The problem of steady laminar mixed convection flow and heat transfer past a moving vertical thin needle in nanofluid for both assisting and opposing cases is analyzed in this paper. Three types of nanoparticles including copper, titania and alumina are taken into consideration. The nonlinear ordinary differential equations for momentum and energy have been transformed by adopting the similarity transformation in linear form. The problem is solved numerically using an implemented package called bvp4c in MATLAB software. The numerical computations are carried out for various parameters of interest, which consists of the velocity ratio parameter, mixed convection parameter, nanoparticle volume fraction parameter and the needle size. A stability analysis of the solution is performed showing that the upper branch solution is stable, while the lower branch solution is unstable. Validation of the present work is done by comparing the current results with those available in the existing literature and found to be in excellent agreement.

In this study, we intend to present the dynamics of a system based on the model of convective heat and mass transfer in magnetohydrodynamics (MHD) flow past a moving vertical thin needle in nanofluid. The problem is formulated in mathematical form by using Buongiorno’s model with the modified boundary condition. The transformed boundary layer ordinary differential equations are solved numerically using the bvp4c function in MATLAB software. The effects of the involved parameters, including, Brownian motion, thermophoresis, magnetic field, mixed convection, needle size and velocity ratio parameter on the flow, heat and mass transfer coefficients are analyzed. The numerical results obtained for the skin friction coefficients, local Nusselt number and local Sherwood number, as well as the velocity, temperature and concentration profiles are graphically presented and have been discussed in detail. The study reveals that the dual solutions appear when the needle and the buoyancy forces oppose the direction of the fluid motion, and the range of the dual solutions existing depends largely on the needle size and magnetic parameter. The presence of the magnetic field in this model reduces the coefficient of the skin friction and heat transfer, while it increases the coefficient of the mass transfer on the needle surface. A stability analysis has been performed to identify which of the solutions obtained are linearly stable and physically relevant. It is noticed that the upper branch solutions are stable, while the lower branch solutions are not.

The steady boundary layer flow of a nanofluid past a thin needle under the influences of heat generation and chemical reaction is analyzed in the present work. The mathematical model has been formulated by using Buongiornos’s nanofluid model which incorporates the effect of the Brownian motion and thermophoretic diffusion. The governing coupled partial differential equations are transformed into a set of nonlinear ordinary differential equations by using appropriate similarity transformations. These equations are then computed numerically through MATLAB software using the implemented package called bvp4c. The influences of various parameters such as Brownian motion, thermophoresis, velocity ratio, needle thickness, heat generation and chemical reaction parameters on the flow, heat and mass characteristics are investigated. The physical characteristics which include the skin friction, heat and mass transfers, velocity, temperature and concentration are further elaborated with the variation of governing parameters and presented through graphs. It is observed that the multiple (dual) solutions are likely to exist when the needle moves against the direction of the fluid flow. It is also noticed that the reduction in needle thickness contributes to the enlargement of the region of the dual solutions. The determination of the stable solution has been done using a stability analysis. The results indicate that the upper branch solutions are linearly stable, while the lower branch solutions are linearly unstable. The study also revealed that the rate of heat transfer is a decreasing function of heat generation parameter, while the rate of mass transfer is an increasing function of heat generation and chemical reaction parameters.

The rotating boundary layer flow over a shrinking permeable surface in nanofluid is numerically studied. The similarity transformation is used to transform the partial differential equations into nonlinear ordinary differential equations. Later, these equations are determined by using bvp4c package in the MATLAB software. The numerical results reveal that there is more than one solution called dual solutions obtained for a certain range of the rotation and suction parameters. A stability analysis is performed to determine which solution is stable by depending on the sign of the eigenvalues. Based on this analysis, the results indicate that the upper branch solution (first solution) is linearly stable, while the lower branch solution (second solution) is linearly unstable. ABSTRAKAliran putaran lapisan sempadan terhadap permukaan mengecut yang telap di dalam nanobendalir dikaji. Penjelmaan keserupaan digunakan untuk menjelmakan persamaan perbezaan separa kepada persamaan perbezaan biasa tak linear. Kemudian, persamaan ini diselesaikan dengan menggunakan pakej bvp4c dalam perisian MATLAB. Keputusan menunjukkan bahawa terdapat lebih daripada satu penyelesaian yang disebut penyelesaian dwi diperolehi untuk julat tertentu bagi parameter putaran dan sedutan. Analisis kestabilan dijalankan untuk menentukan penyelesaian mana yang stabil dengan bergantung kepada tanda nilai eigen. Berdasarkan analisis ini, keputusan menunjukkan bahawa penyelesaian cabang atas (penyelesaian pertama) adalah stabil, sementara penyelesaian cabang bawah (penyelesaian kedua) adalah tidak stabil.

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