Bernoulli Equation, Viscosity and Key Fluid Flow Formulas

Bernoulli's Equation and Energy Balance

Since the fluid is incompressible, A₁Δx₁ = A₂Δx₂ = ΔV.

Work–Energy Principle Derivation

The work done on the fluid is due to the pressure acting on it.

Therefore, the work done, ΔW = F₁Δx₁ - F₂Δx₂

= P₁A₁Δx₁ - P₂A₂Δx₂

= P₁ΔV - P₂ΔV

ΔW = (P₁ - P₂) ΔV

Change in kinetic energy, ΔKE = KE₂ - KE₁

= &frac12 mv₂² - &frac12 mv₁²

ΔKE = &frac12 m (v₂² - v₁²)

= &frac12 ρ ΔV (v₂² - v₁²)

Change in potential energy, ΔPE = PE₂ - PE₁

= m g h₂ - m g h₁ = m g (h₂ - h₁)

Since m = ρ ΔV, ΔPE = ρ ΔV g (h₂ - h₁)

Using the work–energy principle,

ΔW = ΔKE + ΔPE

(P₁ - P₂) ΔV = &frac12 ρ ΔV (v₂² - v₁²) + ρ ΔV g (h₂ - h₁)

Therefore, P₁ - P₂ = &frac12 ρ (v₂² - v₁²) + ρ g (h₂ - h₁)

Rearranging gives:

P₁ + &frac12 ρ v₁² + ρ g h₁ = P₂ + &frac12 ρ v₂² + ρ g h₂

Thus,

P + &frac12 ρ v² + ρ g h = constant

This equation is known as Bernoulli's equation.

Energy Terms per Unit Volume

• P = pressure energy per unit volume
• &frac12 ρ v² = kinetic energy per unit volume
• ρ g h = potential energy per unit volume

Newton's law of viscosity states that the shear stress between two adjacent layers of fluid is directly proportional to the velocity gradient between those adjacent layers (with a negative sign for the gradient direction in some sign conventions).

Flow Types: Laminar and Turbulent

Laminar Flow

If fluid particles move steadily in smooth paths in layers, with each layer sliding past adjacent layers without mixing, such a flow is called laminar flow. In steady laminar flow, streamlines do not cross, and every fluid particle arriving at a given point has the same velocity.

Turbulent Flow

If the motion of fluid particles is irregular, with changing directions and swirling, the flow is called turbulent flow. In turbulent flow, the speed of the fluid at a point continuously changes in both magnitude and direction.

Essential Fluid Flow Equations

1. Equation of Continuity (Volume Flow Rate):

   Q = A₁ v₁ = A₂ v₂

2. Bernoulli's Equation:

   P + &frac12 ρ v² + ρ g h = constant

3. Drag Force (Stokes' Law for a sphere):

   F = 6 π η r v

4. Formula for Viscosity (derived from Stokes' Law at terminal velocity):

   η = &frac{2 g r² (ρ₁ - ρ₂)}{9 v}

5. Height of Capillary Rise/Fall:

   h = &frac{2 γ cos θ}{ρ g r}

Determination of Kinematic Viscosity Using Stokes' Law

Consider the vertical fall of a sphere through a viscous fluid.

The forces acting on the sphere are:

• Weight of the sphere: w = m g = ρ₁ V g = &frac{4}{3} π r³ ρ₁ g
• Buoyant (upward) thrust: F_B = ρ₂ V g = &frac{4}{3} π r³ ρ₂ g
• Drag force: F = 6 π η r v

At terminal (constant) velocity, upward forces equal downward forces:

F_B + F = w

Substituting the expressions:

&frac{4}{3} π r³ ρ₂ g + 6 π η r v = &frac{4}{3} π r³ ρ₁ g

Rearranging gives:

6 π η r v = &frac{4}{3} π r³ (ρ₁ - ρ₂) g

Therefore, solving for viscosity:

η = &frac{2 g r² (ρ₁ - ρ₂)}{9 v}