Question

The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the flow in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is $$\rho\left(\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V} \cdot \nabla) \mathbf{V}\right)=-\nabla p+\mu(\nabla \cdot \nabla) \mathbf{V}.$$ In this notation, \(\mathbf{V}=\langle u, v, w\rangle\) is the three-dimensional velocity field, \(p\) is the (scalar) pressure, \(\rho\) is the constant density of the fluid, and \(\mu\) is the constant viscosity. Write out the three component equations of this vector equation. (See Exercise 40 for an interpretation of the operations.)

Short answer

Question: Write out the three component equations of the given Navier-Stokes equation for incompressible and viscous flow. Answer: u-component: $$ \rho\left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = -\frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right), $$ v-component: $$ \rho\left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right) = -\frac{\partial p}{\partial y} + \mu\left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right), $$ w-component: $$ \rho\left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu\left(\frac{\partial^2 w}{\partial x^2} +\frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right). $$

Step-by-step solution

Identify the equation components

The given Navier-Stokes equation is: $$ \rho\left(\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V} \cdot \nabla) \mathbf{V}\right)=-\nabla p+\mu(\nabla \cdot \nabla) \mathbf{V}, $$ where \(\mathbf{V}=\langle u, v, w\rangle\) is the velocity field, \(p\) is the scalar pressure, \(\rho\) is the fluid density, and \(\mu\) is the viscosity.

Write out the equations for each component

We will rewrite the equation for each component of the velocity field (u, v, w). For the u-component: $$ \rho\left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = -\frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right), $$ For the v-component: $$ \rho\left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right) = -\frac{\partial p}{\partial y} + \mu\left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right), $$ For the w-component: $$ \rho\left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu\left(\frac{\partial^2 w}{\partial x^2} +\frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right). $$ These are the three component equations of the given Navier-Stokes equation.

Key concepts

Fluid Dynamics

Fluid dynamics is the study of fluids (liquids and gases) in motion. It explores how fluids move and the forces acting upon them. The Navier-Stokes equation is a central part of fluid dynamics, modeling how fluids flow in various conditions.

In fluid dynamics, understanding the velocity field is key. Velocity describes how fast and in which direction fluid particles are moving.

The three-dimensional velocity field is usually represented by \( \mathbf{V} = \langle u, v, w \rangle \) where:

• \( u \) corresponds to the velocity in the x-direction
• \( v \) is the velocity component in the y-direction
• \( w \) represents the velocity along the z-axis

These components are crucial for modeling how fluid moves in a 3D space. By breaking down the equations into components, we can analyze each direction separately, making it easier to predict fluid behavior.

Vector Calculus

Vector calculus deals with mathematical operations on vector fields, which are essential tools in fluid dynamics. The Navier-Stokes equation involves several vector calculus operations that describe how forces and velocities are distributed through the fluid.

The key operations in vector calculus used here are:

• The gradient \( abla \) measures how a function changes at each point in space. In the context of Navier-Stokes, it shows how pressure changes across the fluid.
• The divergence \( abla \cdot abla \) applies to a vector, indicating how much a vector field spreads out or converges.
• The expression \( (\mathbf{V} \cdot abla) \mathbf{V} \) defines convective acceleration, describing how fluid velocity changes along the flow path.

These concepts allow us to describe fluid flow not just in terms of magnitude but also direction and change, which are vital for accurately predicting fluid behavior.

Partial Differential Equations

Partial differential equations (PDEs) involve unknown functions with partial derivatives. They describe various phenomena, including fluid flow in the Navier-Stokes equation.

The Navier-Stokes equation is a PDE that consists of:

• Time derivatives \( \frac{\partial \mathbf{V}}{\partial t} \) which relate to how the velocity field changes over time
• Spatial derivatives such as \( \frac{\partial u}{\partial x} \), \( \frac{\partial v}{\partial y} \), depicting how velocity changes in space

By turning the equation into separate component equations for \( u \), \( v \), and \( w \), each direction of flow can be analyzed. This breakdown helps solve complex real-world fluid dynamics problems by addressing specific aspects of flow patterns and effects of pressure and viscosity over time.