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In the realm of fluid dynamics, streamlines are incredibly important. They indicate the direction that a fluid particle will travel at any given point in time. Picture them as a snapshot of the entire flow at a specific instant. Streamlines help us understand fluid motion better by giving a visual depiction of the flow patterns.

To derive a streamline equation, we use the velocity field, which gives the velocity components in the directions of interest. For our exercise, the given two-dimensional unsteady velocity field is \( u = x(1 + 2t) \) and \( v = y \). The first step involves setting up the streamline equation itself using the formula \( \frac{dy}{dx} = \frac{v}{u} \). This relates the changes in the \( y \) direction to changes in the \( x \) direction for the fluid flow, ensuring the slope of the trajectory is accurately captured.

Thus, by substituting the velocity components \( u \) and \( v \) into this formula, we get the streamline differential equation \( \frac{dy}{dx} = \frac{y}{x(1 + 2t)} \), paving the way for further analysis.

Separation of variables is a pivotal technique used to simplify differential equations. When we encounter a differential equation like \( \frac{dy}{dx} = \frac{y}{x(1 + 2t)} \), separation helps us break down the problem into manageable parts.

To apply separation of variables, we rearrange terms to isolate \( y \) and \( x \) on opposite sides of the equation. This means moving all terms involving \( y \) to one side and terms involving \( x \) to the other side: \( \frac{dy}{y} = \frac{dx}{x(1 + 2t)} \).

This separation makes it convenient to integrate each side with respect to its respective variable. This crucial step simplifies solving the equation significantly and leads seamlessly into the integration process.

Integration is one of the cornerstone tools in fluid dynamics, particularly when deriving expressions and understanding changes in the system. Once we've separated the variables, we next focus on integrating each side of the equation obtained: \( \int \frac{dy}{y} = \int \frac{1}{x(1 + 2t)} dx \).

The integration process will yield natural logarithms due to the variable separation method applied earlier. For \( y \), integrating \( \frac{1}{y} \, dy \) gives \( \ln|y| \). For \( x \), integrating \( \frac{1}{x(1 + 2t)} \, dx \) yields \( \frac{1}{1 + 2t} \ln|x| \).

Once we add the integration constant \( C \), we account for the starting condition that these equations hold true for specific scenarios, such as when the streamline passes through point \((x_0, y_0)\). Solving for \( C \) and substituting back into our equations enables us to derive the final time-dependent streamline equation:

• \( \ln \left(\frac{y}{y_0}\right) = \frac{1}{1 + 2t} \ln \left(\frac{x}{x_0}\right) \)
• \( \frac{y}{y_0} = \left(\frac{x}{x_0}\right)^{\frac{1}{1 + 2t}} \)

This final result displays how streamlines transform over time, helping visualize their evolution in varying conditions.