermal activation are given special attention. Appropriate variables are introduced for the transformation of partial differential equations to ordinary differential equations. With the assistance of Runge–Kutta Fehlberg's fourth- fifth-order method with the shooting technique, we determined the prominent result numerically. The prominent examined parameters range is velocity and temperature ratios, heat generation, Dufour, Hartmann, Schmidt numbers (0.1le{{lambda}},{{{theta}}}_{{{w}}},{{Q}},{{{D}}}_{{{u}}},boldsymbol{ }{{M}},{{S}}{{c}}le 0.7), needle thickness (0le {{a}}le 15), radiative parameter (5le {{R}}{{d}}le 8), and Weissenberg number (0.01le {{W}}{{e}}le 0.09), respectively. Graphs for velocity, thermal, concentration, Skin friction coefficient, and heat and mass transport rates are displayed and analyzed for physical parameters. A similar observation of mixed convection and needle thickness parameter is seen on the velocity field. Temperature and heat transfer rate are reverse behavior in the frame of the Dufour effect. Moreover, an enhancement in chemical reaction shows decay to the concentration field.

Flow Reversal of Fully Developed Combined Convection in a Vertical Channel with Boundary Condition of a Third Kind

The heat transfer process of combined convection and its flow pattern in a vertical channel is important, especially in environmental, industrial and engineering applications. There has been some concern regarding the heat transfer process in which it is difficult, expensive and time consuming. There are three main objectives to satisfy the purpose of this study which are to determine the impact on flow and heat transfer of the Robin temperature boundary condition, to verify whether the viscous dissipation plays a classical role in fluid flow and heat transfer and lastly to compare the degree of heat transfer facilitated by the boundary conditions of Dirichlet, Neumann and Robin. In this study, the effects of different dimensional parameters were tested and the flow reversal phenomenon was discussed. The Boundary Value Problem (BVP) was solved numerically using Maple using a built-in routine, dsolve. Validation study to the previously published problem was carried out to verify the accuracy of the present computation. The transformation of partial differential equation to ordinary differential equation involved similarity technique. The numerical results of the flow and temperature variables were presented graphically. The flow reversal occurred when the value of internal heat generation (G) and combined convection parameter (λ) were large. There was no occurrence of flow reversal when the values of local heating exponent (p) and the Biot Numbers (Bi) increase. However, the value of mass transfer coefficient (N) was different, gave no significant impact.

Numerical Solution of Stagnation Point Flow and Heat Transfer over a Nonlinear Stretching/Shrinking Sheet in Hybrid Nanofluid: Stability Analysis

The steady, laminar, stagnation point flow of hybrid nanofluid past a nonlinearly stretching and shrinking sheet is studied. Hybrid nanofluid is regarded by disseminated two distinct nano-sized particles, silver (Ag) and copper oxide (CuO) in pure water. Similarity technique was used for the transformation of partial differential equations (PDEs) into an ordinary differential equations (ODEs). Obtained ODEs were solved using Matlab’s built in function (bvp4c). The results of important governing parameters which are nonlinear parameter, stretching/shrinking parameter and nanoparticle volume fraction are evaluated and discussed in graphical and tabular form for the velocity and temperature profiles, along with local skin friction, local Nusselt number. Nonunique solutions (first and second branch) are visible for some limit of shrinking parameter. It is noticed that nonlinear parameter hastens flow separations. Hence, a stability analysis is executed to identify which solutions are stable and physically feasible.

Numerical treatment of magneto Carreau nanofluid over a stretching sheet considering Joule heating impact and nonlinear thermal ray

AbstractIn this essay, the magnetohydrodynamic flow of a Carreau nanoliquid upon a radiative stretching plate has been reviewed. The impacts of Joule heating and thermal ray are considered. The thermophoresis phenomenon and Brownian motion are applied to model nanoparticles (Buongiorno's model). Governing equations are solved numerically using Runge‐Kutta‐Fehlberg 4.5 after the transformation of partial differential equations into ordinary differential equations. In the obtained outcomes of investigating the impacts of different parameters on the change in velocity, concentration, and temperature profiles for two cases of shear‐thinning liquid and shear thickening liquid are reported as diagrams. Also, in the final segment of this essay, the impacts of diverse parameters on the skin friction coefficient and the local Nusselt number are investigated. The novel findings of current research illustrate that the values of local Nusselt number and surface drag force for shear thickening liquid are higher than shear‐thinning liquid. Also, the temperature profile has direct relationships with thermal radiation and magnetic field.

MHD boundary layer of GO–H2O nanoliquid flow upon stretching plate with considering nonlinear thermal ray and Joule heating effect

AbstractThis essay investigates a steady three‐dimensional laminar boundary layer flow of magnetohydromagnetic radiative of graphene oxide‐water nanofluid over an extensible surface in the attendance of couple stress, thermal ray, and Joule heating impact. Governing equations are solved numerically using the Runge‐Kutta‐Fehlberg 4.5 approach after the transformation of partial differential equations into ordinary differential equations. The main goal of this essay is to check the impacts of variations in the value of numerous parameters on the velocity along x and y‐axis directions () and temperature () profiles, and also on the local skin friction coefficient along x and y‐axis directions in the presence of couple stress (K > 0) and the lack of couple stress (K = 0) and local Nusselt number for the two modes of nonlinear () and linear () thermal ray. The results elucidate that the local Nusselt number for both and modes has a direct connection with radiation parameter (R) and Prandtl number (Pr) and an inverse relation with Eckert number (Ec) and Hartman number (Ha). In addition, the skin friction coefficient has an inverse relation with couple stress (K).

Three dimensional flow and mass transfer analysis of a second grade fluid in a porous channel with a lower stretching wall

This investigation analyses a three dimensional flow and mass transfer of a second grade fluid over a porous stretching wall in the presence of suction or injection. The equations governing the flow are attained in terms of partial differential equations. A similarity transformation has been utilized for the transformation of partial differential equations into the ordinary differential equations. The solutions of the nonlinear systems are given by the homotopy analysis method (HAM). A comparative study with the previous results of a viscous fluid has been made. The convergence of the series solution has also been considered explicitly. The influence of admissible parameters on the flows is delineated through graphs and appropriate results are presented. In addition, it is found that instantaneous suction and injection reduce viscous drag on the stretching sheet. It is also shown that suction or injection of a fluid through the surface is an example of mass transfer and it can change the flow field.

Analytical model for solidification and melting in a finite PCM in steady periodic regime

An analytical solution of the phase change problem, known as the Stefan or Moving Boundary Problem, in a PCM layer (phase change materials) subject to boundary conditions that are variable in time, is presented, in steady periodic regime. The two-phase Stefan Problem is resolved considering periodic boundary conditions of temperature or of heat flux, or even mixed conditions.This phenomenon is present in air-conditioned buildings, the walls of which use PCM layers to reduce thermal loads and energy requirements to be compensated by the plant.The resolution method used is one in which phasors allow the transformation of partial differential equations, describing conduction in the solid and liquid phase, into ordinary differential equations; furthermore the phasors allow transformation of the thermal balance equation at the bi-phase interface into algebraic equations. The Moving Boundary Problem is then reduced to a system of algebraic equations, the solution of which provides the position in time of the bi-phase interface and the thermal field of the layer. The solution obtained provides for different thermodynamic configurations that the layer can assume and makes the position of the bi-phase interface and the thermal field depend on the Fourier number and on the Stefan number calculated in the solid phase and in the liquid phase.In the case of two boundary conditions represented by a single sinusoidal oscillation, a general analysis, addressed in different thermodynamic configurations obtained by varying the Fourier and Stefan number, shows the calculation procedure of the steady and of the oscillating component of the position of the bi-phase interface, of the temperature field and of the heat flux field.In addition, we considered the particular case of a PCM layer with an oscillating temperature boundary condition on one face and a constant temperature on the other face.The analytical procedure was also used for an analysis dedicated to the thermal behaviour of Glauber’s salt subject to independent multi harmonic boundary conditions. This salt hydrate is one of the most studied, having a high latent fusion heat and a melting temperature that is suited for use in the walls of buildings.

Computer facilitated generalized coordinate transformations of partial differential equations with engineering applications

AbstractPartial differential equations (PDEs) play an important role in describing many physical, industrial, and biological processes. Their solutions could be considerably facilitated by using appropriate coordinate transformations. There are many coordinate systems besides the well‐known Cartesian, polar, and spherical coordinates. In this article, we illustrate how to make such transformations using Maple. Such a use has the advantage of easing the manipulation and derivation of analytical expressions. We illustrate this by considering a number of engineering problems governed by PDEs in different coordinate systems such as the bipolar, elliptic cylindrical, and prolate spheroidal. In our opinion, the use of Maple or similar computer algebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommon transformations more frequently at the very least for situations where the transformations provide smarter and easier solutions. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 19: 365–376, 2011

Linearization of Mirror Systems

We demonstrate, through the fourth Painlevé and the modified KdV equations, that the attempt at linearizing the mirror systems (more precisely, the equation satisfied by the new variable θ introduced in the indicial normalization) near movable poles can naturally lead to the Schlesinger transformations of ordinary differential equations or to the Bäcklund transformations of partial differential equations.

TREE METHOD FOR OPTION PRICING UNDER STOCHASTIC VARIANCE

We develop a recombining tree method for pricing of options by using a general two-factor stochastic-variance (SV) diffusion model for asset price dynamics. We show that it is possible to construct a riskless hedge by including additional short-term options in the hedging portfolio. This procedure gives us Partial Differential Equation (PDE) that can be solved by using standard numerical techniques giving us a unique option's price. We show that the option's price does not depend on the long run volatility forecast but only on the parameters of the model, which are related to the volatility of variance. We show one particular transformation of PDE to the Finite Difference Equation (FDE) that leads to the three-dimensional lattice method similar to the standard binomial-tree method. Our tree grows in the price-variance space and similarly to the binomial-tree, the coefficients of the FDE can be interpreted as risk neutral probabilities for jumping between the tree nodes. By investigating an error accumulation in the tree we found the stability criterion and the method that can be applied in order to achieve a stabile procedure. Our option pricing method can be used for European and American options and various payoffs. Since the procedure converges quickly and can be easily implemented, we believe that it could be useful for practitioners. The SV model we used here was shown earlier to be a diffusion limit of various GARCH-type models giving the possibility of using parameters obtained in the discrete-time GARCH framework as an input for our option pricing method.

Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. II. Tzitzéica equation

From the existing methods of singularity analysis only, we derive the two equations which define the Bäcklund transformation of the Tzitzéica equation. This is achieved by defining a truncation in the spirit of the approach of Weiss et al., so as to preserve the Lorentz invariance of the Tzitzéica equation. If one assumes a third-order scattering problem, this truncation admits a unique solution, thus leading to a matrix Lax pair and a Darboux transformation. In order to obtain the Bäcklund transformation (BT), which is the main new result of this paper, one represents the Lax pair by an equivalent two-component Riccati pseudopotential. This yields two different BTs; the first one is a BT for the Hirota–Satsuma equation, while the second one is a BT for the Tzitzéica equation. One of the two equations defining the BT is the fifth ordinary differential equation of Gambier.

Bäcklund transformation of partial differential equations from the Painlevé–Gambier classification. I. Kaup–Kupershmidt equation

Among the 50 nonlinear second-order differential equations of Painlevé and Gambier, those which are linearizable provide a natural scheme for deriving the Lax pair and the Darboux transformation of a nonlinear partial differential equation when the order of the scattering problem is three. This new method allows one to obtain, from singularity analysis only, the Bäcklund transformation of the Kaup–Kupershmidt equation, a result which was missing until now.

On form-preserving point transformations of partial differential equations

New identities are presented relating arbitrary order partial derivatives of and for the general point transformation , . These identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 + 1 partial differential equations. These results facilitate the search for point symmetries, both discrete and continuous (Lie), and assist the search for point transformations which reduce equations to canonical, but similar, form. A simple test for the existence of hodograph-type transformations between equations of similar form is given.

Dynamic mathematical model of deep bed filtration process

Deep bed filtration is commonly applied in clarification of dilute suspensions of particles ranging in size from about 0.1 to 50 μm. A suspension carrying solid particles of different sizes is passed through the porous bed of defined geometrical characteristics. It has been found that sizes of suspended particles and their distribution are very important physical parameters that influence deep bed filter efficiency. During the filtration process the bed porosity decreases, whereas interficial velocity increases due to the particle accumulation in filter bed. Mathematical model has been developed under the assumption that the plug flow model approximates flow of suspension through the bed. The second assumption is that a deposition kinetics is a function of local suspension's particle distribution and locally deposited particle distribution. In order to obtain the experimental data needed for determination of the process kinetics, a set of experiments has been carried out. So obtained experimental data consisted of the local suspension and deposit particle distribution values and also of the local rate values. All three empirical distributions are approximated by standard Logarithm-Normal distribution function. Each distribution is defined by two LN function parameters. Rate distribution parameters are formally dependent on the parameters that define suspension and deposit distribution. This relation has been established using the general regression neural network (GRNN). Thus defined model enables solving the system within given boundary conditions by approximating distribution functions with sums and using orthogonal collocation method for transformation of partial differential equations into a system of ordinary differential equations. Developed method can be applied in process simulation as long as the input concentration and distribution are within the range of experimental values for the kinetics determination. To test the developed method, experiments were conducted on the pilot scale deep bed filter having total height of 1m and diameter of 0.1m. The results show that a very complex process, as is deep bed filtration, can be successfully described using hybrid neural network.