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.

Step-by-Step Explanation:

In a Lagrangian framework, we are given the parametric equations representing the position of a fluid particle at any time t:
$x=x_o{e}^{-kt}$
and
$y=y_o{e}^{kt}$

1. Velocity Components (Eulerian Representation)
To determine the flow properties, we first find the velocity components of the fluid particle by taking the time derivative of the position coordinates.

As depicted in image 0, the velocity component in the x-direction, u, is obtained by differentiating x with respect to t:

$$u=\frac{dx}{dt}=-k{x}_o{e}^{-kt}$$

Substitute $x=x_o{e}^{-kt}$ back into the expression to obtain the velocity in terms of the position coordinate x:

$$u=-kx$$

Similarly, as shown in image 1, the velocity component in the y-direction, v, is found by differentiating y with respect to t:

$$v=\frac{dy}{dt}=k{y}_o{e}^{kt}$$

Substituting $y=y_o{e}^{kt}$ back gives:

$$v=ky$$

2. Determining Flow Characteristics

A. Is the flow steady or unsteady?
A flow is classified as steady if the velocity components at any spatial point are independent of time t. Here, the Eulerian velocity fields are:
$u(x,y)=-kx$
and
$v(x,y)=ky$
Since time t does not explicitly appear in these Eulerian velocity component equations, the velocity field at any point in space is constant over time. Therefore, the flow is steady.

B. Dimensionality of the flow
Since we have two non-zero velocity components u and v that depend on two spatial coordinates (x and y), the flow takes place in a two-dimensional plane. Thus, the flow is two-dimensional.

C. Continuity Check (Image 2)
As verified in image 2, the continuity equation for a two-dimensional incompressible flow is satisfied:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=-k+k=0$
This confirms that the velocity field represents a physically possible, steady, two-dimensional flow.