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The purpose of this paper is to study the problem of two‐dimensional unsteady mixed convective flow a dusty fluid over a stretching sheet in the presence of thermal radiation and space‐dependent heat source/sink.

The equations governing the fluid flow and temperature fields for both the fluid and dust phases are reduced to coupled non‐linear ordinary differential equations by using a suitable set of similarity transformations. Numerical solutions of the resulting equations are obtained using the well known RKF45 method.

The numerical results are benchmarked with previously published studies and found to be in excellent agreement. Finally, the effects of the pertinent parameters which are of physical and engineering interest on the flow and heat transfer characteristics are presented graphically and in tabulated form.

The problem is relatively original as the dusty fluid works for this type of problem are lacking.

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 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.

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.

This study aims to present an analysis on heat transfer attributes of fluid-particle interaction over a permeable elastic sheet. The fluid streaming on the sheet is Casson fluid (CF) with uniform distribution of dust particles.

The basic steady equations of the CF and dust phases are in the form of partial differential equations (PDEs) which are remodeled into ordinary ones with the aid of similarity transformations. In addition to analytical solution, numerical solution is obtained for the reduced coupled non-linear ordinary differential equations (ODEs) to validate the results.

The solution seems to be influenced by significant physical parameters such as CF parameter, magnetic parameter, suction parameter, fluid particle interaction parameter, Prandtl number, Eckert number and number density. The impact of these parameters on flow field and temperature for both fluid and dust phases is presented in the form of graphs and discussed in detail. The effect on skin friction coefficient and heat transfer rate is also presented in tabular form. It has been observed that an increase in the CF parameter curtails the fluid velocity as well as the particle velocity however enhances the heat transfer rate at the wall. Furthermore, comparison of the numerical and analytical solution is also made and found to be in excellent agreement.

Although the analysis of dusty fluid flow has been widely examined, however, the present study obtained both analytical and numerical results of power law temperature distribution in dusty Casson fluid under the influence of magnetic field which are new and original for such type of flow.

The purpose of this paper is to highlight the advantages of a simplified algorithm to solve the problem of heat and mass transfer in porous medium by reducing the number of partial differential equations from four to three.

The approach of the present paper is to develop a simplified algorithm to reduce the number of equations involved in conjugate heat transfer in porous medium.

Developed algorithm/method has many advantages over conventional method of solution for conjugate heat transfer in porous medium.

The current work is applicable to conjugate heat transfer problem.

The developed algorithm is useful in reducing the number of equations to be solved, thus reducing the computational resources required.

Development of simplified algorithm and comparison with conventional method.

The present investigation aims to deal with the study of unsteady, heat-generating/-absorbing and chemically reacting Casson fluid flow over a vertical cone and flat plate saturated with non-Darcy porous medium in the presence of cross-diffusion effects.

A numerical computation for the governing equations has been performed using implicit finite difference method of Crank–Nicolson type.

The influence of various physical parameters on velocity, temperature and concentration distributions is illustrated graphically, and the physical aspects are discussed in detail. Numerical results for average skin-friction, Nusselt number and Sherwood number are tabulated for the pertaining physical parameters. Results indicate that Soret and Dufour effects have notable influence on heat and mass transfer characteristics of the fluid when the temperature and concentration gradients are high. It is also observed that the consideration of heat generation/absorption plays a vital role in predicting the heat transfer characteristics of moving fluids.

Consider a two-dimensional, unsteady, free convective flow of an incompressible Casson fluid over a vertical cone and a flat plate saturated with non-Darcy porous medium. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. The fluid flow is moderate and the permeability of the medium is assumed to be low, so that the Forchheimer flow model is applicable.

The flow of Casson fluids (such as drilling muds, clay coatings and other suspensions, certain oils and greases, polymer melts and many emulsions), in the presence of heat transfer, is an important research area because of its relevance in the optimized processing of chocolate, toffee and other foodstuffs.

In the heat and mass transfer investigations, the Casson fluid model is found to be accurately applicable in many practical situations in the wings of polymer processing industries and biomechanics, etc.; some prominent examples are silicon suspensions, suspensions of bentonite in water and lithographic varnishes used for printing inks.

The motivation of the present study is to bring out the effects of heat source/sink, Soret and Dufour effects on chemically reacting Casson fluid flow over a vertical cone and flat plate saturated with non-Darcy porous medium. The flow of Casson fluids (such as certain oils and greases, polymer melts and many emulsions) in the presence of heat transfer is an important research area because of its relevance in the optimized processing of chocolate, toffee and other foodstuffs. A numerical computation for the governing equations has been performed using implicit finite difference method of the Crank–Nicolson type.

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.

The purpose of this paper is to study the flow, heat and mass transfer of MHD Casson nanofluid due to an inclined stretching sheet using similarity transformation, the governing PDE’S equations of flow, heat and mass transfer are converted into ODE’S. The resulting non-linear ODE’S are solved numerically using an implicit finite difference method, which is known as Kellor-box method. The effects of various governing parameters on velocity, temperature and concentration are plotted for both Newtonian and non-Newtonian cases. The numerical values of skin friction, Nusselt number and Sherwood number are calculated and tabulated in various tables for different values of physical parameters. It is noticed that the effect of angle of inclination enhances the temperature and concentration profile whereas velocity decreases. The temperature decreases due to the increase in the parametric values of Pr and Gr due to thickening in the boundary layer.

Numerical method is applied to find the results.

Flow and heat transfer analysis w.r.t various flow and temperature are analyzed for different values of the physical parameters.

The numerical values of skin friction, Nusselt number and Sherwood number are calculated and tabulated in various tables for different values of physical parameters.

The study of the boundary layer flow, heat and mass transfer is important due to its applications in industries and many manufacturing processes such as aerodynamic extrusion of plastic sheets and cooling of metallic sheets in a cooling bath.

Here in this paper the authors have investigated the MHD boundary layer flow of a Casson nanofluid over an inclined stretching sheet along with the Newtonian nanofluid as a limited.

This paper aims to explore particle shape effect on Cu-H₂O nanoparticles over a moving plate in the presence of nonlinear thermal radiation. To characterize the effect, particle shape and viscous dissipation are considered. Convergent solutions for the resulting nonlinear systems are derived and the effects of embedded parameters of interest on velocity and temperature field are examined.

The Runge–Kutta–Fehlberg fourth-fifth order method along with shooting technique is used to solve the governing equations (6) and (7) with boundary conditions (8). A suitable finite value of η_∞ is considered in such a way that the boundary conditions are satisfied asymptotically.

The results show an increase in both the heat transfer and thermodynamic performance of the system. However, among the three nanoparticle shapes, disk shape exhibited better heat transfer characteristics and heat transfer rate. On the other hand, the velocity profile enhances with increasing values of ϕ in the first solution, but the opposite trend was found in the second solution.

The present paper deals with an exploration of particle shape effect on Cu-H₂O nanoparticles over a moving plate in the presence of nonlinear thermal radiation. To characterize the effect, particle shape and viscous dissipation are considered. Convergent solutions for the resulting nonlinear systems are derived and the effects of embedded parameters of interest on velocity and temperature field are examined. The skin friction coefficient and Nusselt number are numerically tabulated and discussed. The results show an increase in both heat transfer and thermodynamic performance of the system. However, among the three nanoparticle shapes, disk shape exhibited better heat-transfer characteristics and heat-transfer rate. On the other hand, the velocity profile enhances with increasing values of ϕ in the first solution, but the opposite trend was found in the second solution.