Navier – Stokes equations: Why do they look so complex?

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In a nutshell, the Navier and Stokes equations describe a fluid's viscous flow.

The Navier-Stokes equations are a class of partial differential equations that describe the motion of viscous fluids.

Why do Navier-Stokes equations appear so complicated?

In one word, the reason Navier-Stokes equations appear complex is that momentum is a vector. Momentum has mass as well as direction. Navier-Stokes equations deal with four independent and six dependent variables.

Because momentum is a vector quantity with both a magnitude and a direction, dealing with it is more difficult than dealing with mass and energy. Momentum is not only dependent on the mass and speed of the target. In a given direction, velocity is distance, so an object's momentum, therefore, depends on the direction of motion. Since momentum is a vector characterizing momentum on three axes becomes complicated.

In three axes, the Navier-Stokes equations combine mass conservation and momentum conservation for Newtonian fluids. The problem has four independent variables: the x, y, and z spatial coordinates of some domain, as well as the time t. There are six dependent variables: pressure p, density r, and temperature T (which is included in the energy equation via the total energy Et) and three components of the velocity vector: u in the x-direction, v in the y-direction, and w in the z-direction. All four independent variables influence all of the dependent variables. As a result, the differential equations are partial differential equations rather than ordinary differential equations.

Explanation

The motion of viscous fluids

Viscosity

The viscosity of a moving fluid describes its internal friction. A viscous fluid resists motion because its molecular structure causes a lot of internal friction. In other words, due to strong cohesive forces between molecules, any layer in a moving fluid tries to drag the adjacent layer to move at the same speed, resulting in the viscosity effect.

In general, viscosity is affected by the state of a fluid, such as a temperature, pressure, and rate of deformation. However, in some cases, the reliance on some of these properties is negligible. The viscosity of a Newtonian fluid, for example, does not vary significantly with the rate of deformation. A fluid with no viscosity is referred to as ideal or inviscid.

Viscous drag and Reynolds number [N’R]

A resistance force called viscous drag FV is exerted on a moving object as a result of viscosity. This force is usually proportional to the speed of the object (in contrast with simple friction). Experiments have revealed that for laminar flow (N′R less than one), viscous drag is proportional to speed, whereas viscous drag is proportional to speed squared for N′R between about 10 and 10^6.

?The fact that an object falling through a fluid will not continue to accelerate indefinitely is an interesting result of the increase in FV with speed (as it would if we neglect air resistance, for example). Instead, viscous drag increases, slowing acceleration until a critical speed, known as the terminal speed, is reached and the object's acceleration is zero. When this occurs, the object continues to fall at a constant rate (the terminal speed).

The Navier-Stokes equations express momentum and mass conservation mathematically for Newtonian fluids. They are sometimes accompanied by a state equation relating pressure, temperature, and density. They result from applying Isaac Newton's second law to fluid motion, along with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—thus describing the viscous flow.

Explanation

Momentum and mass conservation for Newtonian fluids

The conservation of momentum, like the conservation of energy and the conservation of mass, is a fundamental concept in physics. Momentum is defined as the mass of an object multiplied by its velocity. The conservation of momentum principle states that the amount of momentum within a problem domain remains constant; momentum is neither created nor destroyed, but only changed by the action of forces as described by Newton's laws of motion. Because momentum is a vector quantity with both a magnitude and a direction, dealing with it is more difficult than dealing with mass and energy. At the same time, momentum is conserved in all three physical directions.

It is even more difficult when dealing with gas because forces in one direction can affect momentum in another due to many molecules colliding. On this slide, we will present a very simplified flow problem with only one direction of change. The problem is further simplified by assuming a constant flow that does not change over time and restricting the forces to those associated with pressure. Real-world flow problems are far more complex than this simple example.

The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids [A Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly correlated to the local strain rate—the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.

They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term.

How does the complexity arise?

Explanation

There are four equations

Continuity equation:

It is a time-dependent continuity equation for the conservation of mass. The equation explains how a fluid conserves mass in its motion.

?The differential form of the continuity equation is:

?ρ/?t+▽?(ρu)=0

Where, t = Time, ρ = Fluid density, u = flow velocity vector field.

Momentum equations in X, Y, and Z-axis: Root of complexity

There are four independent variables in the problem, the x, y, and z spatial coordinates of some domain, and the time t. There are six dependent variables; the pressure p, density r, and temperature T (which is contained in the energy equation through the total energy Et) and three components of the velocity vector; u component is in the x-direction, the v component is in the y-direction, and the w component is in the z-direction. All of the dependent variables are functions of all four independent variables. The differential equations are therefore partial differential equations and not ordinary differential equations, as explained right at the beginning of the post.

The terms on the left-hand side of the momentum equations are called the convection terms of the equations. Convection is a physical process that occurs in a flow of gas in which some property is transported by the ordered motion of the flow. The terms on the right-hand side of the momentum equations that are multiplied by the inverse Reynolds number are called diffusion terms. Diffusion is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Turbulence, and the generation of boundary layers, are the result of diffusion in the flow.

Notice that all of the dependent variables appear in each equation. To solve a flow problem, you have to solve all five equations simultaneously; that is why we call this a coupled system of equations.

Credit: Google