This article emphasizes the roles of Joule heating and inclined magnetic field on the Ti-alloy nanofluid towards an exponentially permeable shrinking/stretching surface for the first time. Adopting the Tiwari and Das model, this nanofluid incorporates water as the base fluid and Ti-alloy as the nanoparticles in the mathematical formulation. Non-dimensional form of resultant flow governing equations is solved numerically for dual solutions using the Shooting method along with the Runge-Kutta fourth order technique and also the stability of the system is verified through the eigenvalue approach. The streamlines and eigenvalue patterns are provided to show the stability of obtained solutions and the significance of the problem undertaken. In these solutions, the first solution is found to be realistic and stable whereas the second solution is not stable for each combination of inclined magnetic, Joule heating, stretching/shrinking, Ti-alloy volume fraction, and suction parameters within the limited ranges. The existence of a flow separation point is identified between the shrinking and stretching regions. Finally, the delay of boundary layer separation is pointed out with the enhancing values of volume fraction of Ti-alloy nanoparticles and magnetic parameter in the presence of suction. This kind of analysis plays a very important role in the fields of aerodynamics, medical, and space sciences.

In this article, the authors provided the stability analysis for Darcy porous medium saturated by a Ti-alloy nanofluid in the presence of tilted magnetic field, Joule heating, heat generation, and velocity slip effects. The Tiwari-Das model is implemented and the relevant similarity transformations are identified for mathematical modeling of the proposed fluid flow problem. The set of resultant ODEs has been solved using the shooting method in conjunction with the RK-4th order scheme. The existence of dual solutions is observed along with their critical values for the present setup at and after the bifurcation point but for a limited range of certain fixed parameters. Through the eigenvalue approach, it is shown that the first solutions are stable, but not the second solutions. The streamlines are depicted to understand the flow patterns in addition to the identification of flow separation point and flow reversal. The important findings are as follows: (a) the delay of boundary layer separation is pointed out with the enhancing values of volume fraction of Ti-alloy nanoparticles and the inclined magnetic parameter in the presence of suction, (b) the Ti-alloy nanofluids velocity rises but the temperature decays with rising values of the Ti-alloy nanoparticle volume fraction, and (c) the value of smallest eigenvalues is growing with the growing values of magnetic parameter. Finally, this type of research may be extremely beneficial in the fields of aerodynamics (i.e. production of engine components, aircraft turbines, and high-performance automatic parts) and medicine.

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The impacts of tilted magnetic field and Joule heating on a Ti-alloy nanofluid towards an exponentially permeable stretching/shrinking surface have been looked into in this article. The Tiwari and Das model is adopted for the nanofluid where water is taken as the base fluid and Ti-alloy as the nanoparticles. The dual solutions of the resultant non-dimensional flow equations are evaluated using Shooting and 4th order Runge-Kutta methods and then linear temporal stability analysis is conducted to verify its stability through the smallest eigenvalue approach. The graphical representation of the results for the Ti-alloy/water nanofluid is presented to illustrate interesting features and its stability in the presence of physically effective parameters like inclined magnetic, Joule, volume fraction, and suction parameters. Outcomes of the numerical findings indicate that the dual/multiple solutions are possible only within the limited range of inclined magnetic and suction parameters. Through eigenvalue patterns, it is noticed that the 1st solution is realistic and stable while the 2nd solution is unreliable for each combination. In addition, the streamlines are also displayed to visualize the flow patterns of the Ti-alloy nanofluid. Also, the flow separation point is found in between the shrinking and stretching regions. Finally, the delay of boundary layer separation is pointed out with the enhancing values of volume fraction of Ti-alloy nanoparticles and magnetic parameter in the presence of suction. This kind of analysis performs a very crucial role in the medical sector, aerodynamics and space sciences.