Question

For a two-dimensional flow field given as $\vec V = -x\,\hat{\imath} + y\,\hat{\jmath}$, a streamline passes through points $(2,1)$ and $(5,p)$. The value of $p$ is

• A. $5$
• B. $5/2$
• C. $2/5$
• D. $2$

Hint:

Streamlines satisfy $dy/dx=v/u$. Separate variables and integrate; then use a given point to fix the constant.

Answer 1

The Correct Option is C

Step 1: Write the streamline differential equation.
For $\vec V=(u,v)=(-x,\,y)$, a streamline satisfies \[ \frac{dy}{dx}=\frac{v}{u}=\frac{y}{-x}=-\frac{y}{x}. \]
Step 2: Solve the ODE.
\[ \frac{dy}{y}=-\frac{dx}{x} \;\Rightarrow\; \ln y=-\ln x + C \;\Rightarrow\; \ln(xy)=C' \;\Rightarrow\; xy=C_0. \]
Step 3: Determine the constant from a known point.
Through $(2,1)$: $C_0=2 . 1=2$. Thus the streamline is $xy=2$.
Step 4: Enforce the second point.
At $(5,p)$: $5p=2 \Rightarrow p=\dfrac{2}{5}$. Final Answer: \[ \boxed{\dfrac{2}{5}} \]