Conversion Of Partial Differential Equations Research Articles

Analysis of blood flow of unsteady Carreau-Yasuda nanofluid with viscous dissipation and chemical reaction under variable magnetic field

Blood flow analysis through arterial walls depicts unsteady non-Newtonian fluid flow behavior. Arterial walls are impacted by various chemical reactions and magnetohydrodynamic effects during treatment of malign and tumors, cancers, drug targeting and endoscopy. In this regard, current manuscript focuses on modeling and analysis of unsteady non-Newtonian Carreau-Yasuda fluid with chemical reaction, Brownian motion and thermophoresis under variable magnetic field. The main objective is to simulate the effect of different fluid parameters, especially variable magnetic field, chemical reaction and viscous dissipation on the blood flow to help medical practitioners in predicting the changes in blood to make diagnosis and treatment more efficient. Suitable similarity transformations are used for the conversion of partial differential equations into a coupled system of ordinary differential equations. Homotopy analysis method is used to solve the system and convergent results are drawn. Effect of different dimensionless parameters on the velocity, temperature and concentration profiles of blood flow are analyzed in shear thinning and thickening cases graphically. Analysis reveals that chemical reaction increases blood concentration which enhance the drug transportation. It is also observed that magnetic field elevates the blood flow in shear thinning and thickening scenarios. Furthermore, Brownian motion and thermophoresis increases temperature profile.

Mechanical characteristics of MHD of the non-Newtonian magnetohydrodynamic Maxwell fluid flow past a bi-directional convectively heated surface with mass flux conditions

In engineering and manufacturing industries, stretching flow phenomena have numerous real-world implementations. Real-world applications related to stretched flow models are metalworking, crystal growth processes, cooling of fibers, and plastics sheets. Therefore, in this work, the mechanical characteristics of the magnetohydrodynamics of the non-Newtonian Maxwell nanofluid flow through a bi-directional linearly stretching surface are explored. Brownian motion, thermophoresis, and chemical reaction impacts are considered in this analysis. Additionally, thermal convective and mass flux conditions are taken into consideration. The mathematical framework of the existing problem is constructed on highly non-linear partial differential equations (PDEs). Suitable similarity transformations are used for the conversion of partial differential equations into ordinary differential equations (ODEs). The flow problem is tackled with the homotopy analysis method, which is capable of solving higher-order non-linear differential equations. Different flow profiles against various flow parameters are discussed physically. Heat and mass transference mechanisms for distinct flow factors are analyzed in a tabular form. The outcomes showed that both primary and secondary velocities are the declining functions of magnetic and Maxwell fluid parameters. The heat transfer rate rises with the cumulative values of the Brownian motion and thermal Biot number. In addition, the mass transfer rate decreases with the rising Schmidt number, Brownian motion parameter, and chemical reaction parameter, while it increases with the augmenting thermophoresis parameter. It has been highlighted that streamlines in the current work for Maxwell and Newtonian models are in fact different from one another.

Three Dimensional Modelling of Magnetohydrodynamic Bio-Convective Casson Nanofluid Flow with Buoyancy Effects Over Exponential Stretching Sheet Along with Heat Source & Gyrotactic Micro-Organisms

In current analysis, influence of buoyancy forces in MHD bioconvective non-Newtonian nanofluids over three dimensional exponential sheet has been studied numerically. Additionally, impact of heat source parameter along with convective conditions has been incorporated. Moreover, nanofluid flow with gyrotactic microorganisms has been analyzed in presence of chemical reaction. Initially similarity variables are used for the conversion of partial differential equations into highly non-linear differential equations. Thusly, non-linear behavior of equations makes typical solutions which are not solved analytically. So, computational MATLAB software is used to calculate results and graphs by following shooting algorithm with Runge Kutta Fehlberg technique using ODE45 solver. Present modeling investigates the influence of crucial fluid parameters especially; magnetic M (0.1 ≤ M ≤ 0.9), Casson parameter β (0.09 ≤ β ≤ 0.13), permeability parameter Bθ, Brownian motion Nb (0.5 ≤ Nb ≤ 5.0), thermophoresis Nt (0.2 ≤ Nt ≤ 2.0), thermal Biot number Bt (0.3 ≤ Bt ≤ 0.7), heat generation parameter Q (0.1 ≤ Q ≤ 0.5), Prandtl number Pr (0.1 ≤ Pr ≤ 0.9), concentration Biot number Bc (0.1 ≤ Bc ≤ 0.9), Lewis number Le (1 ≤ Le ≤ 5), chemical reaction parameter Ch (0.1 ≤ Ch ≤ 0.9), bioconvective Lewis number Lb (0.1 ≤ Lb ≤ 2), Peclet number Pe (0.1 ≤ Pe ≤ 5), gyrotactic Biot number Bn (0.1 ≤ Bn ≤ 0.5), stretching ratio parameter c (0.1 ≤ c ≤ 0.3) and microorganism concentration difference parameter Ω (0.1 ≤ Ω ≤ 5.0). Final results are compared for Prandtl number and stretching ratio parameter along with residual errors. It is inferred that motile concentration declines for larger bioconvective Lewis number whereas rises for motile gyrotactic microorganism Biot number.

Soret and Dufour effects on MHD squeezing flow of Jeffrey fluid in horizontal channel with thermal radiation.

The fluid flow with chemical reaction is one of well-known research areas in the field of computational fluid dynamic. It is potentially useful in the modelling of flow on a nuclear reactor. Motivated by the implementation of the flow in the industrial application, the aim of this study is to explore the time-dependent squeeze flow of magnetohydrodynamic Jeffrey fluid over permeable medium in the influences of Soret and Dufour, heat source/sink and chemical reaction. The presence of joule heating, joule dissipation and radiative heat transfer are analyzed. The flow is induced due to compress of two surfaces. Conversion of partial differential equations (PDEs) into ordinary differential equations (ODEs) is accomplished by imposing similarity variables. Then, the governing equations are resolved using Keller-box approach. The present outcomes are compared with previously outcomes in the literature to validate the precision of present outcomes. Both outcomes are shown in close agreement. The tabular and graphical results demonstrate that wall shear stress and velocity profile accelerate with the surfaces moving towards one another. Moreover, the concentration, temperature and velocity profiles decreasing for the increment of Hartmann numbers and Jeffrey fluid parameters. The impacts of heat generation/absorption, joule dissipation and Dufour numbers enhance the heat transfer rate and temperature profile. In contrast, the temperature profile drops and the heat transfer rate boosts when thermal radiation increases. The concentration profile decelerates, and the mass transfer rate elevates with raise in Soret number. Also, the mass transfer rate rises for destructive chemical reaction and contrary result is noted for convective chemical reaction.

Shifted Legendre Collocation Method for the Solution of Unsteady Viscous-Ohmic Dissipative Hybrid Ferrofluid Flow over a Cylinder.

A numerical treatment for the unsteady viscous-Ohmic dissipative flow of hybrid ferrofluid over a contracting cylinder is provided in this study. The hybrid ferrofluid was prepared by mixing a 50% water (HO) + 50% ethylene glycol base fluid with a hybrid combination of magnetite (FeO and cobalt ferrite (CoFeO ferroparticles. Suitable parameters were considered for the conversion of partial differential equations (PDEs) into ordinary differential equations (ODEs). The numerical solutions were established by expanding the unknowns and employing the truncated series of shifted Legendre polynomials. We begin by collocating the transformed ODEs by setting the collocation points. These collocated equations yield a system of algebraic equations containing shifted Legendre coefficients, which can be obtained by solving this system of equations. The effect of the various influencing parameters on the velocity and temperature flow profiles were plotted graphically and discussed in detail. The effects of the parameters on the skin friction coefficient and heat transfer rates were further presented. From the discussion, we come to the understanding that Eckert number considerably decreases both the skin friction coefficient and the heat transfer rate.

Interaction of micropolar fluid structure with the porous media in the flow due to a rotating cone

A model of heat and mass transfer of two dimensional MHD micropolar fluid over a cone is constructed. Similarity transformation is adopted for the conversion of partial differential equations into ordinary differential equations have been linearized by employing the Newton’s linearization technique and then new sets of equations are discretized using the finite difference method. The impact of non-dimensional parameters is further analyzed and the numerical results for profiles of velocity, temperature and concentration are expressed graphically and the results are discussed in detail. For the higher values of Dufour number, temperature field is enhanced graphically but show the opposite behavior in case of Prandlt number. Furthermore, the micro-rotation parameter show decreasing behavior with the enhancing of magnetic and porosity parameters respectively.

Melting Heat Transfer in Thermally Stratified Magnetohydrodynamic Flow of Eyring-Powell Fluid with Homogeneous-heterogeneous Reaction

Features of melting heat transfer as well as homogeneous-heterogeneous reaction in thermally stratified stagnation flow of Eyring-Powell fluid along with heat generation/absorption effects are explored in this article. Fluid flow is examined over a sheet of variable thickness in the presence of stretching phenomena. Variable strength of magnetic field is considered normal to the flow field. Suitable transformations are introduced for the sake of conversion of partial differential equations to ordinary differential equations. Homotopic method is utilized to tackle highly nonlinear problem and series solutions are attained. Behaviors of relevant parameters are portrayed for velocity, thermal and concentration distributions. Graphical results reveal that the concentration profile enhances for higher Schmidt number while it exhibits recessive behavior for increment in homogeneous reaction parameter. Larger values of heterogeneous reaction parameter result in intensified concentration field. Velocity field declines as a result of increment in fluid parameters ε as well as δ*.

Modern aspects of homogeneous-heterogeneous reactions and variable thickness in nanofluids through carbon nanotubes

This article examines homogeneous-heterogeneous reactions and internal heat generation in Darcy-Forchheimer flow of nanofluids with different base fluids. Flow is generated due to a nonlinear stretchable surface of variable thickness. The characteristics of nanofluid are explored using CNTs (single and multi walled carbon nanotubes). Equal diffusion coefficients are considered for both reactants and auto catalyst. The conversion of partial differential equations (PDEs) to ordinary differential equations (ODEs) is done via appropriate transformations. Optimal homotopy approach is implemented for solutions development of governing problems. Averaged square residual errors are computed. The optimal solution expressions of velocity, temperature and concentration are explored through plots by using several values of physical parameters. Further the coefficient of skin friction and local Nusselt number are examined through graphs.

Magnetohydrodynamic three-dimensional nonlinear convective flow of viscoelastic nanofluid with heat and mass flux conditions

The present research focuses on three-dimensional nonlinear convective flow of viscoelastic nanofluid. Here, the flow is generated due to stretching of a impermeable surface. The phenomenon of heat transport is analyzed by considering thermal radiation and prescribed heat flux condition. Nanofluid model comprises of Brownian motion and thermophoresis. An electrically conducting fluid is accounted due to consideration of an applied magnetic field. The dimensionless variables are introduced for the conversion of partial differential equations into sets of ordinary differential systems. The transformed expressions are explored through homotopic algorithm. Behavior of different dimensionless parameters on the non-dimensional velocities, temperature and concentration are scrutinized graphically. The values of skin friction coefficients, Nusselt and Sherwood numbers are also calculated and elaborated. It is visualized that the heat transfer rate increases with Prandtl number and radiation parameter is higher.

In Russian

In this paper heterogeneous subjects are investigated, namely laminated nanoparticles and flat-layered media. Common feature of objects under consideration is presence of surfaces divided into slabs with different physical characteristics. The aim of the work is to demonstrate large possibilities and certain preferences of transfer matrices method for solution different problems of applied physics and generalization of the results taken out by author early. The method is especially convenient for analysis of such systems under action of different external physical fields. The method makes possible to transfer boundary conditions from slab to slab, to avoid the necessity of consideration of high order algebraical systems and matrices inversion problem is reduced to theirs simple multiplication. We presented a general scheme of building transfer matrices based on conversion of partial differential equations to ordinary differential equations with using expansion by a system of orthogonal functions. In addition to known results, the transfer matrices for inhomogeneous differential equations are constructed, i.e. those for physical systems with internal energy sources. Transfer matrix method has been used to solve some applied problems, especially problems of electromagnetic scattering on ellipsoidal nanoparticles (electrostatic approximation), spherical particles (electrostatic and wave approximations), expressions for calculations of electrical energy in slabs are found. Solution of Eigen value problems for multilayer panels is constructed, design formulas for determination of oscillation Eigen frequencies of multilayer panel with using three-dimensional elasticity theory are also obtained. Design formulas and some typical artwork are presented.

Axisymmetric magnetohydrodynamic flow of micropolar fluid between unsteady stretching surfaces

This investigation examines the time dependent magnetohydrodynamic (MHD) flow problem of a micropolar fluid between two radially stretching sheets. Both strong and weak concentrations of microelements are taken into account. Suitable transformations are employed for the conversion of partial differential equations into ordinary differential equations. Solutions to the resulting problems are developed with a homotopy analysis method (HAM). The angular velocity, skin friction coefficient, and wall couple stress coefficient are illustrated for various parameters.

On the Conversion of Partial Differential Equations

This paper outlines the conversion of partial differential equations (PDEs) into the corresponding ordinary differential equations (ODEs) by a complex transformation which is widely used in the exp-function method. The proposed homotopy perturbation method (HPM) is employed to solve the travelling wave solutions. Several examples are given to reveal the reliability and efficiency of the algorithm.