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https://file.aiquickdraw.com/mathbot/file/17406248734228p0toszo.pdf

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using the notes solve q2 part a step by step. use the same notation as the notes and underline vectors 

To solve part (a) of the problem on streamlines for the velocity field \( \mathbf{q}(\mathbf{r}, t) = (x - Vt, y, 0) \), we need to calculate the equation of the streamlines. Streamlines represent the paths along which fluid particles move at any given moment. These are determined by the velocity components.

### Step-by-Step Solution for Streamlines

1. **Understanding the Problem:**
   The velocity field is given by:
   \[
   \mathbf{q}(\mathbf{r}, t) = \left( x - Vt, y, 0 \right)
   \]
   This represents the velocity of a particle at position \( (x, y, z) \) at time \( t \). Specifically, the \( x \)-component is \( x - Vt \), the \( y \)-component is \( y \), and the \( z \)-component is zero.

2. **Streamline Equation:**
   Streamlines are defined by the equation:
   \[
   \frac{dy}{dt} = \frac{v_y}{v_x}
   \]
   where \( v_x \) and \( v_y \) are the components of the velocity in the \( x \)- and \( y \)-directions, respectively. In this case:
   - \( v_x = x - Vt \)
   - \( v_y = y \)

3. **Differential Equation for Streamlines:**
   The velocity field gives us the following system of equations:
   \[
   \frac{dx}{dt} = x - Vt, \quad \frac{dy}{dt} = y
   \]
   From the second equation \( \frac{dy}{dt} = y \), we can solve this as a standard exponential growth equation:
   \[
   y(t) = y_0 e^t
   \]
   where \( y_0 \) is the initial value of \( y \).

4. **Solution for \( x(t) \):**
   To solve for \( x(t) \), we substitute \( y(t) \) into the first equation:
   \[
   \frac{dx}{dt} = x - Vt
   \]
   Solving this equation by separation of variables or using an integrating factor, we get:
   \[
   x(t) = x_0 e^t - \frac{V}{t} + constant
   \]

Hence, the equation for the streamline is expected to be a time-based equation

Compute the streamlines for the velocity field q(r,t) = (x - Vt, y, 0).

A detailed step-by-step solution for computing the streamlines of a given velocity field in fluid dynamics, with key concepts including differential equations and vector calculus.

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