From Navier–stokes Equations to Euler, Bernoulli, Etc

From Navier-Stokes Equations to Euler, Bernoulli, etc

Fluid dynamics is a branch of physics that deals with the study of fluids, their motions and related phenomena. It encompasses a wide range of topics, from the behavior of liquids and gases at rest to their interactions with solid objects and other fluids. The Navier-Stokes equations, named after Claude-Louis Navier and Sir George Gabriel Stokes, are a set of partial differential equations that describe the motion of fluids in a given region of space. These equations are fundamental to the study of fluid dynamics and are widely used in many fields of engineering and science.

The Navier-Stokes Equations

The Navier-Stokes equations describe the motion of fluid particles in a given region of space. They are a set of partial differential equations that relate the rates of change of various properties of the fluid, such as velocity, pressure, density, and temperature, to each other. These equations are based on the principles of conservation of mass, momentum, and energy.

The Navier-Stokes equations in their full form are:

∂ρ/∂t + ∇·(ρv) = 0
ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f
∂e/∂t + ∇·(e + pv) = ∇·(k∇T) + Q
p = (γ − 1)e

Here, ρ is the density of the fluid, v is its velocity, p is the pressure, e is the internal energy per unit mass, T is the temperature, γ is the ratio of specific heats, μ is the viscosity coefficient, k is the thermal conductivity coefficient, and Q is the heat dissipation per unit volume. The symbol ∇ represents the gradient operator.

The Simplified Euler Equations

The Euler equations are a simplified form of the Navier-Stokes equations that describe the motion of an inviscid, adiabatic fluid. In other words, they assume that the fluid has no viscosity and that there is no heat transfer between the fluid and its surroundings. These assumptions make the equations easier to solve and provide a good approximation for many fluid dynamic problems.

The Euler equations in their simplified form can be expressed as:

∂ρ/∂t + ∇·(ρv) = 0
ρ(∂v/∂t + v·∇v) = −∇p
p = (γ − 1)e

Here, the assumptions of zero viscosity and adiabatic conditions have led to the removal of the terms related to viscosity and heat transfer in the Navier-Stokes equations. The remaining terms relate the rates of change of density, velocity, pressure, and internal energy to each other.

Bernoulli’s Equation

Bernoulli’s equation is a simple relationship between the pressure, velocity, and elevation of a fluid in a steady flow. It is based on the principle of conservation of energy and is widely used in many practical applications, such as the design of airplane wings and the calculation of fluid flow in pipes.

Bernoulli’s equation can be expressed as:

p + (1/2)ρv² + ρgh = constant

Here, p is the pressure, v is the velocity, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the elevation above a reference point. The constant on the right-hand side of the equation is the same for any two points along a streamline in a steady flow. This equation shows that as the velocity of the fluid increases, the pressure decreases, and vice versa.

Other Fluid Dynamic Equations

In addition to the Navier-Stokes equations, Euler equations, and Bernoulli’s equation, there are many other fluid dynamic equations that are used in various fields of engineering and science. Some of these equations include:

• The continuity equation, which relates the rate of change of fluid mass to the divergence of the fluid velocity field. This equation is a fundamental principle of fluid dynamics and is used to study the behavior of fluids in various situations.
• The energy equation, which relates the rate of change of fluid energy to heat transfer, work done by the fluid, and changes in the fluid’s internal energy. This equation is used to study the behavior of fluids in systems such as engines and heat exchangers.
• The momentum equation, which relates the rate of change of fluid momentum to the forces acting on the fluid. This equation is used to study the motion of fluids in various situations, such as the flow of air over an airplane wing or the flow of liquid through a pipe.

Conclusion

In conclusion, the Navier-Stokes equations are fundamental to the study of fluid dynamics and describe the motion of fluid particles in a given region of space. The simplified Euler equations assume that the fluid is inviscid and adiabatic, making them easier to solve for many practical applications. Bernoulli’s equation is a simple relationship between pressure, velocity, and elevation in a steady flow and is widely used in many fields of engineering and science. Other fluid dynamic equations, such as the continuity equation, energy equation, and momentum equation, are used to study the behavior of fluids in various situations.