---
title: "Streamline and time in $A/x$, $Ay/x^2$ field"
subject: "Fluid Mechanics"
chapter: 4
key: "semester_3/fluid_mechanics:streamline-time-ax-ayx2-x1-3"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/streamline-time-ax-ayx2-x1-3
site: BCT Engineering
---

# Streamline and time in $A/x$, $Ay/x^2$ field

Fluid Mechanics · Chapter 4: Hydrokinematics

- Asked in: 2076 Asoj, 3+3 marks
- Video: [Introductory Fluid Mechanics L13 p1 - Stream Function - 2D Incompressible Flow](https://www.youtube.com/watch?v=kDO3EcXblwg) — Ron Hugo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/streamline-time-ax-ayx2-x1-3
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-4

## Question

Steady, incompressible flow in xy plane with $\vec V = \frac{A}{x}\hat i + \frac{Ay}{x^2}\hat j$ where A = 2 m²/s and coordinates are in meters. Find (i) equation for streamline through (x,y) = (1,3) (ii) time required for a fluid particle to move from x = 1 m to x = 3 m.

## Short answer

Streamline: y=3x. Time to move from x=1 m to x=3 m is 2.0 s.

## Answer

**Given:** $u = A/x$, $v = Ay/x^2$, $A = 2$ m²/s.

#### (i) Streamline through (1, 3)

$$
\frac{dy}{dx} = \frac{v}{u} = \frac{Ay/x^2}{A/x} = \frac{y}{x}\;\Rightarrow\;\ln y = \ln x + \ln C\;\Rightarrow\; y = Cx
$$

At $(1, 3)$: $C = 3$.

**Streamline: $y = 3x$** (a straight line through the origin).

#### (ii) Time from x = 1 m to x = 3 m

For a particle, $u = dx/dt$:

$$
\frac{dx}{dt} = \frac{A}{x}\;\Rightarrow\; x\,dx = A\,dt
$$

$$
t = \frac{1}{A}\int_1^3 x\,dx = \frac{x_2^2 - x_1^2}{2A} = \frac{9 - 1}{2\times 2} = 2\ \text{s}
$$

**Answer: (i) $y = 3x$; (ii) $t = 2.0$ s.**

## Check yourself

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

1. How is the streamline equation derived?
   A. Solve dy/dx = v/u, giving dy/y = dx/x
   B. Solve dx/dt = u only
   C. Set u = v
   D. Differentiate psi
   Answer: A. v/u = y/x, so ln y = ln x + ln C. (Key point: Streamline equation)

2. What is the streamline through (1,3)?
   A. y = x + 2
   B. y = 3x
   C. y = 3/x
   D. y = x^3
   Answer: B. y = Cx with C = 3. (Key point: Streamline result)

3. How is the time from x=1 to x=3 found?
   A. Divide distance by u at x=1
   B. Integrate dy/dx
   C. Use dx/dt = A/x, so x dx = A dt and integrate
   D. Use the streamline slope
   Answer: C. u varies with x so the path is integrated: t = (x2^2 - x1^2)/(2A). (Key point: Travel time method)

4. What is the time for x from 1 m to 3 m with A = 2?
   A. 4 s
   B. 1 s
   C. 8 s
   D. 2 s
   Answer: D. t = (9-1)/(2*2) = 2 s. (Key point: Travel time value)

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