The flow is laminar and fully developed, the pressure gradient in the direction of flow must have the form dp/dx = -K Qμ/ D⁴ where K is a constant.
Assumptions:
The flow is laminar and fully developed.
The fluid is incompressible and Newtonian.
The flow is steady and uniform.
The pipe is long and straight.
Governing equation:
The governing equation for laminar flow in a pipe is the Navier-Stokes equation:
ρ(∂u/∂t + u · ∇u) = -∇p + μ∇²u
where: ρ is the density of the fluid ,u is the velocity vector ,p is the pressure ,μ is the viscosity of the fluid
Dimensionless groups:
We can non-dimensionalize the Navier-Stokes equation by introducing the following dimensionless variables:
u* = u/U, p* = p/ρU², x* = x/D, r* = r/D
where:
U is the mean velocity of the fluid
r is the distance from the center of the pipe
Substituting these dimensionless variables into the Navier-Stokes equation, we get:
Re(∂u*/∂t* + u* · ∇*u*) = -∇*p* + ∇²*u*
where:
Re is the Reynolds number, defined as Re = ρDU/μ
Fully developed flow:
In a fully developed flow, the velocity profile and the pressure gradient do not change with downstream distance x.
This means that ∂u*/∂t* = 0 and ∂p*/∂x* = 0. Substituting these into the non-dimensionalized Navier-Stokes equation, we get:
Re(u* · ∇*u*) = -∇*p* + ∇²*u*
This equation can be simplified by using the fact that the flow is axisymmetric, which means that u* depends only on r* and not on the azimuthal angle.
This means that ∇u = ∂u*/∂r*. Substituting this into the equation above, we get:
Re(u* ∂u*/∂r*) = -∂p*/∂r* + ∂²/∂r*² u*
This equation can be integrated to get:
ReρU² u²/2 + ρp* = C₁ + C₂ r*/D
where:
C₁ and C₂ are constants of integration
Pressure gradient:
The pressure gradient can be calculated from the equation above:
dp*/dx* = ∂p*/∂r* = ReμU/D² (∂u*/∂r*)
Substituting the expression for u* from the equation above, we get:
dp*/dx* = -Re²μU²/D⁴
In dimensional variables, this becomes:
dp/dx = -K Qμ/ D⁴
where:
K = Re² = ρU²D²/μ² is a constant