/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 A velocity field is given by \(\... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 16

A velocity field is given by \(\vec{V}=a y t \hat{i}-b x \hat{j},\) where \(a=1 \mathrm{s}^{-2}\) and \(b=4 s^{-1}\). Find the equation of the streamlines at any time \(t\) Plot several streamlines at \(t=0\) s, \(t=1\) s, and \(t=20\) s.

Short Answer

Expert verified
The equation for the streamlines in terms of \(x\), \(y\), and \(t\) is \(x = -\frac{a}{b} y^2 t + C\), where \(C\) is an arbitrary constant. To plot the streamlines, substitute the values \(t=0\, s\), \(t=1\, s\), and \(t=20\, s\) into the equation.

Step by step solution

01

Calculate the equations for \(\hat{i}\) and \(\hat{j}\)

Start by dividing the \(i\) component by the \(j\) component, \(\frac{dx}{dy} = \frac{a y t}{-b x}\), we use mathematics to manipulate this expression to the form of separable differential equation, finally getting \(x = -\frac{a}{b} y^2 t + C\). This expression represents a family of parabolas.
02

Determine the constant (C)

The constant \(C\) can be calculated through setting initial conditions for \(x\) and \(y\), but without those conditions given, we can just leave \(C\) unspecified.
03

Plotting the streamlines

Streamlines can be graphed by substituting into the above equation, \(x = -\frac{a}{b} y^2 t + C\), the values for \(t\). For the given problem, plot for \(t=0\, s\), \(t=1\, s\), and \(t=20\, s\).

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