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Streaklines are traced out by neutrally buoyant marker fluid injected into a flow ficld from a fixed point in space. A particle of the marker fluid that is at point \((x, y)\) at time \(t\) must have passed through the injection point \(\left(x_{0}, y_{0}\right)\) at some earlier instant \(t=\tau .\) The time history of a marker particle may be found by solving the pathline equations for the initial conditions that \(x=x_{0}, y=y_{0}\) when \(t=\tau .\) The present locations of particles on the streakline are obtained by setting \(\tau\) equal to values in the range \(0 \leq \tau \leq t\). Consider the flow ficld \(\vec{V}=a x(1+b t) \hat{i}+c y \hat{j},\) where \(a=c=1 \mathrm{s}^{-1}\) and \(b=0.2 \mathrm{s}^{-1}\). Coordinates are measured in meters. Plot the streakline that passes through the initial point \(\left(x_{0}, y_{0}\right)=(1,1),\) during the interval from \(t=0\) to \(t=3\) s. Compare with the streamline plotted through the same point at the instants \(t=0,1,\) and \(2 \mathrm{s}\).

Step by step solution

01

Pathline Equations

To find the path of a single fluid particle, we need to solve the following system of ordinary differential equations, known as the pathline equations: \( \frac{dx}{dt} = a x(1+b t) \) and \( \frac{dy}{dt} = cy \). Integrating these equations with respect to \( t \) will give the pathline equations.

02

Find Pathline Equation

Solving the above equations with initial conditions \( x = x_{0} \) and \( y = y_{0} \) when \( t = \tau \), we get: \(x(t) = x_{0} e^{a t^2/2}\) and \(y(t) = y_{0} e^{ct}\). These are the pathline equations for the fluid particle.

03

Find Streakline Equation

A streakline is a line traced by fluid particles that have passed through a certain point in the past. Substituting \( x_{0} \) and \( y_{0} \) into the pathline equations and varying \( \tau \) from 0 to \( t \) we get: \(x(t,\tau) = e^{ a (t^2-\tau^2)/2} \) and \( y(t,\tau) = e^{c(t-\tau)} \). This gives the streakline of the flow field.

04

Streamline Equation

Streamlines are the instantaneous paths followed by fluid particles and are obtained by the derivative of pathline with respect to time. Therefore, \( \frac{dx}{dt} = x(1+0.2t) \) and \( \frac{dy}{dt} = y \) giving stream line as \( x(t)=e^{(t^2+ t)/2} \) and \( y(t)= e^t \).

05

Plot streakline and streamlines

To find the streaklines and streamlines, plot the equations for \( x(t, \tau) \), \( y(t, \tau) \) and \( x(t) \), \( y(t) \) over the range \( t=0 \) to \( t=3 \) seconds. Compare the plots at time instances \( t=0, 1, 2 \) seconds.

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