The influence of trigonometric cosine, square, sawtooth, and triangular wave types of magnetic-field modulation in nanoliquid within Hele-Shaw cell is studied in this paper utilizing linear/nonlinear explorations. The solvability condition to the third-order solution of the referred model equation has been imposed to get the cubic Ginzburg–Landau equation (GBL-equation) which is utilized to measure the rate of heat (or mass) transfer. In the sequel, the influence of the nondimensional parameters is discussed graphically in detail. It is demonstrated that Prandtl number (Pr)/magnetic Prandtl number (Prm)/Lewis-number (Le)/redefined diffusivity-ratio (NA)/concentration Rayleigh-number (RS1) and magnitude of the magnetic-modulation (δ) destabilize the system, that is, the heat/mass transfer increases. On the other hand, nanoliquid magnetic-number (Q), Hele–Shaw number (Hs), and modulating-frequency (ω) stabilize the system. The outcomes demonstrate that the magnetic-field modulation can be imposed significantly to increase or decrease the heat/mass transfer.

This article uses linear/nonlinear studies to examine the effects of sine, square, and triangular waveforms of magnetic field modulation in Walter-B nanoliquid (electrically conducting) filled in a Hele-Shaw cell with throughflow. The applied magnetic field has periodic components that are both constant and time dependent and changes sinusoidally over time. In order to examine heat and mass transfer in the liquid layer, a local nonlinear concept has been used. Formulation of the autonomous simultaneous ordinary differential equations for the convection amplitude leads to the investigation of the heat/mass transportation factor. This convectional amplitude is calculated using Mathematica's built-in NDSolve tool, and the results have been validated using the Runge-Kutta-Fehlberg (RKF-45) technique. The Nusselt number is found in terms of numerous system parameters, and each parameter's consequence on heat/mass transportation is described in detail. Hele-Shaw number, nanoliquid magnetic number, and frequency of magnetic modulation have a stabilizing impact on the system, whereas elastic parameter, magnetic Prandtl number, nanoliquid Prandtl number, and amplitude of magnetic modulation have a destabilizing impact on the system. Additionally, it has been found that magnetic modulation can be utilized to efficiently regulate the heat and mass transport. The system's basic (conduction) state temperature profile and transportation of mass highly depend on throughflow.

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