Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:Afiuid has velocity components u = (z — 12y) ft/s
and v = — (y + 12x) ft/s, where z and y are in

Question

Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:Afiuid has velocity components u = (z — 12y) ft/s
and v = — (y + 12x) ft/s, where z and y are in feet.

Determine the potential function.
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VIPSolutions · Accepted Answer

Let's solve the problem step by step!

## **Step 1: Understand the Problem**

You are given the velocity components of a fluid:
- $ u = (x - 12y) $ ft/s
- $ v = - (y + 12x) $ ft/s

You are asked to **determine the potential function** $ \phi $.

## **Step 2: Relationship Between Velocity and Potential Function**

For a potential function $ \phi(x, y) $ in a velocity field:
- $ u = \frac{\partial \phi}{\partial x} $
- $ v = \frac{\partial \phi}{\partial y} $

But since $ v $ is often given as negative in potential flow:
- $ u = \frac{\partial \phi}{\partial x} $
- $ v = \frac{\partial \phi}{\partial y} $

But here, the negative sign is already included in $ v $, so **keep the direct sign**.

## **Step 3: Find $ \phi $ Using Integration**

### **Integrate for $ u $:**
Given $ u = x - 12y $:
$$
u = \frac{\partial \phi}{\partial x} = x - 12y
$$

Integrate with respect to $ x $:
$$
\phi(x, y) = \int (x - 12y) \, dx = \frac{1}{2}x^2 - 12y x + f(y)
$$
where $ f(y) $ is a function of $ y $ only.

### **Differentiate for $ v $:**
Given $ v = - (y + 12x) $:
$$
v = \frac{\partial \phi}{\partial y} = - (y + 12x)
$$

Now, use the $ \phi(x, y) $ found above:
$$
\frac{\partial \phi}{\partial y} = -12x + f'(y)
$$

Set equal to above:
$$
-12x + f'(y) = - (y + 12x)
$$
$$
-12x + f'(y) = -y - 12x
$$
$$
f'(y) = -y
$$

Integrate both sides with respect to $ y $:
$$
f(y) = -\frac{1}{2}y^2 + C
$$

## **Step 4: Write the General Potential Function**

Combine all parts:
$$
\phi(x, y) = \frac{1}{2}x^2 - 12x y - \frac{1}{2}y^2 + C
$$

## **Step 5: Match With the Choices**

Looking at the choices, the correct one is:
$$
\boxed{\phi = \frac{1}{2}(x^2 - y^2) - 12xy}
$$

## **Final Answer**

$$
\boxed{\phi = \frac{1}{2}(x^2 - y^2) - 12xy}
$$

---

**Summary of Steps:**
1. Use $ u = \frac{\partial \phi}{\partial x} $ to integrate for $ \phi $.
2. Use $ v = \frac{\partial \phi}{\partial y} $ to find the function of $ y $.
3. Combine results and match with the answer choices.

Let me know if you'd like to see how each choice compares!

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