Laminar Pipe Flow
For steady flow in a pipe (whether laminar or turbulent), a momentum balance on the fluid gives the shear stress at any distance from the pipe centerline.
In Equation (1), Φ = P + ρ gz. The volumetric flowrate Q can be related to the local shear rate by doing an integration by parts of Equation (2).
Newtonian fluid. For a Newtonian fluid, τ _{rx} = µY _{rx}, which gives the following volumetric flowrate, known as the HagenPoiseuille equation.
It can be written in dimensionless form in Equation (4) with the two terms defined in Equations (5) and (6).
Power law. A fluid that follows the power law model obeys the relationship τ _{rx} = – µ(– Y _{rx}) ^{n}. This gives the following equation.
Equation (7) can be rearranged into the following dimensionless form.
Bingham plastic. In this case, there is a solidlike “plug flow” region from the pipe centerline (where τ _{rx} = 0) to the point where – τ _{rx} = τ _{0} (that is, at r = r _{0} = R x τ _{0}/ τ _{w}). The result is a flow integral modified from Equation (2). For a Bingham plastic, – τ _{rx} = τ _{0} + µ _{∞} ( – Y _{rx}). Using this expression and the modified flow integral, the BuckinghamReiner Equation (10) is found.
The equivalent dimensionless form is given by Equations (11), (12) and (13).
Turbulent Pipe Flow
Since most turbulent flows cannot be analyzed from a purely theoretical perspective, data and generalized dimensionless correlations are used.
Newtonian fluid. The friction factor for a Newtonian fluid in turbulent flow is a function of both N _{Re} and the pipe relative roughness, ε /D, which can be read off the Moody diagram [ 5]. The turbulent part of the Moody diagram (for N _{Re} > 4,000) is accurately represented by the Colebrook equation (14).
When N _{Re} is very large, the friction factor depends only on ε /D. This condition is noted with f _{T} as the “fully turbulent” friction factor in Equation (15).
The Churchill Equation [ 2] represents the entire Moody diagram, from laminar, through transition flow, to fully turbulent flow. It is presented here as Equations (16), (17), and (18).
Power law. For a powerlaw fluid, the friction factor depends only upon Equation (9) and the flow index, as represented by Equations (19) – (25) [ 3].
The value of N _{Re} where transition from laminar to turbulent flow occurs ( N _{Re,plc}) is given by Equation (25).
Bingham plastic. For the Bingham plastic, f _{T} is solely a function of N _{Re} _{∞} and N _{He}, as represented by Equations (26) – (29).
DefinitionsNewtonian fluid. A fluid is known to be Newtonian when shear stresses associated with flow are directly proportional to the shear rate of the fluid Power law fluid. A structural fluid has a structure that forms in the undeformed state, but then breaks down as shear rate increases. Such a fluid exhibits “power law” behavior at intermediate shear rates Bingham plastic fluid. A plastic is a material that exhibits a yield stress, meaning that it behaves as a solid below the stress level and as a fluid above the stress level 
Nomenclaturea Dimensionless parameter A Dimensionless parameter B Dimensionless parameter D Diameter, m f Fanning friction factor, dimensionless f _{L} Laminar friction factor, dimensionless f _{T} Fully turbulent friction factor, dimensionless f _{Tr} Transition friction factor, dimensionless g Gravitational acceleration, m/s ^{2} L Length of cylinder or pipe, m m Consistency coefficient, (N)(s)/m ^{2} n Power law fluid flow index, dimensionless N _{He} Hedstrom number, dimensionless N _{Re} Reynolds Number, dimensionless N _{Re,pl} Power law Reynolds Number, dimensionless N _{Re,plc} Power law Reynolds Number at transition from laminar to turbulent flow, dimensionless N _{Re∞} Binghamplastic Reynolds Number, dimensionless P Pressure, Pa Q Volumetric flowrate, m ^{3} /s r Radial position in a pipe or a cylinder, m R Pipe or cylinder radius, m V Velocity, m/s z Vertical elevation above a horizontal reference plane, m α Dimensionless parameter Y _{rx} Shear rate in tube flow, s ^{– 1} ε Wall roughness, m µ Newtonian viscosity, Pa – s µ _{∞} Bingham Plastic limiting viscosity, Pa – s ρ Density, kg/m ^{3} τ _{0} Yield stress, N/m ^{2} τ _{rx} Stress due to force in x direction acting on r surface, N/m ^{2} τ _{w} Stress exerted by fluid on tube wall, N/m ^{2} Φ Flow potential, P + ρ gz, Pa âˆ†Φ Increase in flow potential, Pa 
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