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.

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 effects of viscous dissipation on the boundary layer flow of hybrid nanofluids have been investigated. This study presents the mathematical modelling of steady two dimensional boundary layer flow of Cu-TiO2 hybrid nanofluid. In this research, the surface of the model is stretched and shrunk at the specific values of stretching/shrinking parameter. The governing partial differential equations of the hybrid nanofluid are reduced to the ordinary differential equations with the employment of the appropriate similarity transformations. Then, Matlab software is used to generate the numerical and graphical results by implementing the bvp4c function. Subsequently, dual solutions are acquired through the exact guessing values. It is observed that the second solution adhere to less stableness than first solution after performing the stability analysis test. The existence of viscous dissipation in this model is dramatically brought down the rate of heat transfer. Besides, the effects of the suction and nanoparticles concentration also have been highlighted. An increment in the suction parameter enhances the magnitude of the reduced skin friction coefficient while the augmentation of concentration of copper and titanium oxide nanoparticles show different modes.

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.

The paramagnetic feature of ferrofluid allows it to be utilised in electronic devices and improvise fluid circulation in transformer windings. Hence, the present article aims to conduct the numerical study of ferrofluid boundary layer flow along with the Stefan blowing, velocity and thermal slip, and Soret effects within the stagnation region over a stretching/shrinking surface. The governing equations were solved numerically using the bvp4c function in the MATLAB computing package. Based on the results, a stronger magnetic field of ferrofluid was needed to identify the numerical solutions past the shrinking surface, while the Stefan blowing diminished the solution’s availability. More than one solution is acquired for some specific values of the shrinking parameter, and the stability analysis validated that only one solution is reliable and stable.

The boundary layer flow over a stretching/shrinking cylinder in hybrid nanofluid with the effects of suction, partial slip and convective boundary condition is studied. Hybrid nanoparticles Al₂O₃ and TiO₂ with water as based fluid are considered in the study. The partial differential equations are transformed to ordinary differential equations by employing the similarity variables. The numerical results are obtained using the bvp4c solver in MATLAB software. The influence of nanoparticles volume fraction (Al₂O₃-TiO₂ in water-based fluid), curvature parameter, suction parameter, partial slip parameter and Biot number on the velocity profile, temperature profile, skin friction coefficient and heat transfer rate are discussed. The numerical results indicate that for shrinking surface case, the dual solutions exist for a certain range of curvature parameter and suction parameter.