Magnetohydrodynamic study of eddy formation and heat transfer in a bottom heated two-dimensional lid-driven cavity with the insulated sidewall is reported in this paper. We consider the classic problem of a two-dimensional lid driven cavity in the presence of an applied external magnetic field. In the standard nonlinear Navier–Stokes equation and the energy equation, appropriate terms are added to model the applied magnetic field and thermo-magnetic convection. And we have derived a higher-order numerical scheme that can provide solutions that are fourth-order accurate. The scheme is implemented with suitable fourth-order accurate boundary conditions. The square LDC enclosure is filled with a fluid of Pr = 6.2. The top lid of the cavity is kept moving with constant velocity. The effect of magnetic field on heat transfer and eddy formation is discussed for 0 ≤ Ha ≤ 250, Re = 10 2 ≤ Re ≤ 10 3 and 103 ≤ Gr ≤ 104 and the results are presented in terms of contour plots of streamlines, isotherms, Lorentz force, and the kinetic energy. The physical mechanism for forming multiple recirculation zones is understood and attributed to two opposite forces acting in the fluid about the center-line of the cavity, which acts like a couple (and hence rotational flow). This is discovered by computing the Lorentz force acting in the entire domain of fluid flow. Similarly, from the computation of kinetic energy of the fluid at all the grid points in the fluid region, it is revealed that conduction is the dominant heat transfer mechanism at higher magnetic fields. At the same time, convection is the prime mode at low magnetic fields. It is ascertained that with an increase in magnetic field intensity, velocity profiles are suppressed, and heat transfer stabilizes, giving rise to the formation of recirculation zones in the cavity.

With the use of cubic, quintic, and septic nonlinearities, we demonstrate the influence of modified nonlinear saturation on modulational instability (MI) in a nonlinear complex parity–time (PT)-symmetric fiber Bragg grating (FBG) structure. Using a modified coupled nonlinear Schrodinger equation and linear stability analysis, we derive a dispersion relation for instability gain spectra in a complicated PT-symmetric system. Our main aim is to examine the MI in non-Kerr nonlinearities with nonlinear saturation in three PT-symmetric regimes: below threshold point, at threshold point (breaking point), and above threshold point. The occurrence of MI is known to be problematic at the PT-symmetry threshold point in a standard FBG structure (A.K. Sharma, 2014). At the same time, MI can exist in the normal group velocity dispersion domain when the modified nonlinear saturation effect is used. With the help of a modified form of saturable nonlinearity, we discovered that MI could exist in all three regimes in a complex PT-symmetric FBG structure. In anomalous group velocity dispersion alone, we found bistability behavior in a PT-symmetric FBG structure with higher-order saturable nonlinearity. In the presence of a modified nonlinear saturation effect and higher-order non-Kerr nonlinearities, we found a novel type of dynamics in the PT-symmetric FBG structure. All alterations in the photonic device bandgap directly result from changes in the refractive index of the medium caused by the interaction of PT-symmetric potential with the cubic–quintic–septic and modified form of nonlinear saturation. As a result, we provide approaches for generating and managing the MI in a complex PT-symmetric FBG structure under the influence of the modified nonlinear saturation effect.