This paper investigates the steady, two dimensional, and magnetohydrodynamic flow of copper and alumina/water hybrid nanofluid on a permeable exponentially shrinking surface in the presence of Joule heating, velocity slip, and thermal slip parameters. Adopting the model of Tiwari and Das, the mathematical formulation of governing partial differential equations was constructed, which was then transformed into the equivalent system of non-linear ordinary differential equations by employing exponential similarity transformation variables. The resultant system was solved numerically using the BVP4C solver in the MATLAB software. For validation purposes, the obtained numerical results were compared graphically with those in previous studies, and found to be in good agreement, as the critical points are the same up to three decimal points. Based on the numerical results, it was revealed that dual solutions exist within specific ranges of the suction and magnetic parameters. Stability analysis was performed on both solutions in order to determine which solution(s) is/are stable. The analysis indicated that only the first solution is stable. Furthermore, it was also found that the temperature increases in both solutions when the magnetic parameter and Eckert number are increased, while it reduces as the thermal slip parameter rises. Furthermore, the coefficient of skin friction and the heat transfer rate increase for the first solution when the magnetic and the suction parameters are increased. Meanwhile, no change is noticed in the boundary layer separation for the various values of the Eckert number in the heat transfer rate.

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Enzymes have been widely used as efficient, eco-friendly, and biodegradable catalysts in organic chemistry due to their mild reaction conditions and high selectivity and efficiency. In recent years, the catalytic promiscuity of many enzymes in unnatural reactions has been revealed and studied by chemists and biochemists, which has expanded the application potential of enzymes. To enhance the selectivity and activity of enzymes in their natural or promiscuous reactions, many methods have been recommended, such as protein engineering, process engineering, and media engineering. Among them, the additive approach is very attractive because of its simplicity to use and high efficiency. In this paper, we will review the recent developments about the applications of additives to improve the catalytic performances of enzymes in their natural and promiscuous reactions. These additives include water, organic bases, water mimics, cosolvents, crown ethers, salts, surfactants, and some particular molecular additives.

In this paper, we examine the stochastic version of the D~ng-van der Pol equation. As in [2], [8], [19], [16], we introduce both multiplicative and additive stochastic excitations to the original Duttingvan der Pol equation, i.e.where, a and G are the bifurcation parameters, {1 and {~ are white noise processes with intensities ~1 and ~, respectively. The existence of the extrema of the probability density ~nction is presented for the stochastic system. The method used in this paper is essentiMly the same as that which was used in [19]. We first reduce the above system to a weakly perturbed conservative system by introducing an appropriate scaling. The corresponding unperturbed system is then studied. Subsequently, by transforming the variables and performing stochastic averaging, we obtain a one-dimensionM It6 equation for the Hamiltonian H. The probability density ~nction is found by solving the Fokker-Planck equation. The extrema of the probability density function are then calculated in order to study the so-cMled P-Bifurcation. The bifurcation diagrams for the stochastic version of the D~G~ng-van der Pol oscillator with a -1.0, b -1.0 over the whole (a, G)-plane are given. The related mean exit time problem is also studied.

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