---
title: "Streamlines of $\\psi = x^2 + y^2$"
subject: "Fluid Mechanics"
chapter: 4
key: "semester_3/fluid_mechanics:streamlines-psi-x2-plus-y2"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/streamlines-psi-x2-plus-y2
site: BCT Engineering
---

# Streamlines of $\psi = x^2 + y^2$

Fluid Mechanics · Chapter 4: Hydrokinematics

- Asked in: 2074 Asoj, 3+3 marks
- Video: [Problem No. 1 on Velocity Potential Function and Streamline Function - Fluid Mechanics 1](https://www.youtube.com/watch?v=yLb9tjUXifk) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/streamlines-psi-x2-plus-y2
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-4

## Question

Sketch the streamlines represented by the stream function $\psi = x^2 + y^2$. Find also the velocity and its direction at point (3,4).

## Short answer

Streamlines are concentric circles x^2+y^2=ψ. At (3,4), u=8 m/s, v=-6 m/s, giving 10 m/s at -36.9° to the x-axis.

## Answer

**Given:** $\psi = x^2 + y^2$, with $u = \partial\psi/\partial y$, $v = -\partial\psi/\partial x$.

#### Streamlines

A streamline is $\psi = $ constant, so

$$
x^2 + y^2 = C = r^2
$$

These are concentric circles about the origin with radius $r = \sqrt{\psi}$. For example, $\psi = 1, 4, 9, 16, 25$ give circles of radius $1, 2, 3, 4, 5$ m. For equal steps of $\psi$ (say $\psi = 5, 10, 15, \ldots$) the circles get closer together as $r$ increases, which shows that the velocity increases outward.

```text
             y
             |
         .-"""""-.
       .'  .---.  '.
      /   /     \   \
 ----|---|---O---|---|---- x
      \   \     /   /
       '.  '---'  .'
         '-.....-'
             |
      (flow is clockwise)
```

#### Velocity at (3, 4)

$$
u = \frac{\partial\psi}{\partial y} = 2y = 8\ \text{m/s},\qquad v = -\frac{\partial\psi}{\partial x} = -2x = -6\ \text{m/s}
$$

$$
|V| = \sqrt{8^2 + 6^2} = 10\ \text{m/s},\qquad \theta = \tan^{-1}\frac{-6}{8} = -36.87^\circ
$$

The velocity is $10$ m/s at $36.9^\circ$ below the +x axis. It is tangent to the circle $r = 5$ m through $(3, 4)$ (because $\psi = 25$ there), and the flow is clockwise. In general $|V| = 2r$, so the flow is rotational, like a rigid-body rotation.

**Answer: the streamlines are concentric circles $x^2 + y^2 = \psi$; at (3, 4) the velocity is $(8\hat i - 6\hat j)$ m/s, i.e. 10 m/s at $-36.9^\circ$ to the x-axis.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What do the streamlines of psi = x^2 + y^2 look like?
   A. Concentric circles of radius sqrt(psi)
   B. Parallel lines
   C. Hyperbolas
   D. Radial lines
   Answer: A. psi = const gives x^2 + y^2 = r^2. (Key point: Streamline shape)

2. What is the velocity at (3,4)?
   A. (6, -8) m/s
   B. (8, -6) m/s, 10 m/s
   C. (-8, 6) m/s
   D. (3, 4) m/s
   Answer: B. u = dpsi/dy = 2y = 8, v = -dpsi/dx = -2x = -6. (Key point: Velocity at (3,4))

3. Which statement about direction at (3,4) is correct?
   A. 36.9 degrees above +x, radially outward
   B. Along the radius to the origin
   C. 36.9 degrees below +x, tangent to circle r = 5, clockwise
   D. 53.1 degrees above +x
   Answer: C. tan^-1(-6/8) = -36.9 degrees; velocity is tangent to the streamline (circle). (Key point: Direction)

4. How does speed vary with r?
   A. |V| = 1/r
   B. Constant
   C. |V| = r^2
   D. |V| = 2r, increasing outward
   Answer: D. |V| = sqrt(4x^2+4y^2) = 2r. (Key point: Speed variation)

## Related in Fluid Mechanics

- [Streamlines from stream function](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Astream-function-6ax-8ay) — Fluid Mechanics, chapter 4
- [Pathline and streamlines for velocity field](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Apathline-streamlines-bx-t0) — Fluid Mechanics, chapter 4
- [Streaklines compared with streamlines](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Astreaklines-vs-streamlines-bx) — Fluid Mechanics, chapter 4

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