---
title: "Streamline and time, particle x = 1 to 2"
subject: "Fluid Mechanics"
chapter: 4
key: "semester_3/fluid_mechanics:streamline-time-ax-ayx2-x1-2"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/streamline-time-ax-ayx2-x1-2
site: BCT Engineering
---

# Streamline and time, particle x = 1 to 2

Fluid Mechanics · Chapter 4: Hydrokinematics

- Asked in: 2075 Asoj, 8 marks
- Video: [Streamlines, Pathlines, and Streaklines - Eulerian vs. Lagrangian in 10 Minutes!](https://www.youtube.com/watch?v=AGve4RZ4zjw) — Less Boring Lectures
- 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-x1-2
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-4

## Question

A velocity for a steady, incompressible flow in the xy plane is given by $\vec V = \hat i A/x + \hat j Ay/x^2$, where A = 2 m²/s and the coordinates are measured in meters. Obtain an equation for the streamline that passes through the point (x, y) = (1, 3). Calculate the time required for a fluid particle to move from x = 1 m to x = 2 m in this flow field.

## Short answer

Streamline: y=3x. Time from x=1 m to x=2 m is 0.75 s.

## Answer

**Given:** $\vec V = \hat i\,A/x + \hat j\,Ay/x^2$ with $A = 2$ m²/s.

**Streamline through (1, 3)**

The streamline satisfies $dy/dx = v/u$:

$$
\frac{dy}{dx} = \frac{Ay/x^2}{A/x} = \frac{y}{x}\;\Rightarrow\;\int\frac{dy}{y} = \int\frac{dx}{x}\;\Rightarrow\; y = Cx
$$

Using $(x, y) = (1, 3)$: $C = 3$, so the streamline is $y = 3x$.

**Time for a particle to move from x = 1 m to x = 2 m**

Along its path $dx/dt = u = A/x$, so $x\,dx = A\,dt$:

$$
t = \int_1^2\frac{x}{A}dx = \frac{x_2^2 - x_1^2}{2A} = \frac{4 - 1}{2\times 2} = 0.75\ \text{s}
$$

(The flow is steady, so the particle path and the streamline coincide; the particle moves along $y = 3x$, from $(1, 3)$ to $(2, 6)$.)

**Answer: streamline $y = 3x$; time $= 0.75$ s.**

## Check yourself

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

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

2. Why is the time found by integrating x dx = A dt?
   A. Because the flow is unsteady
   B. u = dx/dt = A/x varies with position
   C. Because v is zero
   D. Because A is constant velocity
   Answer: B. Velocity depends on x, so separating variables and integrating gives the time. (Key point: Time method)

3. What time does a particle need to move from x = 1 m to x = 2 m?
   A. 1.5 s
   B. 0.5 s
   C. 0.75 s
   D. 3 s
   Answer: C. t = (4-1)/(2*2) = 0.75 s. (Key point: Time value)

4. Why does the particle move along y = 3x?
   A. The flow is incompressible
   B. The velocity is constant
   C. y does not change
   D. The flow is steady, so pathlines coincide with streamlines
   Answer: D. In steady flow particles follow streamlines. (Key point: Path vs streamline)

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