---
title: "Acceleration of a fluid particle"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_4/fluid_mechanics_mechanical:particle-acceleration-unsteady-field"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/particle-acceleration-unsteady-field
site: BCT Engineering
---

# Acceleration of a fluid particle

Fluid Mechanics · Chapter 3: Kinematics of Fluid Flow

- Asked in: year not given, 6 marks
- Video: [Acceleration of fluid particle(local and convective) -Fluid Mechanics- series of derivation](https://www.youtube.com/watch?v=YH4xrP3RV3A) — Mukesh Ramani-Engineering 4 free
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/particle-acceleration-unsteady-field
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-3

## Question

The velocity field of a two-dimensional unsteady flow is $u = 2x + t$ and $v = -2y + 2t$ (in m/s, $x$, $y$ in m, $t$ in s). Explain local and convective acceleration. Find the components and magnitude of the acceleration of a fluid particle at the point (1 m, 2 m) at $t = 1$ s, and state whether the flow is steady.

## Short answer

Local acceleration is ∂V/∂t (change at a fixed point); convective acceleration is (V·∇)V (change due to particle motion). At (1 m, 2 m) and t=1 s: a_x = 7 m/s², a_y = 6 m/s², total acceleration = 9.22 m/s². Flow is unsteady because u and v depend on time.

## Answer

#### Local and convective acceleration
Using the Eulerian description, the acceleration of a particle is the material derivative:

$$
\vec a = \frac{D\vec V}{Dt} = \frac{\partial \vec V}{\partial t} + (\vec V\cdot\nabla)\vec V
$$

- **Local (temporal) acceleration** $\partial \vec V/\partial t$: change of velocity with time at a fixed point. It is zero in steady flow.
- **Convective acceleration** $(\vec V\cdot\nabla)\vec V$: change of velocity due to the particle moving to a different position. It is zero in uniform flow.

In 2D:

$$
\begin{aligned}
a_x &= \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \\
a_y &= \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}
\end{aligned}
$$

#### Velocity at the point
At $(1, 2)$ and $t = 1$:

$$
\begin{aligned}
u &= 2(1) + 1 = 3\ \text{m/s} \\
v &= -2(2) + 2(1) = -2\ \text{m/s}
\end{aligned}
$$

#### Derivatives
$\dfrac{\partial u}{\partial t} = 1$, $\dfrac{\partial u}{\partial x} = 2$, $\dfrac{\partial u}{\partial y} = 0$, $\dfrac{\partial v}{\partial t} = 2$, $\dfrac{\partial v}{\partial x} = 0$, $\dfrac{\partial v}{\partial y} = -2$.

#### Acceleration components
$$
\begin{aligned}
a_x &= 1 + (3)(2) + (-2)(0) = 7\ \text{m/s}^2 \\
a_y &= 2 + (3)(0) + (-2)(-2) = 6\ \text{m/s}^2 \\
|a| &= \sqrt{7^2 + 6^2} = 9.22\ \text{m/s}^2
\end{aligned}
$$

#### Steadiness
Since $u$ and $v$ depend on $t$ ($\partial u/\partial t \neq 0$), the flow is **unsteady**.

**Answer: $a_x = 7\ \text{m/s}^2$, $a_y = 6\ \text{m/s}^2$, $|a| = 9.22\ \text{m/s}^2$; flow is unsteady**

## Check yourself

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

1. Convective acceleration is the acceleration due to:
   A. change of velocity with time at a fixed point
   B. the particle moving to a point where the velocity is different
   C. gravity
   D. viscosity
   Answer: B. Local acceleration is dV/dt at a fixed point; convective acceleration (V.grad)V comes from change of position. (Key point: Local and convective acceleration)

2. At (1,2), t = 1 s, u = 3, v = -2. With du/dt = 1, du/dx = 2, du/dy = 0, the value of a_x is:
   A. 1
   B. 5
   C. 7
   D. 9
   Answer: C. a_x = du/dt + u du/dx + v du/dy = 1 + 3(2) + (-2)(0) = 7 m/s^2. (Key point: Acceleration calculation)

3. The flow u = 2x + t, v = -2y + 2t is unsteady because:
   A. the convective acceleration is nonzero
   B. u and v depend on time, so du/dt is not zero
   C. the flow is two-dimensional
   D. the velocity is nonzero at the point
   Answer: B. A flow is steady only if velocity at a point does not change with time; here du/dt = 1. (Key point: Steady or unsteady)

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