Numerous scientific and engineering disciplines, such as aerospace, environmental science, and materials engineering, depend heavily on fluid dynamics. For defining and analysing the intricate behaviour of fluids in these applications, PDEs offer a potent framework. The overview of the basic equations regulating fluid dynamics, such as the continuity equation, Navier-Stokes equations, and energy equation, comes first in the analysis. These equations make up a collection of linked nonlinear PDEs that depict how mass, momentum, and energy are conserved in fluid systems. We look at the mathematical characteristics of these PDEs, including their well-posedness and existence of solutions. Additionally, various numerical approaches, including as spectral, finite element, and finite difference methods, are investigated for solving these PDEs. Fluid dynamics simulations are used to discuss the benefits and drawbacks of each strategy. Additionally, methods for dealing with stability problems, discretization mistakes, and boundary conditions are researched. The investigation of particular applications of fluid dynamics modelling, including flow in porous media, turbulent flows, and multiphase flows, is also covered in this work. Examined are the difficulties brought on by these applications and the resulting adjustments to the governing PDEs.
Key words: Differential Equation, Fluid Dynamics, Nonlinear equation, mathematical modelling.
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