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Chapter 14: Problem 61

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

Expert verified
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

01

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.
02

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Let S be the disk enclosed by the curve \(C: \mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle,\)for \(0 \leq t \leq 2 \pi,\) where \(0 \leq \varphi \leq \pi / 2\) is a fixed angle. Use Stokes' Theorem and a surface integral to find the circulation on \(C\) of the vector field \(\mathbf{F}=\langle-y, x, 0\rangle\) as a function of \(\varphi .\) For what value of \(\varphi\) is the circulation a maximum?

Consider the potential function \(\varphi(x, y, z)=G(\rho),\) where \(G\) is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}} ;\) therefore, \(G\) depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\) b. Let \(S\) be the sphere of radius \(a\) centered at the origin and let \(D\) be the region enclosed by \(S\). Show that the flux of \(\mathbf{F}\) across \(S\) is $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2} G^{\prime}(a) $$ c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use part (c) to show that the flux across \(S\) (as given in part (b)) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.)

Use Green's Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. \(\oint\left(2 x+e^{y^{2}}\right) d y-\left(4 y^{2}+e^{x^{2}}\right) d x,\) where \(C\) is the boundary of the square with vertices \((0,0),(1,0),(1,1),\) and (0,1)

Consider the vector field \(\mathbf{F}=\langle y, x\rangle\) shown in the figure. a. Compute the outward flux across the quarter circle \(C: \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle,\) for \(0 \leq t \leq \pi / 2\) b. Compute the outward flux across the quarter circle \(C: \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle,\) for \(\pi / 2 \leq t \leq \pi\) c. Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b). e. What is the outward flux across the full circle?

Prove Green's Second Identity for scalar-valued functions \(u\) and \(v\) defined on a region \(D\) : $$\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S$$ (Hint: Reverse the roles of \(u\) and \(v\) in Green's First Identity.)
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