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Chapter 4: Problem 9

A two-dimensional, unsteady velocity field is given by \\[ u=5 x(1+t) \text { and } v=5 y(-1+t) \\] where \(u\) is the \(x\) -velocity component and \(v\) the \(y\) -velocity component. Find \(x(t)\) and \(y(t)\) if \(x=x_{0}\) and \(y=y_{0}\) at \(t=0 .\) Do the velocity components represent an Eulerian description or a Lagrangian description?

Short Answer

Expert verified
The position functions obtained by integrating the velocity functions (step 1 and 2) give the trajectories of the fluid elements. Therefore, the given velocity components represent a Lagrangian description of fluid flow.

Step by step solution

01

Find x(t)

The differential equation governing the trajectory in the x direction can be written from \(u= \frac{dx}{dt}=5x(1+t)\). Integrate the above differential equation from \(t=0\) to \(t=t'\) and \(x=x_{0}\) to \(x=x'\), to get the function x(t)
02

Find y(t)

Similarly, for y direction we can write the governing equation from \(v= \frac{dy}{dt}=5y(-1+t)\). Integrate the above differential equation from \(t=0\) to \(t=t'\) and \(y=y_{0}\) to \(y=y'\), to get the function y(t)
03

Eulerian or Lagrangian

Decide whether the description is Eulerian or Lagrangian based on whether it describes the fluid properties at different points in space with respect to time (Eulerian) or follows fluid parcels as they move through space (Lagrangian).

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Most popular questions from this chapter

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