Excerpt from Basic Fluid Dynamics Principles - Application to Percutaneous Intervention

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Introduction: an idealized fluid flow model for mathematical simplicity

The science of fluid flow applies to many facets in the field of radiology, from arteriography and angioplasty to percutaneous drainage. Precise modeling of physiologic fluid flows under real-life flow conditions requires mathematics too complex for practical use in clinical medicine. A workable model may be devised from the first principles of engineering to create a first-order approximation that suits most purposes.

A steady-state laminar flow model of an ideal (Newtonian) fluid in a rigid circular pipe of uniform diameter is used by the author. Frictional (viscosity) energy losses are found for all fluids traveling through a length of conduit (pipe). This causes a pressure drop along the length of the pipe as a function of the pipe size, the type of fluid, and the mean flow velocity or flow rate.

The central formula in the laminar flow model is the Poiseuille equation:

R = (8hL)/pr⁴.

or the Poiseuille law:

Q = (pDPr⁴)/(8hL),

where P = pressure, n = mean flow velocity, Q = volumetric flow rate, L = tube length, h = fluid viscosity, and r = tube radius.

Origins of the Poiseuille equation

The underlying assumption of laminar flow is the condition of uniform viscosity across the diameter of the conduit, ie, each fluid molecule within the pipe is exerting a similar force against its immediate neighbor towards the periphery. This yields the following:

• Fluid adjacent to the conduit wall is motionless (n_o = 0).
• Maximum fluid velocity (n_max) is in the center of the conduit.
• Fluid velocity is related to distance from the center (n_i) by a parabolic function. This is why laminar flow sometimes is termed parabolic flow.

Another way to understand laminar flow conditions is to imagine concentric circles of fluid flow. Flow velocity is uniform within each annulus. Velocity increases as you approach the center of the conduit.

Further simplification: the Ohm law

For clinical applications fluid properties may be taken as constant. The Poiseuille equation can be simplified to resemble the Ohm law of electrical resistance. This law relates voltage drop across an electric circuit (DV) to electric resistance (R) and electric current (I), as follows:

DV = IR.

The greater the current (electron flow) or resistance, the higher the voltage drop required. For the sake of simplicity, DV often is written as V, although it is the change in voltage and not the absolute voltage that matters.

Fluid-flow calculations can be modelled as simple electric circuits. For this analogy (model) we relate DV to DP, defining flow resistance as R = 8Lh/pr⁴ to obtain the following:

DP= QR.

Resistive losses (pressure drops) linearly are related to flow rate and flow resistance. While flow resistance linearly relates to conduit length, it is inversely related to the fourth power of the radius (or diameter). For example, a 1-cm–diameter pipe has 16 times the flow resistance of a 2-cm–diameter pipe of the same length and carrying the same fluid. Stated another way, 16 conduits of 1-cm diameter are needed to handle the same fluid flow at the same pressure as a single 2-cm–diameter conduit; this is an amazing concept.

Related equations

From the Ohm law, equations for series and parallel circuits can be derived (see Image 3).

For a series circuit: R = R₁ + R₂ + R₃ + ....

For a parallel circuit: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ....

For the relative resistance of 2 conduits of equal length for the same fluid: R₁/R₂ = (D₂/D₁)⁴, where d corresponds to the diameter of the conduit.

For conduits of unequal length: R₁/R₂ = (L₁/L₂)(D₂/D₁)⁴.

For the relative flow rate for the same pressure decrease: Q₁/Q₂ = (D₁/D₂)⁴.

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