Purpose 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. Design/methodology/approach 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. Findings 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. Practical implications 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. Originality/value 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.

The paper deals with a stagnation-point boundary layer flow towards a permeable stretching/shrinking sheet in a nanofluid where the flow and the sheet are not aligned. We used the Buongiorno model that is based on the Brownian diffusion and thermophoresis to describe the nanofluid in this problem. The main purpose of the present paper is to examine whether the non-alignment function has the effect on the problem considered when the fluid suction and injection are imposed. It is interesting to note that the non-alignment function can ruin the symmetry of the flows and prominent in the shrinking sheet. The fluid suction will reduce the impact of the non-alignment function of the stagnation flow and the stretching/shrinking sheet but at the same time increasing the velocity profiles and the shear stress at the surface. Furthermore, the effects of the pertinent parameters such as the Brownian motion, thermophoresis, Lewis number and the suction/injection on the flow and heat transfer characteristics are also taken into consideration. The numerical results are shown in the tables and the figures. It is worth mentioning that dual solutions are found to exist for the shrinking sheet.

Purpose 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. Design/methodology/approach 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. Findings 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. Practical implications 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. Originality/value 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 considers the extended problem of the thermosolutal Marangoni forced convection boundary layer by Pop et al. (2001) when the wall is permeable, namely, there is a suction or injection effect. The governing system of partial differential equations is transformed into a system of ordinary differential equations, and the transformed equations are solved numerically using the shooting method. The effects of suction or injection parameterf0on the velocity, temperature, and concentration profiles are illustrated and presented in tables and figures. It is shown that dual solutions exist for the similarity parameterβless than 0.5.

The problem of thermal diffusion and diffusion thermo effects on thermosolutal Marangoni convection flow of an electrically conducting fluid over a permeable surface is investigated. Using appropriate similarity transformations, the governing system of partial differential equation is transformed to a set of nonlinear ordinary differential equations, then solved numerically using the Runge-Kutta-Fehlberg method. The effects of thermal diffusion and diffusion thermo, magnetic field parameter, thermosolutal surface tension ratio, and suction/injection parameter on the flow field, heat transfer characteristic, and concentration are thoroughly examined. Numerical results are obtained for temperature and concentration profiles as well as the local Nusselt and Sherwood numbers are presented graphically and analyzed. It is found that these governing parameters affect the variations of the temperature and concentration and also the local Nusselt and Sherwood numbers.