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The purpose of this paper is to determine both analytically and numerically the kink solutions to a new one-dimensional, viscoelastic generalization of Burgers’ equation, which includes a non-linear constitutive law, and the number of kinks as functions of the non-linearity and relaxation parameters.

An analytical procedure and two explicit finite difference methods based on first-order accurate approximations to the first-order derivatives are used to determine the single- and double-kink solutions.

It is shown that only two parameters characterize the solution and that the existence of a shock wave requires that the (semi-positive) relaxation parameter be less than unity and the non-linearity parameter be less than two. It is also shown that negative values of the non-linearity parameter result in kinks with a single inflection point and strain and dissipation rates with a single relative minimum and a single, relative maximum, respectively. For non-linearity parameters between one and two, it is shown that the kink has three inflection points that merge into a single one as this parameter approaches one and that the strain and dissipation rates exhibit relative maxima and minima whose magnitudes decrease and increase as the relaxation and nonlinearity coefficients, respectively, are increased. It is also shown that the viscoelastic generalization of the Burgers equation presented here is related to an ϕ⁸−scalar field.

A new, one-dimensional, viscoelastic generalization of Burgers’ equation, which includes a non-linear constitutive law and relaxation is proposed, and its kink solutions are determined both analytically and numerically. The equation and its solutions are connected with scalar field theories and may be used to both studies the effects of the non-linearity and relaxation and assess the accuracy of numerical methods for first-order, non-linear partial differential equations.

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

The purpose of this study is to investigate the effects of the shear-thinning viscoelastic behavior of the surrounding matrix on droplet deformation by weakly compressible smoothed particle hydrodynamics (WC-SPH). Also, the effect of the presence of another droplet is examined.

A modified consistent weakly compressible SPH method is proposed. After code verification, a complete parameter study is performed for a drop under the simple shear flow of a Giesekus liquid. The investigated parameters are 0.048≤Ca ≤ 14.4, 0.1≤c ≤ 10, 0.04≤De ≤ 10, 0≤α ≤ 1 and 0.12≤Re ≤ 12.

It is demonstrated that the rheological behavior of the surrounding fluid could dramatically affect the droplet deformation. It is shown that the droplet deformation is increased by increasing Re and Ca. In contrast, the droplet deformation is decreased by increasing a, De and polymer content. Also, it is indicated the presence of another droplet could drastically affect the flow field, and the primary stress difference (N1) is resonated between two droplets.

The main originality of this paper is to introduce a new consistent WC-SPH algorithm. The proposed method is very versatile for tackling the shear-thinning viscoelastic multiphase problems. Furthermore, a complete parameter study is performed for a drop under the simple shear flow of Giesekus liquid. Another novelty of the current paper is studying the effect of the presence of a second droplet. To the best of the authors’ knowledge, this is performed for the first time.

The need for precise synthesis of customized designs has resulted in the development of advanced coating processes for modern nanomaterials. Achieving accuracy in these processes requires a deep understanding of thermophysical behavior, rheology and complex chemical reactions. The manufacturing flow processes for these coatings are intricate and involve heat and mass transfer phenomena. Magnetic nanoparticles are being used to create intelligent coatings that can be externally manipulated, making them highly desirable. In this study, a Keller box calculation is used to investigate the flow of a coating nanofluid containing a viscoelastic polymer over a circular cylinder.

The rheology of the coating polymer nanofluid is described using the viscoelastic model, while the effects of nanoscale are accounted for by using Buongiorno’s two-component model. The nonlinear PDEs are transformed into dimensionless PDEs via a nonsimilar transformation. The dimensionless PDEs are then solved using the Keller box method.

The transport phenomena are analyzed through a comprehensive parametric study that investigates the effects of various emerging parameters, including thermal radiation, Biot number, Eckert number, Brownian motion, magnetic field and thermophoresis. The results of the numerical analysis, such as the physical variables and flow field, are presented graphically. The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as fluid parameter increases. An increase in mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid.

Intelligent materials rely heavily on the critical characteristic of viscoelasticity, which displays both viscous and elastic effects. Viscoelastic models provide a comprehensive framework for capturing a range of polymeric characteristics, such as stress relaxation, retardation, stretching and molecular reorientation. Consequently, they are a valuable tool in smart coating technologies, as well as in various applications like supercapacitor electrodes, solar collector receivers and power generation. This study has practical applications in the field of coating engineering components that use smart magnetic nanofluids. The results of this research can be used to analyze the dimensions of velocity profiles, heat and mass transfer, which are important factors in coating engineering. The study is a valuable contribution to the literature because it takes into account Joule heating, nonlinear convection and viscous dissipation effects, which have a significant impact on the thermofluid transport characteristics of the coating.

The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as the fluid parameter increases. An increase in the mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid. Increasing the strength of the magnetic field promotes an increase in the density of the streamlines. An increase in the mixed convection parameter results in a decrease in the isotherms and isoconcentration.