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We present a new exact solution of the thermal diffusion equations for steady-state shear flows of a binary fluid. Shear fluid flows are used in modeling and simulating large-scale currents of the world ocean, motions in thin layers of fluid, fluid flows in processes, and apparatuses of chemical technology. To describe the steady shear flows of an incompressible fluid, the system of Navier–Stokes equations in the Boussinesq approximation is redefined, so the construction of exact and numerical solutions to the equations of hydrodynamics is a very difficult and urgent task. A non-trivial exact solution is constructed in the Lin-Sidorov-Aristov class. For this class of exact solutions, the hydrodynamic fields (velocity field, pressure field, temperature field, and solute concentration field) were considered as linear forms in the x and y coordinates. The coefficients of linear forms depend on the third coordinate z. Thus, when considering a shear flow, the two-dimensional velocity field depends on three coordinates. It is worth noting that the solvability condition given in the article imposes a condition (relation) only between the velocity gradients. A theorem on the uniqueness of the exact solution in the Lin–Sidorov–Aristov class is formulated. The remaining coefficients of linear forms for hydrodynamic fields have functional arbitrariness. To illustrate the exact solution of the overdetermined system of Oberbeck–Boussinesq equations, a boundary value problem was solved to describe the complex convection of a vertical swirling fluid without its preliminary rotation. It was shown that the velocity field is highly stratified. Complex countercurrents are recorded in the fluid. [ABSTRACT FROM AUTHOR]

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn from one and apply it to the other. The objective of this note is to convert ordinary stochastic processes into interval stochastic processes due to the wide range of applications in various disciplines. We have developed Hermite–Hadamard ( H. H ), Ostrowski-, and Jensen-type inequalities using interval h-convex stochastic processes. Our main results can be applied to a variety of new and well-known outcomes as specific situations. The results of this study are expected to stimulate future research on inequalities using fractional and fuzzy integral operators. Furthermore, we validate our main findings by providing some non-trivial examples. To demonstrate their general properties, we illustrate the connections between the examined results and those that have already been published. The results discussed in this article can be seen as improvements and refinements to results that have already been published. This is a fascinating subject that can be investigated in the future to identify equivalent inequalities for various convexity types. [ABSTRACT FROM AUTHOR]