Question

In a Lagrangian system,the position of a fluid particle in a flow is described as $x=x_0{e}^{-kt}$ and $y=y_0{e}^{kt}$ where t is the time while $x_0,y_0$, and k are constants. The flow is

• A. unsteady and one-dimensional
• B. steady and two-dimensional
• C. steady and one-dimension
• D. unsteady and two-dimensional

Answer 1

The correct option is B steady and two-dimensional

x direction scalar of velocity field,

$u=\frac{dx}{dt}$

$u=-k{x}_0{e}^{-kt}$

y direction scalar of velocity field

$v=\frac{dy}{dt}$

$v=k{y}_0{e}^{kt}$

$\overrightarrow{V}=u\hat{i}+v\hat{j}$

$\overrightarrow{V}=-k\left(x_0{e}^{-kt}\right)\hat{i}+k\left(y_0{e}^{kt}\right)\hat{j}$

$\overrightarrow{V}=-kx\hat{i}+ky\hat{j}$

So flow is steady $\overrightarrow{V}\ne f\left(t\right)$

u & v are non zero scalar t$\ge$0 so it is 2D flow.
2D possible flow field.

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

$\frac{\partial }{\partial x}(-k{x}_0{e}^{-kt})+\frac{\partial }{\partial y}\left(k{y}_0{e}^{kt}\right)=0$

0+0 = 0 continuity satisfied.