Final answer:
The pressure gradient dp/dx for laminar flow in a pipe can be derived from Poiseuille's Law and is found to be proportional to the flow rate Q and inversely proportional to the fourth power of the pipe diameter D, with the constant K encapsulating viscosity and other constants.
Step-by-step explanation:
The question you've asked revolves around determining the pressure gradient ∂p/∂x in a fully developed laminar pipe flow given the density ρ, viscosity μ, pipe diameter D, and volumetric flow rate Q. The relationship for the pressure gradient in laminar flow for incompressible fluids is given by Poiseuille's Law, which can be derived from the Navier-Stokes equations under certain assumptions about the nature of the flow.
According to Poiseuille's Law, the volumetric flow rate Q is directly proportional to the pressure difference (P₂ - P₁) over the length l of the pipe, and inversely proportional to fluid viscosity μ. Moreover, it also depends on the fourth power of the pipe radius (r). The law is generally expressed as Q = (π(P₂ - P₁)r⁴)/(8μl), where π is the mathematical constant pi and r is the radius of the pipe.
To express the pressure gradient as a function of flow rate, we first acknowledge that the volumetric flow rate Q is equal to the product of the cross-sectional area A of the pipe and the average velocity u, which leads to Q = Au. Additionally, we know that A = (π/4)D² when expressed in terms of the pipe diameter D. Integrating these relationships into Poiseuille's formula and solving for the pressure gradient ∂p/∂x, we find that ∂p/∂x = -(8μQ)/( πr⁴). Since r = D/2, the equation simplifies to ∂p/∂x = -KQ/D⁴, where K is a constant that includes terms involving π and μ.