Navier Stokes Equation

Overview

Engineering reference data for Navier Stokes Equation in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Physical Meaning and Derivation

The Navier-Stokes equations are derived from applying Newton's second law to fluid motion, assuming that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term. They describe how the velocity field of a fluid substance evolves over time and space.

Vector Form and Components

The equations are often expressed in vector form for generality. For an incompressible fluid with constant density (ρ) and viscosity (μ), they are:

Continuity Equation (Conservation of Mass):

$$\nabla \cdot \mathbf{u} = 0$$

Momentum Equation:

$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$$

Where:

• \mathbf{u} is the velocity vector field
• p is the pressure field
• \mathbf{f} represents body forces (e.g., gravity, often \rho \mathbf{g})

This vector equation encapsulates the three scalar momentum equations (one for each spatial dimension) referenced in the original Key Formulas section.

Non-Dimensional Form and Reynolds Number

For engineering analysis, the equations are often non-dimensionalized. Using characteristic length (L), velocity (U), and time (L/U) scales, the momentum equation becomes:

$$\frac{\partial \mathbf{u}^*}{\partial t^*} + \mathbf{u}^* \cdot \nabla^* \mathbf{u}^* = -\nabla^* p^* + \frac{1}{Re} \nabla^{*2} \mathbf{u}^* + \mathbf{f}^*$$

The Reynolds Number (Re = \frac{\rho U L}{\mu}) emerges as the single governing parameter that balances inertial and viscous forces. This is critical for predicting flow regimes (laminar vs. turbulent).

Governing Equation Forms

The equations can be written in different coordinate systems depending on the problem geometry. Common engineering forms include:

Cartesian Coordinates (x, y, z): The x-momentum component, for example, is:

$$\rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + f_x$$

Cylindrical Polar Coordinates (r, θ, z): Often used for pipe flow or axisymmetric problems. The radial (r) momentum component is:

$$\rho \left( \frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_\theta}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_\theta^2}{r} + u_z \frac{\partial u_r}{\partial z} \right) = -\frac{\partial p}{\partial r} + \mu \left( \nabla^2 u_r - \frac{u_r}{r^2} - \frac{2}{r^2}\frac{\partial u_\theta}{\partial \theta} \right) + f_r$$

Applications and Simplifications

Engineers rarely solve the full equations directly. Key simplifications for analytical or computational solutions include:

• Stokes Flow (Re less than 1): Neglects the non-linear inertia term (\mathbf{u} \cdot \nabla \mathbf{u}), leading to linear equations.
• Euler Equations (Inviscid, μ=0): Neglects viscous terms, suitable for high-speed external flows.
• Boundary Layer Equations: Simplifies the full equations for thin viscous regions near surfaces.

References

• Original Source