This work analyzes the unsteady two-dimensional nanofluid flow over a vertical stretching permeable surface in the presence of an inclined magnetic field and nonuniform heat source/sink. Four different types of nanoparticles, namely silver Ag, copper Cu, alumina Al 2 O 3 , and titania TiO 2 , are considered by using water as a base fluid with the Prandtl number Pr = 6.785. The governing partial differential equations are transformed to coupled non-linear ordinary differential equations by appropriate similarity transformation. Furthermore, the similarity equations are solved numerically by using the fourth-order Runge-Kutta integration scheme with Newton Raphson shooting method. A comparison of obtained numerical results is made with previously published results in some special cases, and excellent agreement is noted. Numerical results for velocity and temperature profiles as well as skin friction coefficient and local Nusselt number are discussed for various values of physical parameters. It tends to be discovered that, the magnetic field inclination angle γ has the capability to strengthens the magnetic field and reduce the velocity profile of the flow. Also, it can be found that, by using various types of nanofluids, velocity and temperature distributions change, which means that the nanofluids are important in the cooling and heating processes. The thermal boundary layer thickness is related to the increased thermal conductivity of different types of nanofluids, i.e., the minimum (maximum) value of the temperature is obtained by adding titanium oxide (silver) to the fluid as the nanoparticles.

Purpose This paper aims to investigate a mathematical model with numerical simulation for bacterial growth in the heart valve. Design/methodology/approach For antibacterial activities and antibodies properties, nanoparticles have been used. As antibiotics are commonly thought to be homogeneously dispersed through the blood, therefore, non-Newtonian fluid of Casson micropolar blood flow in the heart valve for two dimensional with variable properties is used. The heat transfer with induced magnetic field translational attraction under the influence of slip is considered for the resemblance of the heart valve prosthesis. The numeral results have been obtained by using the Chebyshev pseudospectral method. Findings It is proven that vascular resistance decreases for increasing blood velocity. It is noted that when the magnetic field will be induced from the heart valve prosthesis then it may cause a decrease in vascular resistance. The unbounded molecules and antibiotic concentration that are able to penetrate the bacteria are increased by increasing values of vascular resistance. The bacterial growth density cultivates for upswing values of magnetic permeability and magnetic parameters. Originality/value To the best of the authors’ knowledge, this is the first study to investigate a mathematical model with numerical simulation for bacterial growth in the heart valve.

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

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