In a two dimensional steady flow the velocity of fluid particle at (x,y) is given by `vecV=(u_(0)+bx)hati-` by `hatj, u_(0)` and b are positive constants. Write the equation of streamilnes. Draw few streamlines for `xgt0`.

Answer

To find the equation of streamlines for the given velocity field in a two-dimensional steady flow, we start with the velocity vector provided:

\[ \vec{V} = (u_0 + bx) \hat{i} - by \hat{j} \]

Here, \(u_0\) and \(b\) are positive constants. The components of the velocity vector are:

- \(u = u_0 + bx\)
- \(v = -by\)

### Step 1: Understand the relationship between velocity and streamlines

The streamlines in a flow represent the paths that fluid particles follow. The slope of the streamline at any point is given by the ratio of the velocity components:

\[ \frac{dy}{dx} = \frac{v}{u} \]

### Step 2: Substitute the velocity components

Substituting the expressions for \(u\) and \(v\):

\[ \frac{dy}{dx} = \frac{-by}{u_0 + bx} \]

### Step 3: Rearrange the equation

Rearranging gives:

\[ dy = -\frac{by}{u_0 + bx} dx \]

### Step 4: Separate variables

We can separate the variables \(y\) and \(x\):

\[ \frac{dy}{y} = -b \frac{dx}{u_0 + bx} \]

### Step 5: Integrate both sides

Now we integrate both sides:

\[ \int \frac{dy}{y} = -b \int \frac{dx}{u_0 + bx} \]

The left side integrates to:

\[ \ln |y| = -b \left(\frac{1}{b} \ln |u_0 + bx|\right) + C \]

This simplifies to:

\[ \ln |y| = -\ln |u_0 + bx| + C \]

### Step 6: Exponentiate to eliminate the logarithm

Exponentiating both sides gives:

\[ |y| = \frac{C}{u_0 + bx} \]

Let \(C\) be a constant that can take both positive and negative values. Thus, we can write:

\[ y = \frac{C}{u_0 + bx} \]

This is the equation of the streamlines.

### Step 7: Draw the streamlines

To draw the streamlines for \(x > 0\), we can choose different values of \(C\) (like \(C = 1, 2, -1, -2\)) and plot the corresponding curves. The streamlines will approach the x-axis as \(x\) increases, and they will diverge as \(y\) approaches zero.

### Summary of the Solution

The equation of the streamlines is given by:

\[ y = \frac{C}{u_0 + bx} \]