In a Lagrangian system, the position of a fluid particle in a flow is described as 𝑥 = 𝑥_𝑜𝑒 ^−𝑘𝑡 and 𝑦 = 𝑦_𝑜𝑒^𝑘𝑡 where t is the time while 𝑥_𝑜, 𝑦_𝑜, and k are constants. The flow is

The correct answer is steady and two-dimensional.

To determine the nature of the flow, we need to find the velocity components of the fluid particle from its given position coordinates in the Lagrangian system and express them in terms of the spatial coordinates (Eulerian system).

The position of the fluid particle is given by:
$x=x_0{e}^{-kt}$
and
$y=y_0{e}^{kt}$

First, we find the velocity component in the x-direction, u, by differentiating x with respect to time t:
$u=\frac{dx}{dt}=-k{x}_0{e}^{-kt}$

Substituting $x=x_0{e}^{-kt}$ into the equation for u gives:
$u=-kx$

Next, we find the velocity component in the y-direction, v, by differentiating y with respect to time t:
$v=\frac{dy}{dt}=k{y}_0{e}^{kt}$

Substituting $y=y_0{e}^{kt}$ into the equation for v gives:
$v=ky$

Now, let's analyze the velocity field:
1. Dimension of the flow: Since the velocity components exist in both the x and y directions (i.e., both u and v are non-zero and depend on the spatial coordinates x and y), the flow is two-dimensional.
2. Steadiness of the flow: A flow is steady if the velocity at any point does not change with time. The Eulerian velocity field components are:
$u=-kx$
and
$v=ky$
Since neither u nor v contains the time variable t explicitly, we have:
$\frac{\partial u}{\partial t}=0$
and
$\frac{\partial v}{\partial t}=0$
Thus, the velocity at any spatial position remains constant over time, making the flow steady.

Combining these two characteristics, the flow is steady and two-dimensional.