Stability Analysis of Gravity‐Modulated Thermohaline Convection in Viscoelastic Fluid Saturating Porous Medium

ABSTRACT This paper investigates the modified stability analysis of thermohaline convection in a horizontal Oldroyd‐B fluid using the Darcy model, with applications in geophysics, environmental engineering, material science, and biological systems. The basic equations of motion and heat conduction in the present model are based on the Darcy model and the modified Boussinesq approximation. The perturbation equations are derived using the power series perturbation method to study two‐dimensional convective roll instability with finite amplitude disturbances. Linear stability analysis, as a first‐order stability problem, provides expressions for the Rayleigh numbers for stationary and oscillatory convection. The influence of various parameters, including the Darcy–Prandtl number, solutal Rayleigh number, stress and strain retardation parameters, and variations in the coefficient of specific heat, on the stability of the considered system, is studied numerically. It is observed that an approximate 67% increase in the coefficient of specific heat variations and a 23% increase in the solutal Rayleigh number result on average increases of 80% and 8%, respectively, in the value of the oscillatory Darcy–Rayleigh number. Conversely, an average 42% increase in the Darcy–Prandtl number leads to an average decrease of 8% in the oscillatory Darcy–Rayleigh number. In the weakly nonlinear oscillatory analysis, second‐ and third‐order stability problems are discussed, and the Landau equation is derived, which describes the amplitudes of the convection cells. The effects of the parameters on heat and mass transfer rates, as described by the Nusselt and Sherwood numbers, are studied numerically. Furthermore, the weakly nonlinear stability analysis evaluates the effects of the Darcy–Prandtl number, specific heat coefficient, stress relaxation time, salinity Rayleigh number, strain retardation time, diffusivity ratio, and gravity modulation characteristics on heat and mass transfer rates. The effects of amplitude and frequency of gravity modulation on heat and mass transport are analyzed and depicted graphically. The study establishes that heat and mass transport can be effectively controlled by a mechanism external to the system. The results are validated by comparison with previous work, which considered a special case without the velocity gradient term in the momentum equation.