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