Deriving Poiseuille's Law From Dimensional Analysis
Poiseuille’s law tells you the rate of laminar flow through a cylindrical pipe.
The flow rate depends on the fluid’s viscosity, the cross-sectional dimensions of the pipe, and the pressure gradient.
Poiseuille's law can be determined from a simple procedure involving dimensional analysis.
Poiseuille’s law describes flow rate in a pipe and applies to areas like fluid flow in a capillary tube
Describing laminar flow starts from a set of complex partial differential equations, but it reduces to some simple relations describing the flow rate, particularly in a closed system. The imposition of an enclosed boundary on laminar flow, such as laminar flow in a pipe, allows one to derive an important relation describing the flow rate, known as Poiseuille’s law. This simple empirical law tells you the laminar flow rate through a cylindrical pipe in terms of the fluid viscosity, the pipe’s radius, and the pressure gradient that drives laminar flow.
While one would normally start from the Navier-Stokes equations to derive Poiseuille’s law, it can also be derived using dimensional analysis under some simple assumptions in laminar flow. In this article, we’ll show how to arrive at Poiseuille’s law from dimensional analysis.
Poiseuille's Law Definition
Poiseuille’s law states that the flow rate through a circular pipe is:
- Proportional to the fourth power of the pipe radius (or square of the cross-sectional area).
- Inversely proportional to the dynamic viscosity.
- Proportional to the pressure gradient.
This is summarized in a simple equation for the volumetric flow rate:
Poiseuille’s law describing the volumetric flow rate of a fluid in the laminar regime
The various parameters in Poiseuille’s law are shown graphically below. The above equation technically describes the increase in flow rate along a pipe due to an applied pressure gradient, which is assumed to be constant along the length of the pipe.
Important parameters in Poiseuille’s law
Analytical Derivation of Poiseuille's Law
To derive the above result for Poiseuille’s law, we start by reducing the derivative terms in the Navier-Stokes equation of motion given incompressible laminar flow (i.e., using the divergenceless property of the flow velocity). This gives a simple relationship between the radial component of the flow velocity and the pressure gradient along the direction of the flow.
Since the differential element in the gradient on the LHS is rdr, we can integrate over the pipe radius to get the following function:
This assumes that the pressure gradient in the pipe cross-section is constant (not a function of r). A final integration through the cross-sectional area gives the volumetric flow rate as described by Poiseuille’s law:
Poiseuille's Law From Dimensional Analysis
In general, because the Navier-Stokes equation of motion is a nonlinear partial differential equation, it could produce a flow rate result that is nonlinear in terms of the pressure gradient, viscosity, and pipe geometry. Therefore, to proceed from dimensional analysis, we simply assume that the flow rate is proportional to some power of the flow parameters:
Given the SI unit definitions of dynamic viscosity, pressure gradient, and flow rate (volume per second), one immediately finds that we must have 𝛽 = -Ɣ in order to eliminate force from the result. Finally, we would have 𝛼 = 4 in order to reach the units for flow rate on the LHS of the above equation. This elementary exercise omits the proportionality constant (π/8) because that value is particular to a circular cross-sectional area. In general, for a non-circular cross section, the proportionality constant could be different and will depend on the coordinate system used to describe the cross-sectional area of the pipe.
Extension to Non-Circular Cross Sections
The extension of this relation to non-circular cross-sectional pipes is performed with a functional transformation. This is a standard technique in analysis of differential equations that is used to transform a derivative between two different coordinate systems. For example, we could transform from a circular cross section to an elliptical cross section, square cross section, or rectangular cross section. Each of these pipe shapes are standard in different practical systems.
To do this, one must determine a relationship between the differential elements in the dv/dr term in the pressure gradient. This requires some functional relationship between the coordinate definitions in the two coordinate systems:
A functional relationship between two coordinate systems can be used to derive Poiseuille’s law for non-circular cross sections
From here, a relationship between the differential elements can be determined and substituted into the derivative relationship shown above.
Where Poiseuille's Law and Dimensional Analysis Fail
The use of dimensional analysis and Poiseuille’s law fails in two situations that are not mutually exclusive:
- In turbulent flow where the velocity variation across the pipe cross section occurs in 3D.
- In compressible flow, such as with gases, whether they are laminar or turbulent.
For example, gases are compressible, so they might not have the same strict fourth power dependence between flow rate and pipe radius. Similarly, the flow rate may not be a linear function of pressure gradient; this is known to be the case for gases undergoing isothermal compression. The gas could compress without a significant increase in dynamic viscosity, so we would not have a fourth-power change in flow rate. In turbulent flow, we would, at minimum, have a gradient defining the velocity change across the pipe cross section. Accounting for both factors requires CFD simulations to determine the flow rate and its dependence on pipe cross section in enclosed flows.
In more complicated systems, you can’t determine complex turbulent flow behaviors because Poiseuille's law and dimensional analysis cannot treat vortical flow. Instead, use the complete set of CFD simulation features in Omnis from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented in Cadence’s CFD simulation tools.
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