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The purpose of this paper was to study laminar fluid flow and convective heat transfer in a conical gap at small conicity angles up to 4° for the case of disk rotation with a fixed cone.

In this paper, the improved asymptotic expansion method developed by the author was applied to the self-similar Navier–Stokes equations. The characteristic Reynolds number ranged from 0.001 to 2.0, and the Prandtl numbers ranged from 0.71 to 10.

Compared to previous approaches, the improved asymptotic expansion method has an accuracy like the self-similar solution in a significantly wider range of Reynolds and Prandtl numbers. Including radial thermal conductivity in the energy equation at small conicity angle leads to insignificant deviations of the Nusselt number (maximum 1.23%).

This problem has applications in rheometry to experimentally determine viscosity of liquids, as well as in bioengineering and medicine, where cone-and-disk devices serve as an incubator for nurturing endothelial cells.

The study can help design more effective devices to nurture endothelial cells, which regulate exchanges between the bloodstream and the surrounding tissues.

To the best of the authors’ knowledge, for the first time, novel approximate analytical solutions were obtained for the radial, tangential and axial velocity components, flow swirl angle on the disk, tangential stresses on both surfaces, as well as static pressure, which varies not only with the Reynolds number but also across the gap. These solutions are in excellent agreement with the self-similar solution.

The purpose of the present article is to obtain the similarity solution for the shock wave generated by a piston propagating in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows.

The Lie group theoretic method given by Sophus Lie is used to obtain the similarity solution in the present article.

Similarity solution with exponential law shock path is obtained for both ideal and nonideal gas cases. The effects on the flow variables, density ratio at the shock front and shock strength by the variation of the shock Cowling number, adiabatic index of the gas, gravitational parameter and nonidealness parameter are investigated. The shock strength decreases with an increase in the shock Cowling number, nonidealness parameter and adiabatic index, whereas the strength of the shock wave increases with an increase in gravitational parameter.

Propagation of shock wave with spherical geometry in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows has not been studied by any author using the Lie group theoretic method.

Thus, the purposes of this study are to study the limits of applicability of the self-similar solution to the problem of fluid flow, heat and mass transfer in conical gaps with small conicity angles, to substantiate the impossibility of using a self-similar formulation of the problem in the case of large conicity angles and to substantiate the absence of the need to take into account the radial thermal conductivity in the energy equation in its self-similar formulation for the conicity angles up to 4°.

In the present work, an in-depth and extended analysis of the features of fluid flow and heat transfer in a conical gap at small angles of conicity up to 4° is performed. The Couette-type flow arising, in this case, was modeled using a self-similar formulation of the problem. A detailed analysis of fluid flow calculations using a self-similar system of equations showed that they provide the best agreement with experiments than other known approaches. It is confirmed that the self-similar system of flow and heat transfer equations is applicable only to small angles of conicity up to 4°, whereas, at large angles of conicity, this approach becomes unreasonable and leads to significantly inaccurate results. The heat transfer process in a conical gap with small angles of conicity can be modeled using the self-similar energy equation in the boundary layer approximation. It was shown that taking into account the radial thermal conductivity in the self-similar energy equation at small conicity angles up to 4° leads to maximum deviations of the Nusselt number up to 1.5% compared with the energy equation in the boundary layer approximation without taking into account the radial thermal conductivity.

It is confirmed that the self-similar system of fluid flow equations is applicable only for small conicity angles up to 4°. The inclusion of radial thermal conductivity in the model unnecessarily complicates the mathematical formulation of the problem and at small conicity angles up to 4° leads to insignificant deviations of the Nusselt number (maximum 1.5%). Heat transfer in a conical gap with small conicity angles up to 4° can be modeled using the self-similar energy equation in the boundary layer approximation.

This paper investigates the question of the validity of taking into account the radial heat conduction in the energy equation.

This paper aims to study the two-dimensional steady magneto-hydrodynamic flow of a second-grade fluid in a porous channel using the homotopy perturbation method (HPM).

The governing Navier–Stokes equations of the flow are reduced to a third-order nonlinear ordinary differential equation by a suitable similarity transformation. Analytic solution of the resulting differential equation is obtained using the HPM. Mathematica software is used to visualize the flow behavior. The effects of the various parameters on velocity field are analyzed through appropriate graphs.

It is found that x component of the velocity increases with the increase of the Hartman number when the transverse direction variable ranges from 0 to 0.2 and the reverse behavior is observed when transverse direction variable takes values between 0.2 and 0.5. It is noted that the y component of the velocity increases rapidly with the increase of the transverse direction variable. The y component of the velocity increases marginally with the increase of the Hartman number M. The effect of the Reynolds number R on the x and y components of the velocity is quite opposite to the effect of the Hartman number on the x and y components of the velocity and the effect of the parameter on the x and y components of the velocity is similar to that of the Reynolds number.

To the best of the author’s knowledge, nobody had tried before two-dimensional steady magneto-hydrodynamic flow of a second-grade fluid in a porous channel using the HPM.

The problem of finding the possible classes of solution of different nonlinear equations seems to be of a great importance for many applications. In the context of the theory of self‐organization it is interpreted as finding all possible structures which arise and preserve themselves in the corresponding unbounded nonlinear medium. First, results on the numerical realization of a class of blow‐up invariant solutions of a nonlinear heat‐transfer equation with a source are presented in this article. The solutions considered describe a spiral propagation of the inhomogeneities in the nonlinear heat‐transfer medium. We have found initial perturbations which are good approximations to the corresponding eigen functions of combustion of the nonlinear medium. The local maxima of these initial distributions evolve consistent with the self‐similar law up to times very close to the blow‐up time.

The reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesian coordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. The Ladyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to the governing equation of interest. The reduction is based on a self-similar transformation as demonstrated in existing literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi-infinite domain. In our search for classical solutions, existence and uniqueness will be determined depending on the signs of two parameters with physical interpretation in the equation. Illustrations are included to highlight some of the main results.

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

This paper aims to theoretically and numerically investigate the steady two-dimensional (2D) Hiemenz flow with heat transfer of Reiner-Rivlin fluid over a linearly stretching/shrinking sheet.

The Navier–Stokes equations are transformed into self-similar equations using appropriate similarity transformations and then solved numerically by using shooting technique. A simple but effective mathematical analysis has been used to prove the existence of a solution for stretching case (λ> 0). Moreover, an attempt has been laid to carry the asymptotic solution behavior for large stretching. The obtained asymptotic solutions are compared with direct numerical solutions, and the comparison is quite remarkable.

It is observed that the self-similar equations exhibit dual solutions within the range [λ_c, −1] of shrinking parameter λ, where λ_c is the turning point from where the dual solutions bifurcate. Unique solution is found for all stretching case (λ > 0). It is noticed that the effects of cross-viscous parameter L and shrinking parameter λ on velocity and thermal fields show opposite character in the dual solution branches. Thus, a linear temporal stability analysis is performed to determine the basic feasible solution. The stability analysis is based on the sign of the smallest eigenvalue, where positive or negative sign leading to a stable or unstable solution. The stability analysis reveals that the first solution is stable that describes the main flow. Increase in cross-viscous parameter L resulting in a significant increment in skin friction coefficient, local Nusselt number and dual solutions domain.

This work’s originality is to examine the combined effects of cross-viscous parameter and stretching/shrinking parameter on skin friction coefficient, local Nusselt number, velocity and temperature profiles of Hiemenz flow over a stretching/shrinking sheet. Although many studies on viscous fluid and nanofluid have been investigated in this field, there are still limited discoveries on non-Newtonian fluids. The obtained results can be used as a benchmark for future studies of higher-grade non-Newtonian flows with several physical aspects. All the generated results are claimed to be novel and have not been published elsewhere.

Considers the motion of a viscous fluid within the narrow gap under a floating disk above an infinite porous plate. The flow is caused by a uniform blowing from the porous surface. If we ignore edge effects, after scaling the variables, an expression for the vertical component of the velocity is found with two particularities in the sense of an asymptotic expansion and for a self‐similar solution. We obtain an expression for the pressure distribution under the disk. Then, with this solution, Navier‐Stokes equations are reduced into one non‐linear equation of the fourth order with two points boundary conditions. First, we used a development of this equation by Newton’s method. The purpose of this paper is to show that the numerical scheme of quasilinearization gives rapid convergence to solution of this boundary layer problem. The vertical force balance gives the prediction for the height of the disk floating above the porous surface when the mass of the disk is known. Without any previous hypothesis, “TRIO”, a general computer code for thermal and fluid flow analysis developed at the CEA (Commissariat à l’Energie Atomique), confirms our main hypothesis and all our results. These two numerical solutions are well in keeping with the analytical and experimental solutions of Hinch and Lemaitre in 1994.

The purpose of this paper is the stagnation-point flow driven by a permeable stretching/shrinking surface with convective boundary condition and heat generation.

It is known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, for the present case it is demonstrated that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is constant.

The governing boundary layer equations are transformed to self-similar nonlinear ordinary differential equations using similarity transformations. Numerical results of the resulting equations are obtained using the function bvp4c from Matlab for different values of the governing parameters. In addition an analytical solution has been obtained for the energy equation when heat generation is absent. The streamlines for the upper branch solution show that the pattern is almost similar to the normal stagnation-point flow, but because of the existence of suction and shrinking effect, the flow seems like suck to the permeable wall.

Dual solutions are found for negative values of the moving parameter. A stability analysis has been also performed to show that the first upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible. The streamlines for the lower branch solution are also graphically shown.