Navier Strokes Equations

Navier strokes equations describe the motion of fluid substances. It is named after Claude Louis navier and George Gabriel strokes. Navier stroke equation arise from applying Newton’s second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term, plus a pressure term.

A solution of the Navier stokes equations is called the velocity field of flow fluid, it describes the velocity of fluid at a given point in space and time, it dictates not position but rather velocity. The other quantities of interest may be found once the velocity field is solved. Instead of position, studying velocity makes more sense for a fluid; one can compute various trajectories for visualization purposes.

Uses

 This equation will describe the physics of many things of educational and economic interest. This is used to model the weather, ocean currents, water flow in pipe and air flow around a wing. The navier strokes equations in their full forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. When they are coupled with Maxwell’s equations they can be used to model and study magnetohydrodynamics.

Nonlinearity

In almost every real situation, navier strokes equations are nonlinear partial differential equations. The equation can be simplified to linear equations in some cases, such as one dimensional flow and strokes flow. The non linearity makes the most problem difficult to solve and main contributor to the turbulence that the equations model. The nonlinearity is because of convective acceleration, it means the acceleration along with the change in velocity over position. The convective flow can be either turbulent or laminar it will involve nonlinearity. For example convective but laminar flow would be the passage of a viscous fluid (oil) through a small converging nozzle.

Turbulence

Turbulence is nothing but, it is the time dependent disordered behavior that can be seen in many fluid flows. The numerical solution of the navier strokes equations for the turbulent flow is extremely difficult, and due to the significantly different mixing length scales are involved in turbulent flow, the stable solution of this requires a fine mesh resolution that the computational time becomes significantly infeasible for calculation. To solve the turbulent flow using the laminar solver typically results in a time unsteady solutions and fails to converge approximately. Reynolds averaged Navier strokes equations supplemented with turbulence models, are used in practical navier strokes equations to counter this.

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