The Navier-Stokes equations describe the motion of fluid substances such as liquids and gases. They are a set of nonlinear partial differential equations that arise from applying Newton's second law to fluid motion, incorporating the effects of viscosity. Below, we will derive the Navier-Stokes equation and solve it for a simple case.

1. Derivation of the Navier-Stokes Equation

The Navier-Stokes equations can be derived from the principles of conservation of momentum and mass. The general form for an incompressible fluid is given by:

        ρ(∂u/∂t + u · ∇u) = -∇p + μ∇²u + f
    

Where:

• ρ = density of the fluid
• u = velocity vector of the fluid (u, v, w)
• p = pressure
• μ = dynamic viscosity
• f = body forces per unit volume (e.g., gravity)

The equation can be broken down into the following components:

• Convective acceleration:u · ∇u
• Local acceleration:∂u/∂t
• Pressure gradient:-∇p
• Viscous forces:μ∇²u
• External forces:f

2. Simplifying Assumptions

For illustrative purposes, let's simplify the Navier-Stokes equations by considering a two-dimensional, incompressible, steady flow in a rectangular channel with no external forces. We can express the Navier-Stokes equations as follows:

        ρ(∂u/∂t + u ∂u/∂x + v ∂u/∂y) = -∂p/∂x + μ(∂²u/∂x² + ∂²u/∂y²)
    

Assuming steady state (∂u/∂t = 0) and uniform density, we can simplify this to:

        0 = -∂p/∂x + μ(∂²u/∂x² + ∂²u/∂y²)
    

Rearranging gives us:

        ∂p/∂x = μ(∂²u/∂x² + ∂²u/∂y²)
    

3. Solving the Equation

To solve this equation, we can apply boundary conditions. For instance, if we consider a flow in a channel where:

• The velocity at the wall (y=0 and y=h) is zero (no-slip condition).
• The velocity profile is uniform across the width of the channel.

Let’s take a simple case where the velocity profile is parabolic, typical of laminar flow in a channel:

        u(y) = Umax(1 - (y/h)²)
    

where Umax is the maximum velocity at the center of the channel.

Example Calculation

Using the parabolic velocity profile, we can substitute into our equation:

        ∂²u/∂y² = -2Umax/h²
    

Now substituting this back into our rearranged Navier-Stokes equation:

        ∂p/∂x = μ(-2Umax/h²)
    

This implies that the pressure gradient is constant along the channel:

        p(x) = -2μUmax/h² * x + C
    

where C is a constant determined by boundary conditions.

4. Conclusion

The Navier-Stokes equations are fundamental to fluid mechanics and can be quite complex depending on the flow regime and boundary conditions. The example provided illustrates a basic scenario of laminar flow in a channel, leading to a parabolic velocity profile and a linear pressure gradient.

References

• Batchelor, G.K. (2000). Introduction to Fluid Dynamics. Cambridge University Press.
• White, F.M. (2011). Fluid Mechanics. McGraw-Hill.
• Landau, L.D., & Lifshitz, E.M. (1987). Fluid Mechanics. Pergamon Press.