Fluid Mechanics Reference Guide

Definition of Fluid

A fluid is a substance which deforms continuously under the action of shear force. A fluid is a substance with zero shear modulus.

Classification of Fluids

1. Based on Flow Characteristics

Steady Fluid: Velocity remains constant at each point while flowing.

Unsteady Fluid: Velocity differs between any two points while flowing.

2. Based on Compressibility

Compressible Fluid: Density changes significantly with temperature and pressure.

ρ = f(P,T) where ρ is density, P is pressure, T is temperature

Incompressible Fluid: Density remains constant regardless of pressure or temperature.

ρ = constant

3. Based on Viscosity

Viscous Fluid: Exhibits resistance to flow due to internal friction.

Non-Viscous Fluid: No internal resistance to flow (idealized concept).

4. Based on Rotation

Rotational Flow: Angle between intersecting fluid element boundaries changes.

Irrotational Flow: Fluid rotates as a whole with no angle changes.

Types of Fluids

Type	Description	Key Formula
Ideal Fluid	Hypothetical fluid with no viscosity and incompressible	μ = 0, ρ = constant
Real Fluid	Actual fluid with viscosity and compressibility	μ > 0, ρ = f(P,T)
Newtonian Fluid	Obeys Newton's law of viscosity	τ = μ(du/dy)
Non-Newtonian Fluid	Doesn't obey Newton's law of viscosity	τ = K(du/dy)ⁿ
Dilatant Fluid	Shear stress increases with velocity gradient (n > 1)	n > 1 in τ = K(du/dy)ⁿ
Pseudoplastic Fluid	Shear stress decreases with velocity gradient (n < 1)	n < 1 in τ = K(du/dy)ⁿ

Properties of Fluids

1. Density (ρ)

Mass per unit volume

ρ = m/V [kg/m³]

2. Specific Weight (γ)

Weight per unit volume

γ = ρg [N/m³]

3. Specific Gravity (SG)

Ratio of fluid density to water density

SG = ρ_fluid/ρ_water

4. Specific Volume (v)

Volume per unit mass

v = 1/ρ [m³/kg]

5. Viscosity (μ)

Resistance to flow

τ = μ(du/dy) [Pa·s or N·s/m²]

6. Kinematic Viscosity (ν)

Ratio of dynamic viscosity to density

ν = μ/ρ [m²/s]

7. Surface Tension (σ)

Force per unit length

σ = F/L [N/m]

8. Capillarity

Height of liquid rise/fall in tube

h = (4σcosθ)/(ρgd)

Pressure Types

Type	Description	Relationship
Atmospheric Pressure	Pressure exerted by atmosphere	~101.325 kPa at sea level
Gauge Pressure	Pressure relative to atmospheric	P_gauge = P_absolute - P_atm
Absolute Pressure	Total pressure including atmospheric	P_absolute = P_gauge + P_atm
Vacuum Pressure	Pressure below atmospheric	P_vacuum = P_atm - P_absolute

Hydrostatic Force

Horizontally Immersed Surface

P = γAħ

where ħ is depth of centroid

Vertically Immersed Surface

P = γAħ
h* = (I_G/Aħ) + ħ

where h* is center of pressure

Inclined Immersed Surface

P = γAħ
h* = (I_Gsin²θ/Aħ) + ħ

Curved Surface

P = √(P_H² + P_V²)
θ = tan⁻¹(P_V/P_H)

Buoyancy

The upward force exerted by a fluid on a submerged or floating body.

F_b = ρ_fgV_disp

where V_disp is volume of displaced fluid

Lock Gates

Reaction between gates: R = P/(2sinθ)

Buoyancy and Floating Bodies

Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. It occurs due to the pressure difference between the top and bottom of the submerged object.

Buoyant Force (F_b) = ρ_fluid × g × V_displaced

Center of Buoyancy

The point through which the buoyant force acts, coinciding with the centroid of the displaced fluid volume.

Metacenter (M)

The point about which a floating body oscillates when tilted.

Metacentric height (GM) = BM - BG
BM = I/V_displaced

Equilibrium Conditions

Stable: M above G (GM > 0)
Unstable: M below G (GM < 0)
Neutral: M coincides with G (GM = 0)

Experimental Determination of Metacentric Height

GM = (w × x)/(W × tanθ)
where w = movable weight, x = distance moved, W = total weight

Liquid in Relative Equilibrium

For horizontal acceleration (a): tanθ = a/g
For vertical acceleration: h = h₀(1 ± a/g)
For rotating fluids: z = (ω²r²)/(2g)

Hydrostatic Forces

Surface Orientation	Total Force	Center of Pressure
Vertical	F = ½ρgh² × width	h* = ⅔h from surface
Horizontal	F = ρgh × area	At centroid
Inclined (θ)	F = ρgh̅A	h* = (I_Gsin²θ)/(Ah̅) + h̅

Flow Measurement Devices

Venturimeter

Q = C_d(A₁A₂/√(A₁²-A₂²))√(2gh)
h = x(S_m/S_f - 1) for heavier manometer fluid
h = x(1 - S_m/S_f) for lighter fluid

Pitot Tube

v = √(2gh)
where h = (P_total - P_static)/ρg

Flow Classification

Classification	Type	Description
Time Variation	Steady	∂/∂t = 0 (no change with time)
Time Variation	Unsteady	∂/∂t ≠ 0 (varies with time)
Space-Time	Steady Uniform	No change in space or time
	Steady Non-uniform	Varies in space but not time
	Unsteady Uniform	Varies in time but not space
	Unsteady Non-uniform	Varies in both space and time
Flow Nature	Laminar (Re < 2000)	Smooth, orderly flow
	Transitional (2000 < Re < 4000)	Intermittent turbulence
	Turbulent (Re > 4000)	Chaotic, eddying flow

Reynolds Number: Re = ρvD/μ = vD/ν
Froude Number: Fr = v/√(gL)

Flow Visualization

Streamline

Tangent gives instantaneous velocity direction

Pathline

Actual path followed by a fluid particle

Streakline

Locus of all particles passing through a point

Fundamental Equations

Continuity Equation

1D: ρ₁A₁v₁ = ρ₂A₂v₂
For incompressible: A₁v₁ = A₂v₂
3D: ∂(ρu)/∂x + ∂(ρv)/∂y + ∂(ρw)/∂z = -∂ρ/∂t

Bernoulli's Equation

P/ρg + v²/2g + z = constant
Assumptions:
1. Ideal fluid (inviscid)
2. Steady flow
3. Incompressible
4. Along a streamline

Important Notes

Key Points to Remember:

Kinematic viscosity in CGS: 1 stoke = 1 cm²/s = 10⁻⁴ m²/s
For capillary tubes: h ∝ 1/r and mass ∝ r
Viscosity of water ≈ 55 × viscosity of air
Inclined manometers provide better sensitivity for small pressure differences
Vapor pressure increases with temperature
Maximum pressure in a bubble: ΔP = 4σ/R

Approaches in Fluid Mechanics:

Lagrangian: Follows individual fluid particles

Eulerian: Observes flow at fixed points in space

Orifice Flow

An orifice is a small opening of any cross-section on the side or bottom of a tank through which fluid flows.

Actual velocity (v) = C_v√(2gh)
where C_v = coefficient of velocity (0.95-0.99)
h = head above orifice center

Small Orifice

Head from center > 5× depth of orifice

Large Orifice

Head from center < 5× depth of orifice

Fundamental Principles

Principle	Statement	Equation
Conservation of Mass	Mass can neither be created nor destroyed	ρ₁A₁v₁ = ρ₂A₂v₂
Conservation of Energy	Energy can neither be created nor destroyed	P₁/ρg + v₁²/2g + z₁ = P₂/ρg + v₂²/2g + z₂ + h_L
Conservation of Momentum	Change in momentum equals force × time	F = d(mv)/dt = ρQ(v₂ - v₁)

Flow Potential Functions

Velocity Potential (ϕ)

u = -∂ϕ/∂x
v = -∂ϕ/∂y
w = -∂ϕ/∂z

Stream Function (ψ)

u = ∂ψ/∂y
v = -∂ψ/∂x
Relation: ∂ϕ/∂x = ∂ψ/∂y and ∂ϕ/∂y = -∂ψ/∂x

Energy Heads

Potential Head

Energy due to elevation (z)

Pressure Head

P/ρg

Velocity Head

v²/2g

Total Head = P/ρg + v²/2g + z
Hydraulic Gradient Line (HGL) = P/ρg + z
Energy Gradient Line (EGL) = P/ρg + v²/2g + z

[Diagram would show HGL and EGL relationships]

Hydraulic Coefficients

Coefficient	Definition	Formula	Typical Value
C_v (Velocity)	Actual velocity/Theoretical velocity	C_v = v_actual/√(2gh)	0.95-0.99
C_c (Contraction)	Jet area/Orifice area	C_c = A_jet/A_orifice	0.61-0.69
C_d (Discharge)	Actual discharge/Theoretical discharge	C_d = Q_actual/Q_theoretical = C_v×C_c	0.61-0.65

Actual discharge through orifice: Q = C_dA√(2gh)

Flow Measurement Devices

Orifice Meter

Q = C_dA_o/√(1-(A_o/A_p)²) × √(2gΔh)

Cheaper alternative to venturimeter

Venturimeter

Q = C_dA₁A₂/√(A₁²-A₂²) × √(2gΔh)
Head loss: h_L = Δh(1-C_d²)

Pitot Tube

v = √(2g(P_total-P_static)/ρg) = √(2gΔh)

Weirs and Notches

Rectangular weir: Q = (2/3)C_dL√(2g)H^3/2
V-notch (90°): Q = (8/15)C_d√(2g)tan(θ/2)H^5/2 ≈ 2.364H^5/2

Tank Emptying/Filling

Without Inflow

Time to empty: T = 2A(√H₁-√H₂)/(C_da√(2g))

With Inflow (Q_in)

dt = A·dh/(Q_in - C_da√(2gh))

Momentum Principles

Linear momentum: F = ρQ(v₂ - v₁)
Angular momentum: T = ρQ(v₂r₂ - v₁r₁)
Force on plate by jet: F = ρA(v-u)²

Boundary Layer Concepts

The thin layer near a boundary where velocity varies from zero (at wall) to free-stream velocity.

Drag Force (F_D)

F_D = C_D½ρAv²

Parallel to flow direction

Lift Force (F_L)

F_L = C_L½ρAv²

Perpendicular to flow direction

Key Notes:

Free stream velocity is the undisturbed flow velocity
Dynamic pressure = ½ρv²
Convective acceleration occurs due to position change
Weirs measure river flow rates
Bernoulli's energy constant varies between streamlines

Fluid Mechanics Reference Guide

Definition of Fluid

Classification of Fluids

1. Based on Flow Characteristics

2. Based on Compressibility

3. Based on Viscosity

4. Based on Rotation

Types of Fluids

Properties of Fluids

1. Density (ρ)

2. Specific Weight (γ)

3. Specific Gravity (SG)

4. Specific Volume (v)

5. Viscosity (μ)

6. Kinematic Viscosity (ν)

7. Surface Tension (σ)

8. Capillarity

Pressure Types

Hydrostatic Force

Horizontally Immersed Surface

Vertically Immersed Surface

Inclined Immersed Surface

Curved Surface

Buoyancy

Lock Gates

Buoyancy and Floating Bodies

Center of Buoyancy

Metacenter (M)

Equilibrium Conditions

Experimental Determination of Metacentric Height

Liquid in Relative Equilibrium

Hydrostatic Forces

Flow Measurement Devices

Venturimeter

Pitot Tube

Flow Classification

Flow Visualization

Streamline

Pathline

Streakline

Fundamental Equations

Continuity Equation

Bernoulli's Equation

Important Notes

Orifice Flow

Small Orifice

Large Orifice

Fundamental Principles

Flow Potential Functions

Velocity Potential (ϕ)

Stream Function (ψ)

Energy Heads

Potential Head

Pressure Head

Velocity Head

Hydraulic Coefficients

Flow Measurement Devices

Orifice Meter

Venturimeter

Pitot Tube

Weirs and Notches

Tank Emptying/Filling

Without Inflow

With Inflow (Qin)

Momentum Principles

Boundary Layer Concepts

Drag Force (FD)

Lift Force (FL)

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With Inflow (Q_in)

Drag Force (F_D)

Lift Force (F_L)