Simplifying the Navier-Stokes Equations to Inviscid Flow and Bernoulli's Law

Understanding fluid dynamics involves dealing with the Navier-Stokes (N-S) equations, a fundamental set of equations in fluid mechanics. These equations describe the motion of fluid substances and encompass both compressible and incompressible flow, taking into account factors such as viscosity. However, in many practical scenarios, it is useful to simplify these equations to their inviscid form, dropping the viscous terms. In this article, we will explore the process of simplifying the N-S equations to the incompressible inviscid form and derive Bernoulli's law from it.

From Navier-Stokes to Inviscid Flow

The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. The momentum equation, one of the main equations in the N-S set, is given by:

F ρ(?v/?t v·?v) - ?p μ?2v f

where:

ρ is the fluid density, which can be variable v is the fluid velocity ?v/?t is the substantial derivative ?p is the pressure gradient μ is the dynamic viscosity ?2v is the Laplacian of the velocity field f is the external non-viscous force per unit mass

To simplify this to an inviscid flow form, we drop the viscous terms:

F ρ(?v/?t v·?v) - ?p f

Now, we divide this equation by ρ:

(?v/?t v·?v) - (1/ρ) ?p (1/ρ) f

This form is known as the Euler equations. The key advantage of these equations is that they describe the flow without the dissipative effects of viscosity, making them useful in a wide range of applications where these effects can be neglected.

From Euler Equations to Bernoulli's Law

The next step is to simplify the Euler equations even further, leading us to Bernoulli's law. To achieve this, we first consider the tensor identity for the gradient of velocity (v):

?v (?v (?v)T) (?v - (?v)T)

where:

(?v (?v)T) is symmetric (the rotational part) (?v - (?v)T) is anti-symmetric (the vorticity part)

For incompressible flow, the vorticity part (?v - (?v)T) is perpendicular to the direction of streamline (denoted as l), hence it vanishes. This simplifies the tensor identity to:

?v (?v (?v)T)

Considering only the component of the tensor ?v along the streamline (l), we get:

div(?x(v · l)) (v · ?) (v · l)

From the Euler equations, we know that:

(?v/?t v·?v) - (1/ρ) ?p (1/ρ) f

Multiplying both sides by the direction of streamline l and integrating along a streamline, we obtain:

dp -g dl

where g is the conservative external force per unit mass. This is Bernoulli's law in its most general form, applicable to any fluid (gas or liquid).

Energy Interpretation and Steady-State Considerations

Bernoulli's law can be interpreted in terms of mechanical energy. The left-hand side of the equation can be interpreted as the change in mechanical energy along the streamline. For steady flow, the first term on the left-hand side vanishes, and the equation simplifies to:

(1/2)ρv2 (zg) constant

This is the well-known Bernoulli's equation, which is commonly used in fluid dynamics and physics to analyze pressure and velocity profiles in fluid systems.

Practical Applications and Limitations

Bernoulli's law is particularly useful in practical applications where the effects of viscosity can be neglected. For example, in the case of frontal stagnation, the pressure in the viscous boundary layer is uniform and directly transmitted to the wall. However, for internal pipe flow, the total load of fluid pressure, kinetic energy, and potential energy is not constant but decreases due to viscous effects. This is reflected in the loss of energy in the system, which engineers refer to as "pressure" or "head" losses.

Conclusion and Further Reading

In this article, we have explored the process of simplifying the Navier-Stokes equations to an inviscid flow form and derived Bernoulli's law from it. The inviscid flow and Bernoulli's law provide valuable insights into fluid dynamics, making them important tools in both theoretical analysis and practical applications. For those interested in delving deeper into the analytical side of fluid mechanics, I recommend consulting some classic texts on the subject. These old books often provide more in-depth derivations and theoretical analyses, which are available in two notable references:

Fluid Mechanics by Frank M. White Applied Hydrodynamics by Brian D. Smith

While modern engineering books often focus on practical applications, these older texts can provide a richer understanding of the fundamental principles of fluid dynamics.