Obtain the governing differential equation for unidirectional flow in a circular pipe. Use N-S equations and employ cylindrical coordinates.
To obtain the governing differential equation for unidirectional flow in a circular pipe using the Navier-Stokes (N-S) equations and cylindrical coordinates, we'll follow these steps:
Step 1: Identify the flow assumptions
In unidirectional flow in a circular pipe, we assume that the flow is steady (does not change with time) and fully developed (the velocity profile does not change along the length of the pipe).
Step 2: Write down the N-S equations in cylindrical coordinates
The N-S equations for an incompressible fluid in cylindrical coordinates are:
Continuity equation:
∂ρ/∂t + (1/r)(∂(rρu)/∂r) + (1/r)(∂(ρv)/∂θ) + (∂(ρw)/∂z) = 0
Momentum equations:
∂(ρu)/∂t + (1/r)(∂(rρu^2)/∂r) + (1/r)(∂(ρuv)/∂θ) + (∂(ρuw)/∂z) = (-1/ρ)(∂p/∂r) + ∂(τ_rr)/∂r + (1/r)(∂τ_θr)/∂θ + (∂τ_rz)/∂z + ρg_r
(1/r)(∂(rρv)/∂r) + (1/r)(∂(ρv^2)/∂θ) + (∂(ρvw)/∂z) = (-1/ρ)(∂p/∂θ) + ∂(τ_rθ)/∂r + (1/r)(∂τ_θθ)/∂θ + (∂τ_θz)/∂z + ρg_θ
∂(ρw)/∂t + (1/r)(∂(rρuw)/∂r) + (1/r)(∂(ρvw)/∂θ) + (∂(ρw^2)/∂z) = (-1/ρ)(∂p/∂z) + ∂(τ_rz)/∂r + (1/r)(∂τ_θz)/∂θ + (∂τ_zz)/∂z + ρg_z
Step 3: Apply simplifying assumptions
- Since the flow is unidirectional, we assume there is no velocity component in the θ (azimuthal) direction. Thus, v = 0 and (∂(ρv)/∂θ) = 0 in all equations.
- In fully developed flow, the velocity does not change along the pipe length, so (∂(ρu)/∂z) = (∂(ρw)/∂z) = 0.
- Since it is assumed that the pipe is circular, there are no variations in the radial direction, so (∂(τ_rθ)/∂r) and (∂(τ_θr)/∂θ) become zero.
Step 4: Simplify the equations
Applying the above assumptions, the simplified continuity equation becomes:
(1/r)(∂(rρu)/∂r) + (∂(ρw)/∂z) = 0
The simplified momentum equations become:
(1/r)(∂(rρu^2)/∂r) = (-1/ρ)(∂p/∂r) + ∂(τ_rr)/∂r + ρg_r
∂(ρw^2)/∂z = (-1/ρ)(∂p/∂z) + ∂(τ_zz)/∂z + ρg_z
Step 5: Apply the no-slip condition and symmetry
At the pipe walls, there is no slip, meaning the velocity is zero. This condition results in u = w = 0 at r = R, where R is the radius of the pipe.
Step 6: Simplify further by neglecting body forces
If we neglect body forces (such as gravity), the simplified momentum equations become:
(1/r)(∂(rρu^2)/∂r) = (-1/ρ)(∂p/∂r) + ∂(τ_rr)/∂r
∂(ρw^2)/∂z = (-1/ρ)(∂p/∂z) + ∂(τ_zz)/∂z
So, after simplifying and neglecting body forces, the governing differential equations for unidirectional flow in a circular pipe using N-S equations and cylindrical coordinates are:
(1/r)(∂(rρu)/∂r) + (∂(ρw)/∂z) = 0 (Continuity equation)
(1/r)(∂(rρu^2)/∂r) = (-1/ρ)(∂p/∂r) + ∂(τ_rr)/∂r (Radial momentum equation)
∂(ρw^2)/∂z = (-1/ρ)(∂p/∂z) + ∂(τ_zz)/∂z (Axial momentum equation)
These equations can be further simplified based on the specific conditions of the flow or pipe geometry, if necessary.