---
title: "Pressure variation in a static fluid"
subject: "Fluid Mechanics"
chapter: 2
key: "semester_4/fluid_mechanics_mechanical:pressure-variation-static-fluid"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/pressure-variation-static-fluid
site: BCT Engineering
---

# Pressure variation in a static fluid

Fluid Mechanics · Chapter 2: Fluid Statics

- Asked in: year not given, 3+5 marks
- Video: [What is Pressure Variation in a Fluid at Rest -Pressure and Pressure Measurement - Fluid Mechanics 1](https://www.youtube.com/watch?v=7Tg7zqKZJZw) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/pressure-variation-static-fluid
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-2

## Question

(a) State Pascal's law and explain atmospheric, gauge, absolute and vacuum pressure with a sketch of their relationship.
(b) Derive the basic equation for the variation of pressure with depth in a static incompressible fluid, starting from an elementary fluid prism or cylinder.

## Short answer

Pascal’s law: pressure at a point in a static fluid is same in all directions. Absolute pressure = gauge + atmospheric; vacuum = atmospheric – absolute. Hydrostatic equation: dp/dz = –ρg; for incompressible fluid, p = p0 + ρgh. Pressure increases linearly with depth.

## Answer

#### (a) Pascal's law and pressure terms
**Pascal's law:** the pressure intensity at a point in a fluid at rest is the same in all directions, i.e. $p_x = p_y = p_z$. Any pressure applied to a confined fluid is transmitted equally in all directions.

- **Atmospheric pressure:** pressure due to the weight of the air above; standard value $101.325\ \text{kPa}$ at sea level.
- **Absolute pressure:** measured from absolute zero (perfect vacuum).
- **Gauge pressure:** measured above local atmospheric pressure; $p_{gauge} = p_{abs} - p_{atm}$.
- **Vacuum pressure:** pressure below atmospheric; $p_{vac} = p_{atm} - p_{abs}$.

```text
 p_abs ^
       |   ______ point A (above atm)
       |  |  gauge pressure
 p_atm |--|-------------------------
       |  |  vacuum pressure
       |  |______ point B (below atm)
       |
 0     +--- absolute zero ----------
```

$$
p_{abs} = p_{atm} + p_{gauge}
$$

#### (b) Derivation of hydrostatic equation
Consider a vertical cylinder of fluid of cross-section $dA$ and height $dz$ in equilibrium. Take $z$ upward. Pressure at the bottom is $p$ and at the top is $p + dp$.

Forces in the vertical direction:
- Upward force on bottom: $p\,dA$
- Downward force on top: $(p + dp)\,dA$
- Weight: $\rho g\, dA\, dz$ (downward)

For equilibrium, $\sum F_z = 0$:

$$
\begin{aligned}
p\,dA - (p + dp)\,dA - \rho g\,dA\,dz &= 0 \\
\frac{dp}{dz} &= -\rho g = -\gamma
\end{aligned}
$$

The negative sign shows that pressure decreases as $z$ increases. For horizontal directions there is no net force, so $\partial p/\partial x = \partial p/\partial y = 0$: pressure is constant on a horizontal plane.

For an incompressible fluid ($\rho$ constant), integrating between two points 1 and 2:

$$
p_2 - p_1 = -\rho g (z_2 - z_1) \quad \text{or} \quad \frac{p}{\rho g} + z = \text{constant}
$$

At depth $h$ below a free surface where the pressure is $p_0$:

$$
p = p_0 + \rho g h
$$

Here $p/\rho g$ is the pressure head. The sum of pressure head and elevation head is constant throughout a static fluid.

## Check yourself

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

1. Pascal's law states that, for a fluid at rest:
   A. pressure increases linearly with height
   B. pressure intensity at a point is the same in all directions
   C. pressure acts only perpendicular to gravity
   D. pressure is the same at all points in the fluid
   Answer: B. At a point in a static fluid, px = py = pz; pressure applied to a confined fluid is transmitted equally. (Key point: Pascal's law)

2. A point has absolute pressure below atmospheric. Its vacuum pressure is:
   A. p_abs - p_atm
   B. p_atm - p_abs
   C. p_atm + p_abs
   D. p_abs / p_atm
   Answer: B. Vacuum pressure is the amount by which pressure falls below atmospheric: p_vac = p_atm - p_abs. Gauge pressure is p_abs - p_atm. (Key point: Gauge, absolute and vacuum pressure)

3. Balancing forces on a vertical fluid cylinder (z upward) gives dp/dz = -rho g. What does the negative sign mean?
   A. Pressure increases as z increases
   B. Pressure decreases as z increases (pressure rises with depth)
   C. Pressure is constant on a vertical line
   D. Weight acts upward
   Answer: B. The weight of the fluid above must be carried by pressure, so pressure decreases with height and increases with depth. (Key point: Basic hydrostatic equation)

4. For an incompressible static fluid with pressure p0 at the free surface, the pressure at depth h is:
   A. p = p0 - rho g h
   B. p = p0 + rho g h
   C. p = rho g h / p0
   D. p = p0 + rho h
   Answer: B. Integrating dp/dz = -rho g gives p = p0 + rho g h; pressure is also constant on a horizontal plane. (Key point: Pressure at depth)

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