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A fluid of density p and viscosity u flows through a long pipe of diameter D at the volume flow rate Q. (a) Demonstrate that if the flow is laminar (ie, totally steady) and fully developed (the velocity profile and the pressure gradient no longer change with downstream distance x), the pressure gradient in the direction of flow must have the form dp/dx = -K Qμ./ D⁴ where K is constant,

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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

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