System Of Nonlinear ODEs Research Articles

Melting heat transfer and entropy optimization owing to carbon nanotubes suspended Casson nanoliquid flow past a swirling cylinder-A numerical treatment

The astounding physiognomies of carbon nanotubes such as lightweight, mechanical stability, excellent thermal and electrical conductivities, and physicochemical compatibility make them perfect material for electrochemical gadgets. Having such amazing characteristics in mind our intention is to provide a numerical solution of Casson nanofluid flow containing melting heat transfer past a swirling cylinder. Analysis of multi-wall carbon nanotubes (MWCNTs) with water as a base fluid is performed. Effects of entropy generation, magnetohydrodynamics, and heat generation/absorption are also contemplated. The proposed physical model is erected and through feasible transformations, the obtained coupled nonlinear system of ODEs is solved via bvp4c function of MATLAB software numerically. The impacts of Casson nanofluid comprises of MWCNTs on skin friction, entropy generation coefficient and heat transfer rate are analyzed. Graphs of physical parameters of interest, Casson parameter (0≤β≤2), magnetic parameter (0≤M≤6), nanoparticle volume fraction (0≤ϕ≤0.3), melting parameter (0.1≤M*≤2.0), dimensionless temperature difference (0≤α≤0.3), Brinkman number (0.7≤Br≤2.0), versus swirling velocity, temperature and entropy optimization are also illustrated and discussed in detail. Tabulated numerical values of skin friction and heat transfer rate with requisite discussion are also a part this analysis. It is finally summarized that entropy optimization is an escalating function of magnetic parameter.

Flow of Eyring-Powell dusty fluid in a deferment of aluminum and ferrous oxide nanoparticles with Cattaneo-Christov heat flux

This is a speculative investigation of the magnetohydrodynamic flow of Eyring-Powell liquid under suspension of nano-particles and dust. A simulation is executed by fraternization of Ferrous oxide (Fe3O4) and aluminum oxide (Al2O3) nanoparticles in Eyring-Powell dusty fluid. Dispersion of ferrous oxide (Fe3O4) and aluminum oxide (Al2O3) nano-particles in dusty fluid have applications in heat exchanger system, biocompatibility, biosensors, nuclear reactor heating process, detection and cancer treatment, in monitoring stem cells differentiation etc. Ferrous oxide (Fe3O4) and aluminum oxide (Al2O3) mixtures are substantially useful in optimizing the heat transport occurrences. Implementation of similarity variables leads to the systems of ordinary differential expressions. These nonlinear systems of ODEs are tackled with the use of Runge-Kutta-Fehlberg Scheme (RKFS). The analysis of dimensionless temperature and velocity fields is given via plots. The numerical benchmarks of friction-factors and heat transport rate are for different constraints are given and examined. Obtained results are matched with previously published material and noted to be satisfactory. This model expresses that the rate of heat transportation is more in aluminum oxide nanofluid compared to ferrous oxide nanofluid with existing of viscous variation parameter. The presence of thermal and momentum slips correspond the enhancement in local Nusselt number in case of ferrous oxide nanoparticles when compared to aluminum oxide nanoparticles.

Significance of static–moving wedge for unsteady Falkner–Skan forced convective flow of MHD cross fluid

The purpose of current communication is to analyze the characteristics of static–moving wedge for unsteady 2D Falkner–Skan flow of cross fluid in the presence of nonlinear thermal radiation. Furthermore, revised Buongiorno’s relation of nanomaterials is implemented for mathematical modeling of Falkner–Skan flow of cross fluid. Nanofluid characteristics for Brownian movement and thermophoresis are deliberated in this communication. In the considered model, we have utilized the aspects of constructive–destructive phenomenon to elaborate the mechanism of mass transfer. The transformed nonlinear system of ODEs are computed numerically by implementing bvp4c scheme. Physics related to the assumed flow model is described graphically. Furthermore, the graphical analysis revealed that the considered physical model has a significant impact on the physical parameter involved in this problem. It is anticipated from graphical data that declining conduct is observed with the impact of chemical reaction parameter temperature profile for n = 0.5, for static–moving wedge.

Aspects of improved heat conduction relation and chemical processes in 3D Carreau fluid flow

This article communicates the numerical consideration of 3D Carreau liquid flow under the impact of chemical responses over a stretched surface. Moreover, the heat transfer exploration is carried out with a view to improve the heat flux relation. This phenomenon is established upon the theory of Cattaneo–Christov heat flux relation that contributes by the thermal relaxation. On exploitation of an appropriate transformation a system of nonlinear ODEs is attained and then elucidated numerically by means of bvp4c scheme. The descriptions of temperature and concentration fields equivalent to the frequent somatic parameters are graphically scrutinised. Our analysis carries that the concentration of the Carreau liquid displays similar tendency and decline as the heterogeneous–homogeneous reaction parameters $$(k_2 , k_1 )$$ augment. Furthermore, it is notable that for shear thinning $$(n<1)$$ liquid, the influence of local Weissenberg numbers $$(\mathrm{We}_1 ,\mathrm{We}_2 )$$ are absolutely conflicting compared with the instance of shear thickening $$(n>1)$$ liquid. Additionally, validation of numerical results is done via benchmarking with previously stated limiting cases with two different schemes namely, homotopy analysis method (HAM) and bvp4c scheme. These comparisons initiate a superb correspondence with these outcomes.

Numerical Techniques for Unsteady Nonlinear Burgers Equation Based on Backward Differentiation Formulas

AbstractWe introduce new numerical techniques for solving nonlinear unsteady Burgers equation. The numerical technique involves discretization of all variables except the time variable which converts nonlinear PDE into nonlinear ODE system. Stability of the nonlinear system is verified using Lyapunov’s stability criteria. Implicit stiff solvers backward differentiation formula of order one, two and three are used to solve the nonlinear ODE system. Four test problems are included to show the applicability of introduced numerical techniques. Numerical solutions so obtained are compared with solutions of existing schemes in literature. The proposed numerical schemes are found to be simple, accurate, fast, practical and superior to some existing methods.

Numerical simulation of flow in smectic liquid crystals

Our aim is to simulate a nonlinear system of ODEs describing the flow in smectic liquid crystals. The nonlinear system is first linearized. We present a direct approach to compute the exact analytic solution of this linear system and use this solution as a starting profile in the Matlab package bvpsuite2.0 to obtain the approximate solution to the nonlinear system. Although, the solution of the nonlinear system has steep boundary layers and therefore is difficult to resolve, we demonstrate that bvpsuite2.0 can cope with the problem and provide an approximation with reasonable accuracy.

A non-Fourier approach towards the analysis of heat transfer enhancement with water based nanofluids through a channel

In this article, a non-Fourier approach to model the heat transfer phenomenon in nanofluids having application to automotive industry is studied. In this respect, a recently proposed hyperbolic heat flux equation is embedded into the heat energy equation and thereby incorporating the effect of thermal relaxation time. Nanofluids are formed by considering copper oxide (CuO), Titanium dioxide (TiO2) and Aluminum oxide (Al2O3) nano-solid particles in the base fluid. The flow governing system of PDEs along with boundary conditions is transformed into its respective coupled system of nonlinear ODEs using suitable similarity functions. Runge-Kutta-Fehlberg (RK-5) numerical scheme embedded with shooting method is implemented and used to solve the obtained boundary value problem. Numerical simulations are performed and tabulated to analyze the influence of solid volume fraction on local coefficient of skin-friction and Nusselt number. A comparison is made between the results by Fourier and present heat flux model. We conclude that the presented new approach is more general and thus allows predicting the influence of thermal relaxation time on the heat transfer characteristics. Moreover, consideration of present model over the Fourier model helps to predict the actual temporal behavior of solution.

Modeling Chemically Reactive Flow of Sutterby Nanofluid by a Rotating Disk in Presence of Heat Generation/Absorption

In this article we investigate the flow of Sutterby liquid due to rotating stretchable disk. Mass and heat transport are analyzed through Brownian diffusion and thermophoresis. Further the effects of magnetic field, chemical reaction and heat source are also accounted. We employ transformation procedure to obtain a system of nonlinear ODE’s. This system is numerically solved by Built-in-Shooting method. Impacts of different involved parameter on velocity, temperature and concentration are described. Velocity, concentration and temperature gradients are numerically computed. Obtained results show that velocity is reduced through material parameter. Temperature and concentration are enhanced with thermophoresis parameter.

Biomechanics of stem cells

Stem cells play a key role in the healthy development and maintenance of organisms. They are also critically important in medical treatments of various diseases. It has been recently demonstrated that the mechanical factors such as forces, adhesion, stiffness, relaxation, etc. have significant effects on stem cell functions. Under physiological conditions, cells (stem cells) in muscles, heart, and blood vessels are under the action of externally applied strains. We consider the stem cell microenvironment and performance associated with their conversion (differentiation) into skeletal muscle cells. Two problems are studied by using mathematical models whose parameters are then optimized by fitting experiments. First, we present our analysis of the process of stem cell differentiation under the application of cyclic unidirectional strain. This process is interpreted as a transition through several (six) stages where each of them is defined in terms of expression of a set of factors typical to skeletal muscle cells. The stem cell evolution toward muscle cells is described by a system of nonlinear ODEs. The parameters of the model are determined by fitting the experimental data on the time course of expression of the factors under consideration. Second, we analyse the mechanical (relaxation) properties of a scaffold that serves as the microenvironment for stem cells differentiation into skeletal muscle cells. This scaffold (surrounded by a liquid solution) is composed of unidirectional fibers with pores between them. The relaxation properties of the scaffold are studied in an experiment where a long cylindrical specimen is loaded by the application of ramp displacement until the strain reaches a prescribed value. The magnitude of the corresponding load is recorded. The specimen is considered as transversely isotropic poroelastic cylinder whose force relaxation is associated with liquid diffusion through the pores. An analytical solution for the total force applied to the cylinder in terms of the mechanical properties of the scaffold (longitudinal and lateral Young’s moduli, two Poisson’s ratios, and typical time of liquid diffusion) is used. The number of constant is then reduced to three by estimating the longitudinal Young’s modulus and one of Poisson’s ratios from an earlier experiment. Finally, three remaining parameters are estimated by fitting the relaxation curve corresponding to strain rate of loading of 0.01 s−1. The developed mathematical solution is then tested by comparing the theoretical and experimental results for another strain rate of 0.0025 s−1. The scaffold relaxation properties can be important for differentiation of stem cells inside the pores.

A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations

PurposeIn this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations.Design/methodology/approachTwo schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples.FindingsA comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method.Originality/valueFor the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

Impacts of variable thermal conductivity on stagnation point boundary layer flow past a Riga plate with variable thickness using generalized Fourier’s law

Abstract This article explores the problem of two-dimensional, laminar, steady and boundary layer stagnation point slip flow over a Riga plate. The incompressible upper-convected Maxwell fluid has been considered as a rheological fluid model. The heat transfer characteristics are investigated with generalized Fourier’s law. The fluid thermal conductivity is assumed to be temperature dependent in this study. A system of partial differential equations governing the flow of an upper-convected Maxwell fluid, heat and mass transfer using generalized Fourier’s law is developed. The main objective of the article is to inspect the impacts of pertinent physical parameters such as the stretching ratio parameter ( 0 ⩽ A ⩽ 0.3 ) , Deborah number ( 0 ⩽ β ⩽ 0.6 ) , thermal relaxation parameter ( 0 ⩽ γ ⩽ 0.5 ) , wall thickness parameter ( 0.1 ⩽ α ⩽ 3.5 ) , slip parameter ( 0 ⩽ R ⩽ 1.5 ) , thermal conductivity parameter ( 0.1 ⩽ δ ⩽ 1.0 ) and modified Hartmann number ( 0 ⩽ Q ⩽ 3 ) on the velocity and temperature profiles. Suitable local similarity transformations have been used to get a system of non-linear ODEs from the governing PDEs. The numerical solutions for the dimensionless velocity and temperature distributions have been achieved by employing an effective numerical method called the shooting method. It is seen that the velocity profile shows the reduction in the velocity for the higher values of viscoelastic parameter and the thermal relaxation parameter. In addition, to enhance the reliability at the maximum level of the obtained numerical results by shooting method, a MATLAB built-in solver bvp4c has also been utilized.

Numerical treatment for Carreau nanofluid flow over a porous nonlinear stretching surface

Abstract The impact of magnetic field and nanoparticles on the two-phase flow of a generalized non-Newtonian Carreau fluid over permeable non-linearly stretching surface has been analyzed in the existence of all suction/injection and thermal radiation. The governing PDEs with congruous boundary condition are transformed into a system of non-linear ODEs with appropriate boundary conditions by using similarity transformation. It solved numerically by using 4th–5th order Runge-Kutta-Fehlberg method based on shooting technique. The impacts of non-dimensional controlling parameters on velocity, temperature, and nanoparticles volume concentration profiles are scrutinized with aid of graphs. The Nusselt and the Sherwood numbers are studied at the different situations of the governing parameters. The numerical computations are in excellent consent with previously reported studies. It is found that the heat transfer rate is reduced with an increment of thermal radiation parameter and on contrary of the rising of magnetic field. The opposite trend happens in the mass transfer rate.

Chemically reacting on MHD boundary-layer flow of nanofluids over a non-linear stretching sheet with heat source/sink and thermal radiation

In this paper, steady 2-D MHD free convective boundary-layer flows of an electrically conducting nanofluid over a non-linear stretching sheet taking into account the chemical reaction and heat source/sink are investigated. The governing equations are transformed into a system of non-linear ODE using suitable similarity transformations. Analytical solution for the dimensionless velocity, temperature, concentration, skin friction coefficient, heat and mass transfer rates are obtained by using homotopy analysis method. The obtained results show that the flow field is substantially influenced by the presence of chemical reaction, radiation, and magnetic field.

Exploring the Cattaneo-Christov heat flux phenomenon on a Maxwell-type nanofluid coexisting with homogeneous/heterogeneous reactions

This specific article unfolds the efficacy of Cattaneo-Christov heat flux on the heat and mass transport of Maxwell nanofluid flow over a stretched sheet with changeable thickness. Homogeneous/heterogeneous reactions in the fluid are additionally considered. The Cattaneo-Christov heat flux model is initiated in the energy equation. Appropriate similarity transformations are taken up to form a system of nonlinear ODEs. The impact of related parameters on the nanoparticle concentration and temperature is inspected through tables and diagrams. It is renowned that temperature distribution increases for lower values of the thermal relaxation parameter. The rate of mass transfer is enhanced for increasing in the heterogeneous reaction parameter but the reverse tendency is ensued for the homogeneous reaction parameter. On the other side, the rate of heat transfer is getting enhanced for the Cattaneo-Christov model compared to the classical Fourier’s model for some flow factors. Thus the implication of the current study is to delve its unique effort towards the generalized version of traditional Fourier’s law at nano level.

Application of the hypodifferential descent method to the problem of constructing an optimal control

In this paper the problem of optimal control of a nonlinear ODE system with given boundary conditions and the integral restriction on control is considered. With the help of the theory of exact penalty functions the original problem is reduced to the problem of unconstrained minimization of a nonsmooth functional. The necessary minimum conditions in terms of hypodifferentials are found. A class of problems for which these conditions are also sufficient is distinguished. On the basis of these conditions the hypodifferential descent method is applied to the considered problem. Under some additional assumptions the hypodifferential descent method converges in a certain sense.

The Stochastic Quasi-chemical Model for Bacterial Growth: Variational Bayesian Parameter Update

We develop Bayesian methodologies for constructing and estimating a stochastic quasi-chemical model (QCM) for bacterial growth. The deterministic QCM, described as a nonlinear system of ODEs, is treated as a dynamical system with random parameters, and a variational approach is used to approximate their probability distributions and explore the propagation of uncertainty through the model. The approach consists of approximating the parameters’ posterior distribution by a probability measure chosen from a parametric family, through minimization of their Kullback–Leibler divergence.

Boundary layer flow of nanofluid over a moving surface in a flowing fluid using revised model with stability analysis

This article reveals the boundary layer flow analysis of nanofluid past a moving surface in a uniform free stream with the physically more realistic approach by imposing both temperature and concentration fraction on the surface in such a way that nanoparticle concentration adjusts and flux becomes zero on the surface. In other word, nanoparticle fraction adjusts accordingly on the boundaries. The system of nonlinear ODEs is solved numerically using bvp4c function and shooting technique. With the help of various initial guesses, dual solutions are found up to a certain limit when the free stream and the plate are in the opposite directions. Then examined the stability of the solutions obtained using the method of stability analysis. The impact of various flow parameters on the skin friction coefficient and the local Nusselt number, the velocity, temperature and concentration distributions are conferred in detail. The reduced Nusselt number is estimated with the help of linear and quadratic regressions. For the validity, the comparison is made. Results indicate that the first (upper branch) solution is stable and thus, physically realizable. There is no effect on the reduced Nusselt number for any value of the Brownian motion parameter due to zero nanoparticle flux on the surface. On the other hand, the heat transfer rate decreases for higher values of the Schmidt number and thermophoresis parameter. The flow pattern behaved same as the stagnation point flow for the first (upper branch) solution but separated into two regions for the second (lower branch) solution.

Radiative nonlinear 3D flow of ferrofluid with Joule heating, convective condition and Coriolis force

Characteristics of heat transport mechanism in three-dimensional ferrofluid flow past a deformed surface subjected to the Coriolis and Lorentz forces are analyzed. The impacts of Joule heating, nonlinear thermal radiation, viscous dissipation and convective condition are also accounted. The carrier fluid (water) is embedded by Fe3O4 nanoparticles. The boundary layer approximations are employed in problem statement. Stretching transformations are utilized to form nonlinear ODE system from governed PDE system. The subsequent system is treated numerically via Runge-Kutta-Fehlberg method. Effects of relevant parameters on different flow fields are discussed comprehensively with help of graphs. It is established that the heat transfer rate is enhanced due to Coriolis and Lorentz forces. Furthermore, Fe3O4 nanoparticles enhance the Nusselt number significantly in comparison with Al2O3 nanoparticles.

Analytic solutions of a microstructure PDE and the KdV and Kadomtsev–Petviashvili equations by invariant Painlevé analysis and generalized Hirota techniques

Truncated Painlevé expansions, invariant Painlevé analysis, and generalized Hirota expansions are employed in combination to solve (‘partially reduce to quadrature’) the integrable KdV and KP equations, and a nonintegrable generalized microstructure (GMS) equation. Although the multisolitons of the KdV and KP equations are very well-known, the solutions obtained here for all the three NLPDEs are novel and non-trivial. The solutions obtained via invariant Painlevé analysis are all complicated rational functions, with arguments which themselves are confluent hypergeometric (KdV) or trigonometric (GMS) functions of various distinct non-traveling (KdV) and traveling wave variables. In some cases, this is slightly reminiscent of doubly-periodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained by the use of recently-generalized Hirota-type expansions in the truncated Painlevé expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the three NLPDEs considered here.

On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

ABSTRACTA canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.