Variation of Pressure with Depth Hydrostatics

Hydrostatics fluid mechanics ka ek fundamental branch hai, jo fluids at rest ke behavior ko study karta hai. Ek of the most important concepts is variation of pressure with depth, jo engineering, oceanography, meteorology, and fluid design ke liye crucial hai.

1. Introduction

Fluid:

A substance that deforms continuously under the application of even the smallest shear stress.

Hydrostatics: Study of fluid at rest.

• Applications: dams, tanks, submarines, hydraulic systems, water towers.

Key Concept:

• Pressure in a fluid increases with depth due to weight of overlying fluid.
• Leads to hydrostatic pressure distribution:

P=f(h)P = f(h)P=f(h)

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2. Pressure in Fluids

2.1 Definition of Pressure

Pressure is the force exerted per unit area by a fluid:

P=FAP = \frac{F}{A}P=AF​

Where:

• PPP = pressure (Pa)
• FFF = normal force (N)
• AAA = area (m²)

Units:

• SI: Pascal (Pa) = N/m²
• CGS: dyne/cm²

Characteristics of Fluid Pressure:

1. Acts normal to any surface in contact with fluid
2. Acts equally in all directions at a point (Pascal’s Law)

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2.2 Hydrostatic Pressure Concept

• Hydrostatic pressure arises due to weight of the overlying fluid column
• Acts equally in all directions at a point in the fluid

Key assumptions:

1. Fluid at rest
2. Incompressible or compressible fluid depending on problem
3. Negligible viscosity

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3. Derivation of Hydrostatic Pressure Equation

3.1 Consider a Fluid Element

• Small vertical fluid element of cross-sectional area A and height dh
• Fluid density = ρ, gravity = g

For equilibrium in vertical direction: P+dP=P+increase in pressure due to weightP + dP = P + \text{increase in pressure due to weight} P+dP=P+increase in pressure due to weight

• Weight of fluid slice: dW=ρgAdhdW = \rho g A dhdW=ρgAdh
• Force balance:

dP⋅A=ρgAdh  ⟹  dP=ρgdhdP \cdot A = \rho g A dh \implies dP = \rho g dhdP⋅A=ρgAdh⟹dP=ρgdh

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3.2 Integration

dP=ρgdh  ⟹  ∫P0PdP=∫0hρgdhdP = \rho g dh \implies \int_{P_0}^{P} dP = \int_{0}^{h} \rho g dhdP=ρgdh⟹∫P0​P​dP=∫0h​ρgdh P−P0=ρgh  ⟹  P=P0+ρghP – P_0 = \rho g h \implies P = P_0 + \rho g hP−P0​=ρgh⟹P=P0​+ρgh

Where:

• P0P_0P0​ = pressure at reference point (usually surface)
• hhh = depth below the surface

This is the basic hydrostatic pressure formula.

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3.3 Variation for Depth

• Pressure increases linearly with depth for incompressible fluid
• Graph: P vs h → straight line with slope ρg

P
|
|      *
|     *
|    *
|   *
|  *
|_________________ h

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4. Hydrostatic Pressure in Different Scenarios

4.1 Open Tank of Liquid

• Depth h below free surface:

P=Patm+ρghP = P_{atm} + \rho g hP=Patm​+ρgh

• P_atm = atmospheric pressure

Example: Water at 10 m depth: P=101.3kPa+1000⋅9.81⋅10=101.3+98.1=199.4kPaP = 101.3 kPa + 1000 \cdot 9.81 \cdot 10 = 101.3 + 98.1 = 199.4 kPaP=101.3kPa+1000⋅9.81⋅10=101.3+98.1=199.4kPa

4.2 Multiple Liquids in Contact

• Liquid A (ρ₁), Liquid B (ρ₂), heights h₁ and h₂:

P=P0+ρ1gh1+ρ2gh2P = P_0 + \rho_1 g h_1 + \rho_2 g h_2P=P0​+ρ1​gh1​+ρ2​gh2​

Applications: Oil-water interfaces, stratified tanks

4.3 Submerged Objects

• Pressure at depth h: P=P0+ρghP = P_0 + ρ g hP=P0​+ρgh
• Buoyant force: Fb=ρgVdisplacedF_b = ρ g V_{displaced}Fb​=ρgVdisplaced​

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5. Pressure Distribution

• Hydrostatic pressure acts equally in all directions (Pascal’s Law)
• Pressure at a point:

Px=Py=PzP_x = P_y = P_zPx​=Py​=Pz​

• Implication: Force on walls of container can be calculated:

F=P⋅AF = P \cdot AF=P⋅A

• Pressure at bottom of tank = maximum
• Pressure at surface = P₀

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6. Variation with Fluid Density

• Denser fluids → higher pressure at same depth
• Mercury (ρ ≈ 13600 kg/m³) vs water (ρ ≈ 1000 kg/m³):
• 1 m of mercury → P = 13600 × 9.81 × 1 ≈ 133 kPa
• 1 m of water → P ≈ 9.81 kPa

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7. Compressible Fluids

• For gases, density ρ varies with pressure → hydrostatic pressure is non-linear
• Using ideal gas law: ρ = P/(RT)

dP=ρgdh=PgRTdhdP = \rho g dh = \frac{P g}{R T} dhdP=ρgdh=RTPg​dh

• Integration for isothermal atmosphere:

P=P0e−ghRTP = P_0 e^{- \frac{g h}{R T}}P=P0​e−RTgh​

• Exponential decrease of pressure with height → explains atmospheric pressure variation

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8. Gauge Pressure and Absolute Pressure

• Absolute pressure: total pressure including atmospheric

Pabs=P0+ρghP_{abs} = P_0 + \rho g hPabs​=P0​+ρgh

• Gauge pressure: relative to atmospheric pressure

Pgauge=P−P0=ρghP_{gauge} = P – P_0 = \rho g hPgauge​=P−P0​=ρgh

• Important for engineering applications like hydraulics

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9. Manometers and Pressure Measurement

• U-tube manometer: measures pressure difference between two points
• Differential manometer: uses density difference
• Hydrostatic equation used to relate height difference to pressure:

ΔP=(ρ2−ρ1)gh\Delta P = (\rho_2 – \rho_1) g hΔP=(ρ2​−ρ1​)gh

• Applications: fluid system monitoring, chemical processing

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10. Applications of Hydrostatic Pressure

1. Dams and Water Tanks:
   • Wall thickness and strength determined by maximum hydrostatic pressure
2. Submarines:
   • Hull designed to withstand pressure at operating depth
3. Barometers:
   • Atmospheric pressure measurement using mercury column
4. Hydraulic Systems:
   • Pressure transmission via incompressible fluids (Pascal’s principle)
5. Marine Engineering:
   • Pressure distribution affects ship design and stability

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11. Examples and Calculations

Example 1: Pressure at Depth

• Water tank, depth h = 5 m, ρ = 1000 kg/m³, P₀ = 101 kPa

P=101000+1000⋅9.81⋅5=101000+49050=150050 Pa≈150kPaP = 101000 + 1000 \cdot 9.81 \cdot 5 = 101000 + 49050 = 150050 \, Pa \approx 150 kPaP=101000+1000⋅9.81⋅5=101000+49050=150050Pa≈150kPa

Example 2: Pressure at Bottom of Oil-Water Tank

• Oil ρ₁ = 900 kg/m³, water ρ₂ = 1000 kg/m³, heights h₁ = 2 m, h₂ = 3 m, P₀ = 101 kPa

P=P0+ρ1gh1+ρ2gh2P = P_0 + \rho_1 g h_1 + \rho_2 g h_2P=P0​+ρ1​gh1​+ρ2​gh2​ P=101000+900⋅9.81⋅2+1000⋅9.81⋅3P = 101000 + 900 \cdot 9.81 \cdot 2 + 1000 \cdot 9.81 \cdot 3P=101000+900⋅9.81⋅2+1000⋅9.81⋅3 P=101000+17658+29430≈148088 PaP = 101000 + 17658 + 29430 \approx 148088 \, PaP=101000+17658+29430≈148088Pa

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12. Graphical Representation

• Pressure vs Depth in a single fluid: Linear
• Pressure vs Depth in layered fluids: Piecewise linear
• Atmospheric pressure with altitude: Exponential

P
|
|       *
|      *
|     *
|    *
|   *
|__*_______________ h

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13. Summary Table

Concept	Formula	Notes
Hydrostatic pressure	P = P₀ + ρ g h	Linear with depth for incompressible fluid
Pressure difference	ΔP = ρ g Δh	Gauge pressure
Multiple fluids	P = P₀ + ρ₁ g h₁ + ρ₂ g h₂	Layered fluids
Compressible fluid	P = P₀ e^(-gh/RT)	Isothermal atmosphere
Buoyant force	F_b = ρ g V	Upward force on submerged object