Questions: Consider a fluid whose flow is constrained to be in the shape of a square, but the size of the square is free to expand and contract. This fluid is initially not moving and is located at 45° S. After some time, the tangential velocity measured along each 10-m side of the square is 25 m s^-1, and the circulation is in the cyclonic direction. What is the size of the initial square (length of one side)? Clearly state both assumptions required to answer this question.

Consider a fluid whose flow is constrained to be in the shape of a square, but the size of the square is free to expand and contract. This fluid is initially not moving and is located at 45° S. After some time, the tangential velocity measured along each 10-m side of the square is 25 m s^-1, and the circulation is in the cyclonic direction. What is the size of the initial square (length of one side)? Clearly state both assumptions required to answer this question.

Transcript text: Consider a fluid whose flow is constrained to be in the shape of a square, but the size of the square is free to expand and contract. This fluid is initially not moving and is located at $45^{\circ} \mathrm{S}$. After some time, the tangential velocity measured along each $10-\mathrm{m}$ side of the square is $25 \mathrm{~m} \mathrm{~s}^{-1}$, and the circulation is in the cyclonic direction. What is the size of the initial square (length of one side)? Clearly state both assumptions required to answer this question.

Solution

Solution Steps

Step 1: Understand the Problem

The problem involves a fluid constrained to be in the shape of a square, with the size of the square able to expand and contract. The fluid is initially not moving and is located at 45°N. Over time, the tangential velocity measured along each 10-m side of the square is 25 m/s, and the circulation is in the cyclonic direction. We need to determine the size of the initial square (length of one side).

Step 2: Define Circulation and Tangential Velocity

Circulation ($\Gamma$) is defined as the line integral of the tangential velocity ($v_t$) around a closed curve: \[ \Gamma = \oint_C \mathbf{v} \cdot d\mathbf{l} \] Given that the tangential velocity is 25 m/s along each 10-m side, we can calculate the circulation.

Step 3: Calculate Circulation

Since the square has four sides, each 10 meters long, and the tangential velocity is 25 m/s: \[ \Gamma = 4 \times (25 \, \text{m/s} \times 10 \, \text{m}) = 1000 \, \text{m}^2/\text{s} \]

Step 4: Relate Circulation to Vorticity

Circulation is also related to vorticity ($\zeta$) and the area ($A$) of the square: \[ \Gamma = \zeta \times A \] Given that the area of the square is $A = L^2$, where $L$ is the length of one side of the square.

Step 5: Solve for the Length of One Side

Assuming the vorticity is constant and using the given circulation: \[ 1000 \, \text{m}^2/\text{s} = \zeta \times L^2 \] To find $L$, we need to know the vorticity $\zeta$. However, since the problem does not provide $\zeta$, we assume it is a known constant. For simplicity, let's assume $\zeta = 1 \, \text{s}^{-1}$: \[ 1000 = 1 \times L^2 \implies L^2 = 1000 \implies L = \sqrt{1000} \approx 31.62 \, \text{m} \]