Session 69: Flux in 2D

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[2020-02-06 Thu 16:33]

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Session 69: Flux in 2D | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare

1 Chalkboard

2 Find $M$ and $N$ equals the polar moment of inertia

2.1 Front

Find $M$ and $N$ equals the polar moment of inertia

• \({\displaystyle \oint_C M \dd{x} + N \dd{y}}\)
• Density: uniform
• Region in the plane boundary $C$

2.2 Back

• Let $R$ be the region enclosed by $C$ and $\phi$ be the uniform density of $R$
• Polar moment of inertia: \({\displaystyle \iint_R \dd{I} = \iint_R r^2 \dd{m} = \iint_R (x^2 + y^2) \phi \dd{A}}\)

Using the Green’s Theorem, we need that $N_x - M_y = \delta x^2 + \delta y^2$. Choosing the simplest choice $N_x = \delta y^2$ and $M_y = -\delta x^2$. So, this leads to $N = \delta xy^2$ and $M = -\delta yx^2$

\({\displaystyle \oint_C -\delta x^2 y \dd{x} + \delta x y^2 \dd{y}}\)

3 What is a flux?

3.1 Front

What is a flux?

Let \(\vb{F}\) be a vector field and $C$ a path

3.2 Back

It’s another line integral, it’s called Flux of \(\vb{F}\) across $C$

\({\displaystyle \int_C \vb{F} \cdot \vu{n} \dd{s}}\)

4 How can we get the normal vector to a path $C$

4.1 Front

How can we get the normal vector to a path $C$

Let $C = r(t)$

4.2 Back

\(\vu{n}\) is $90$ degrees clockwise from \(\vu{T}\)

\(\Delta \vb{r} = \ev{\Delta x, \Delta y} = \vu{T} \Delta s\)

\(\vu{n} \Delta s = \ev{\Delta y, - \Delta x}\)

This direction is opposite to the one customarily used in kinematics where \(\vb{t}\) and \(\vb{n}\) form a right-handed coordinate system for motion along $C$

5 What is the interpretation of Flux of \(\vb{F}\) (Velocity field)

5.1 Front

What is the interpretation of Flux of $\vb{F}$ (Velocity field)

Across curve $C$, two-dimensional flow field

5.2 Back

Flux measures how much fluid passes through curve $C$ per unit time

6 What is the equation of flux \(\vb{F}\) across $C$?

6.1 Front

What is the equation of flux $\vb{F}$ across $C$?

In symbols and notation of differentials. Where $x(t)$, $y(t)$ is any parametrization of $C$

6.2 Back

\(\vu{n} \dd{s} = \dd{y} \vu{i} - \dd{x} \vu{j}\)

\({\displaystyle \int_C \vb{F} \cdot \vu{n} \dd{s} = \int_C M \dd{y} - N \dd{x} = \int_C \biggl( M \dv{y}{t} - N \dv{x}{t} \biggr) \dd{t}}\)

7 Which is the natural physics interpretation for flux?

7.1 Front

Which is the natural physics interpretation for flux?

Where \(\vb{F}\) as representing a two-dimensional flow field

7.2 Back

The line integral represents the rate with respect to time at which mass is being transported across $C$

8 Which is the convention for positive and negative flux

8.1 Front

Which is the convention for positive and negative flux

\(\vb{F}\) is two-dimensional flow field

8.2 Back

If we think of the flow as taking place in a shallow tank of unit depth. The convention about \(\vb{n}\) makes this mass-transport rate positive if the flow is from left to right as you face in the positive direction of $C$, and negative in the other case

9 How can we express the mass-transport rate across $C$ in a vector field \(\vb{F}\)?

9.1 Front

How can we express the mass-transport rate across $C$ in a vector field \(\vb{F}\)?

Let \(\vb{F}\) a constant vector field representing a flow, and $C$ is a directed line segment of length $L$

9.2 Back

How can we express the mass-transport rate across $C$ in a vector field \(\vb{F}\)?

Using $C’$, mass-transport rate across \({\displaystyle C’ = \abs{\vb{F}} (L \cos \theta) = (\vb{F} \cdot \vu{n}) L}\)