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The purpose of this paper is to conduct a detailed investigation of the two-dimensional natural convection flow of a dusty fluid. Therefore, the incompressible boundary layer flow of a two-phase particulate suspension is investigated numerically over a semi-infinite vertical flat plate. Comprehensive flow formations of the gas and particle phases are given in the boundary layer region. Primitive variable formulation is employed to convert the nondimensional governing equations into the non-conserved form. Three important two-phase mechanisms are discussed, namely, water-metal mixture, oil-metal mixture and air-metal mixture.

The full coupled nonlinear system of equations is solved using implicit two point finite difference method along the whole length of the plate.

The authors have presented numerical solution of the dusty boundary layer problem. Solutions obtained are depicted through the characteristic quantities, such as, wall shear stress coefficient, wall heat transfer coefficient, velocity distribution and temperature distribution for both phases. Results are interpreted for wide range of Prandtl number Pr (0.005-1,000.0). It is observed that thin boundary layer structures can be formed when mass concentration parameter or Prandtl number (e.g. oil-metal particle mixture) are high.

The results of the study may be of some interest to the researchers of the field of chemical engineers.

To explore the fusion of dust particles and of polymers in a viscous liquid is the main purpose of this article. Newtonian fluid as a base fluid is considered and the mutual presence of polymers and dusty bodies is investigated. It discusses the steady laminar flow and heat transportation of a polymeric dusty liquid induced by a uniformly heated, penetrable and stretchable surface inside the boundary layer.

The mathematical system incorporates separate equations of energy and momentum for dusty bodies and for fluid. The classical Oldroyd-B model is chosen for exploring polymer presence. For the fluid phase, this model adds another stress to the conservation law of momentum. Appropriate similarity variables are introduced to transform the system of partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs). The problem is solved by introducing a numerical iterative procedure which turned out to be fastly converging.

Expeditious changes inside the boundary layer cause polymers to deform. No changes outside the boundary layer are noticed on account of polymer stretching. The dependence of heat transfer rate and skin friction on the parameter of polymer concentration and Weissenberg number is analyzed and displayed graphically against interaction parameters for temperature and velocity, dust particles’ mass concentration, Eckert and Prandtl numbers. Combining effects of polymers and dust particles cause skin friction to decrease and heat transfer rate to increase. Increasing values of interaction parameter for velocity, dust particles’ mass concentration and Eckert number reduces the drag coefficient and local Nusselt number. On the other hand, the Prandtl number and interaction parameter of temperature magnify the heat flux at the wall.

This article studies the infinite extensibility of polymers. FENE and FENE-P models can be used to investigate the polymer presence in dusty fluids in the future.

In this article, the authors’ aim is to study the combined presence of polymers and dusty bodies. Keeping the existing literature in view, this type of fusion is not studied yet. Polymer inclusion in a viscous dusty fluid is studied and the behavior of fluid flow and heat transportation is investigated within the boundary layer over a permeable linearly stretching sheet.

Natural gas leak from underground pipelines could lead to serious damage and global warming, whose spreading in soil should be systematically investigated. This paper aims to propose a three-dimensional numerical model to analyze the methane–air transportation in soil. The results could help understand the diffusion process of natural gas in soil, which is essential for locating leak source and reducing damage after leak accident.

A numerical model using finite element method is proposed to simulate the methane spreading process in porous media after leaking from an underground pipe. Physical models, including fluids transportation in porous media, water evaporation and heat transfer, are taken into account. The numerical results are compared with experimental data to validate the reliability of the simulation model. The effects of methane leaking direction, non-uniform soil porosity, leaking pressure and convective mass transfer coefficient on ground surface are analyzed.

The methane mole fraction distribution in soil is significantly affected by the leaking direction. Horizontally and vertically non-uniform soil porosity has a stronger effect. Increasing leaking pressure causes increasing methane mole flux and flow rate on the ground surface.

Most existing gas diffusion models in porous media are for one- or two-dimensional simulation, which is not enough for predicting three-dimensional diffusion process after natural gas leak in soil. The heat transfer between gas and soil was also neglected by most researchers, which is very important for predicting the gas-spreading process affected by the soil moisture variation because of water evaporation. In this paper, a three-dimensional numerical model is proposed to further analyze the methane–air transportation in soil using finite element method, with the presence of water evaporation and heat transfer in soil.

This paper aims to present dual solutions for the two-dimension copper oxide with silver (CuO–Ag) and zinc oxide with silver (ZnO–Ag) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip.

The governing partial differential equations for the two dust particle phases are reduced to the pertinent ordinary differential equations using a similarity transformation. Closed-form analytical solutions for the reduced skin friction and reduced Nusselt number, as well as for the velocity and temperature profiles, were presented, both graphically and in tables, under specific non-dimensional physical parameters such as the suction parameter, Prandtl number, slip parameter and shrinking parameter, which are also presented in both figures and tables.

The results indicate that for the shrinking flow, the wall skin friction is higher in the dusty fluid when compared with the clear (viscous) fluid. In addition, the effect of the fluid–particle interaction parameter to the fluid phase can be seen more clearly in the shrinking flow. Furthermore, multiple (dual, upper and lower branch solutions) are found for the governing similarity equations and the upper branch solution expanded with higher values of the suction parameter. It can be confirmed that the lower branch solution is unstable.

In practice, the study of the stretching/shrinking flow is crucially important and useful. Both the problems of steady and unsteady flow of a dusty fluid have a wide range of possible applications in practice, such as in the centrifugal separation of particles, sedimentation and underground disposal of radioactive waste materials.

Even though the problem of dusty fluid has been broadly investigated, very limited results can be found for a shrinking sheet. Indeed, this paper has succeeded to obtain analytically dual solutions. The stability analysis can be performed by following many published papers on stretching/shrinking sheets. Finally, the critical values and plotting curves for obtaining single or dual solution are successfully presented.

This paper aims to examine the Cu-Al2O3/water hybrid nanofluid flow over a shrinking sheet in the presence of the magnetic field and dust particles.

The governing partial differential equations for the two-phase flow of the hybrid nanofluid and the dust particles are reduced to ordinary differential equations using a similarity transformation. Then, these equations are solved using bvp4c in MATLAB software. The bvp4c solver is a finite-difference code that implements the three-stage Lobatto IIIa formula. The numerical results are gained for several values of the physical parameters. The effects of these parameters on the flow and the thermal characteristics of the hybrid nanofluid and the dust particles are analyzed and discussed. Later, the temporal stability analysis is used to determine the stability of the dual solutions obtained as time evolves.

The outcome shows that the flow is unlikely to exist unless satisfactory suction strength is imposed on the shrinking sheet. Besides, the heat transfer rate on the shrinking sheet decreases with the increase of . However, the increase in and lead to enhance the heat transfer rate. Two solutions are found, where the domain of the solutions is expanded with the rising of, and. Consequently, the boundary layer separation on the surface is delayed in the presence of these parameters. Implementing the temporal stability analysis, it is found that only one of the solutions is stable as time evolves.

The dusty fluid problem has been widely studied for the flow over a stretching sheet, but only limited findings can be found for the shrinking counterpart. Therefore, this study considers the problem of the dusty fluid flow over a shrinking sheet containing Cu-Al₂O₃/water hybrid nanofluid with the effect of the magnetic field. In fact, this is the first study to discover the dual solutions of the dusty hybrid nanofluid flow over a shrinking sheet. Also, further analysis shows that only one of the solutions is stable as time evolves.

The purpose of this paper is to model the boundary layer flow and heat transfer of dusty fluid with suspended nanoparticles over a stretching surface. The effect of multiple slip and nonlinear thermal radiation is taken into the account. Adequate similarity transformations are used to obtain a set of nonlinear ordinary differential equations to govern formulated problem. The resultant non-dimensionalized boundary value problem is solved numerically using the RKF-45 method. The profiles for velocity and temperature, which are controlled by thermophysical parameters, are presented graphically. Based on these plots, the conclusion is given and the obtained numerical results are tabulated. Observed interesting fact is that the SiO₂-water nanoparticles show a thicker thermal boundary layer than TiO₂-water nanoparticles.

The governing partial differential equations are approximated to a system of nonlinear ordinary differential equations by using suitable similarity transformations. An effective fourth–fifth-order Runge–Kutta–Fehlberg integration scheme numerically solves these equations along with a shooting technique. The effects of various pertinent parameters on the flow and heat transfer are examined.

Present results have an excellent agreement with previous published results in the limiting cases. The values of skin friction and wall temperature for different governing parameters are also tabulated. It is demonstrated that the SiO₂-water nanoparticles show a thicker thermal boundary layer than TiO₂-water nanoparticles. It is interesting to note that the dusty nanofluids are found to have higher thermal conductivity.

This paper is a new work related to comparative study of TiO₂ and SiO₂ nanoparticles in heat transfer of dusty fluid flow.

The purpose of this paper is to numerically study the boundary layer problem for the case of two-dimensional flow of dusty fluid over a shrinking surface in the presence of the fluid suction at the surface.

The governing equations of the problem are reduced to the system of ordinary differential equations using the similarity transformation and then solved using the bvp4c method in the Matlab software.

The effects of the drag coefficient parameter L, the fluid–particle interaction parameter δ, the suction parameter s and the particle loading parameter ω on the flow of the permeable shrinking sheet are investigated. It is found that the aforementioned parameters have different effects in the shrinking sheet flow. This study has also succeeded in discovering the second solution, and through the stability analysis, it is suggested that the solution is unstable and not physically realizable in practice.

The current findings add to a growing body of literature on the boundary layer problem in the dusty fluid. The dusty fluid is significant in various practical applications such as in the transporting suspended powdered materials through pipes, propulsion and combustion in rockets, the flow of blood in arteries, wastewater treatment and as corrosive particles in engine oil flow.

Even though the dusty fluid problem has been extensively studied in the flow of the stretching sheet, limited findings can be found over a shrinking flow. In fact, this is the first study to discover the second solution in the dusty fluid problem.

This present aims to present the numerical study of the unsteady stretching/shrinking flow of a fluid-particle suspension in the presence of the constant suction and dust particle slip on the surface.

The governing partial differential equations for the two phases flows of the fluid and the dust particles are reduced to the pertinent ordinary differential equations using a similarity transformation. The numerical results are obtained using the bvp4c function in the Matlab software.

The results revealed that in the decelerating shrinking flow, the wall skin friction is higher in the dusty fluid when compared to the clean fluid. In addition, the effect of the fluid-particle interaction parameter to the fluid-phase can be seen more clearly in the shrinking flow. Other non-dimensional physical parameters such as the unsteadiness parameter, the mass suction parameter, the viscosity ratio parameter, the particle slip parameter and the particle loading parameter are also considered and presented in figures. Further, the second solution is discovered in this problem and the solution expanded with higher unsteadiness and suction values. Hence, the stability analysis is performed, and it is confirmed that the second solution is unstable.

In practice, the flow conditions are commonly varying with time; thus, the study of the unsteady flow is very crucial and useful. The problem of unsteady flow of a dusty fluid has a wide range of possible applications such as in the centrifugal separation of particles, sedimentation and underground disposable of radioactive waste materials.

Even though the problem of dusty fluid has been broadly investigated, limited discoveries can be found over an unsteady shrinking flow. Indeed, this paper managed to obtain the second (dual) solutions, and stability analysis is performed. Furthermore, the authors also considered the artificial particle-phase viscosity, which is an important term to study the particle-particle and particle-wall interactions. With the addition of this term, the effects of the particle slip and suction parameters can be investigated. Very few studies in the dusty fluid embedded this parameter in their problems.

This paper aims to study the magneto-radiative gas (water vapor) on an unsmooth boundary.

This paper provided a numerical treatment via the implicit Chebyshev pseudo-spectral method to investigate unsteady compressible magneto-radiative gas (water vapor Pr = 1) flow near a heated vertical wavy wall through porous medium in the presence of inclined magnetic field. The impacts of viscous dissipation, temperature-dependent fluid properties, thermal conductivity and viscosity in the presence of nonlinear thermal radiation are studied. The sinusoidal surface is transformed into a flat one using a suitable transformation. The comparison figures of published data with the present outcomes illustrate a good match. The present steady-state outcomes are presented for the temperature, velocity, Nusselt number and the shearing stress through figures for several interested physical parameters, namely, compressibility, magnetic, radiation, viscosity–temperature variation, thermal conductivity–temperature variation, surface sinusoidal waveform and porous parameters.

The present numerical outcomes confirm the importance of applying nonlinear thermal radiation cases in all studies that investigate heat transfer under the influence of thermal radiation.

A mathematical model is established for a wavy boundary, and Chebyshev pseudo-spectral method is adopted for the numerical study.

The main purpose of this study, reflecting mainly the content of the authors’ plenary lecture, is to make a brief overview of several approaches developed by the author and his colleagues to the solution to ill-posed inverse heat transfer problems (IHTPs) with their possible extension to a wider class of inverse problems of mathematical physics and, most importantly, to show the wide possibilities of this methodology by examples of aerospace applications. In this regard, this study can be seen as a continuation of those applications that were discussed in the lecture.

The application of the inverse method was pre-tested with experimental investigations on a special test equipment in laboratory conditions. In these studies, the author used the solution to the nonlinear inverse problem in the conjugate (conductive and convective) statement. The corresponding iterative algorithm has been developed and tested by a numerical and experimental way.

It can be stated that the theory and methodology of solving IHTPs combined with experimental simulation of thermal conditions is an effective tool for various fundamental and applied research and development in the field of heat and mass transfer.

With the help of the developed methods of inverse problems, the investigation was conducted for a porous cooling with a gaseous coolant for heat protection of the re-entry vehicle in the natural environment of hypersonic flight. Moreover, the analysis showed that the inverse methods can make a useful contribution to the study of heat transfer at the surface of a solid body under the influence of the hypersonic heterogeneous (dusty) gas stream and in many other aerospace applications.