Fluid Mechanics: Laplacian Operator and Laminar Flow Analysis
What is the Laplacian Operator?
The 3D balance equation in ψ terms is:
∂ψ/∂t + (v·∇)ψ = ψ̇a + (∇·δ∇ψ) − ψ(∇·v)
If diffusivity δ is constant, then:
(∇·δ∇ψ) = δ(∇·∇ψ)
Substituting this, we get:
∂ψ/∂t + (v·∇ψ) = ψ̇a + δ(∇·∇ψ) − ψ(∇·v)
The term (∇·∇ψ) is defined as:
∇[ i(∂ψ/∂x) + j(∂ψ/∂y) + k(∂ψ/∂z) ] = ∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z²
The dot product (∇·∇) operating on a scalar is given the special symbol ∇², called the Laplacian Operator.
Concentration Variation in a Cylindrical Tube
For a fluid flowing down a cylindrical tube with soluble walls (steady state, no chemical reaction, flat velocity profile U), we apply the Navier-Stokes equations:
- x-direction: Vx(∂Vx/∂x) + Vy(∂Vx/∂y) = gx + ν(∂²Vx/∂x² + ∂²Vx/∂y²)
- y-direction: Vx(∂Vy/∂x) + Vy(∂Vy/∂y) = gy + ν(∂²Vy/∂x² + ∂²Vy/∂y²)
For mass transfer:
- x-direction: Vx(∂C/∂x) + Vy(∂C/∂y) = gx + D(∂²C/∂x² + ∂²C/∂y²)
- y-direction: Vx(∂C/∂y) + Vy(∂C/∂y) = gy + D(∂²C/∂x² + ∂²C/∂y²)
Laminar Flow in a Tube
Assumptions: (i) Steady state, (ii) No entrance/exit effects, (iii) Constant density, (iv) Constant viscosity, (v) Laminar flow.
The velocity profile derivation results in:
u₂ = [(-ΔP)/(4μL)] (r₀² – r²)
Key results include:
- Maximum velocity: u₂max = (-ΔP · r₀²)/(4μL)
- Average velocity: u₂av = u₂max / 2
- Volumetric flow rate: Q = (-ΔP · πr₀⁴)/(8μL)
Tangential Flow Between Co-axial Cylinders
For an incompressible fluid between two vertical co-axial cylinders (outer rotating at angular velocity ω), the velocity distribution Uθ is derived from the Navier-Stokes equations in cylindrical coordinates:
Uθ = [ωr₀²/(r₀² – r₁²)] [ r – (r₁²/r) ]
The radial pressure drop is given by:
(∂P/∂r) = ρω²r₀⁴(r² – r₁²)² / [ r³(r₀² – r₁²)² ]
The shear stress at the inner cylinder (r₁) is:
τrθ|r₁ = -μ [ 2ωr₀²/(r₀² – r₁²) ]