Hagen Poiseuille Equation

6 Key excerpts on "Hagen Poiseuille Equation"

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  Computational Bioengineering: Current Trends And Applications

  Current Trends and Applications

  • Miguel Cerrolaza, Manuel Doblare, Gabriela Martinez(Authors)
  • 2004(Publication Date)
  • ICP

    (Publisher)

  He only pointed out a possible connection with temperature changes. This formula was also obtained by Wiedeman and Hagenbach who independently published in 1856 and 1860 that: (1) V = K -11 K=- 1287 i.e. as expected by Poiseuille, it is a parameter depending on temperature since it is a function of the dynamic viscosity coefficient q. Therefore it is reasonable according to the formula, i.e. Hagen - Poiseuille’s flow used nowadays, another name should be assigned - namely Gotthilf Hagen. Despite the fact that Poiseuille first considered the connection (l), Hagen also confirmed experimentally that: 163 (2) 164 J. Wojnarowski - d 4 The Hagen - Poiseuille’s rule is considered at present as so obvious that its importance cannot be overestimated. It should be admitted that the problem was, among others, considered by Claude Luis Navier, recognised as the founder of fluid mechanics, who established that: V - d’ (4) but as we know - this was wrong. The model of Hagen - Poiseuille’s flow provides a theoretical and basic, but very usefil theory for the analysis of flow phenomena in blood vessels. This model is confirmed by the works of Wormersly [3,4], McDonald [l] or more recently by Rodkiewicz [5]. In the present paper the Hagen - Poiseuille’s model has been used to analyse the influence of human blood state parameters, such as the hemocritic number and level of protein in the plasma (more precisely the fibrinogen and globulin level), on the behaviour of the velocity field in the cross-section of blood vessels. The values of pressure drop that arises during the blood flow through a vessel are of great importance. The formulation of the problem allows us to solve it analytically. Simultaneously, it allows focusing on the connection between the flow and hematological parameters which are the source of several uncertainties and controversies.

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  Mechanics of Fluids

  • John Ward-Smith(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  The Hagen–Poiseuille formula was developed on the assumption that the centre-line of the pipe was straight. Slight curvature of the centre-line, in other words, a radius of curvature large compared with the radius of the pipe, does not appreciably affect the flow through the pipe. For smal-ler radii of curvature, however, the flow is not accurately described by the Hagen–Poiseuille formula. 196 Laminar flow between solid boundaries Example 6.1 Oil, of relative density 0.83 and dynamic viscosity 0.08 kg · m -1 · s -1 , passes through a circular pipe of 12 mm diameter with a mean velocity of 2.3 m · s -1 . Determine: (a) the Reynolds number; (b) the maximum velocity; (c) the volumetric flow rate; (d) the pressure gradient along the pipe. Solution Denote the mean velocity by u . ( a ) Re = ρ ud μ = 0.83 × 1000 kg · m -3 × 2.3 m · s -1 × 12 mm 0.08 kg · m -1 · s -1 × 1000 mm / m = 286 This value of Reynolds number is well within the laminar range, so the relations for laminar flow may be used throughout the remainder of the question. ( b ) u max = 2 u = 2 × 2.3 m · s -1 = 4.6 m · s -1 ( c ) Q = π 4 d 2 u = π 4 12 10 3 m 2 × 2.3 m · s -1 = 260 × 10 -6 m 3 · s -1 ( d ) d p * d x = -128 Q μ π d 4 -128 × ( 260 × 10 -6 ) m 3 · s -1 × 0.08 kg · m -1 · s -1 × ( 10 3 mm / m ) 4 π × 12 4 mm 4 = -40.9 × 10 3 kg · m -2 · s -2 = -40.9 × 10 3 Pa · m -1 The negative pressure gradient indicates that the pressure decreases with distance along the pipe axis. 6.2.1 Laminar flow of a non-Newtonian liquid in a circular pipe Results corresponding to those in Section 6.2 may be obtained for a non-Newtonian liquid. We again consider steady flow and thus suppose that the rate of shear is a function of τ only, say f (τ) . Then, for fully developed flow in a circular pipe, d u d r = f (τ) (6.11)

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  Introduction to Biomedical Engineering

  Biomechanics and Bioelectricity - Part I

  • Douglas Christensen(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  Figure 3.4 shows the arrangement analyzed. 3.2. POISEUILLE’S LAW 31 volume flow rate Q length = l pressure P 1 pressure P 2 radius a Figure 3.4: Tube for illustrating Poiseuille’s Law. In Fig. 3.4, a volume flow rate Q of fluid passes through a tube of length l under a pressure difference P = P 1 − P 2 . Hagen and Poiseuille found the following relationship: Q = πa 4 8μ P l , Poiseuille’s Law (3.2) where Q = volume flow rate (m 3 /s) P = P 1 − P 2 = pressure difference (Pa or N/m 2 or kg/m·s 2 ) P 1 = fluid pressure at entrance to tube (Pa) P 2 = fluid pressure at exit from tube (Pa) a = tube radius (m) μ = fluid viscosity (kg/m·s or Pa·s), and l = length over which the pressure drop is measured (m). The existence of the various terms in (3.2) can be qualitatively explained by a ranging check using each of the variables in turn (except for the factor 8, which is only found by a mathematical derivation beyond the scope of this chapter). That P appears in the numerator seems reasonable, since for a given length of tube, the higher the driving pressure, the higher the flow should be. The location of l in the denominator also makes sense, since the longer the tube, the less the flow for a given pressure drop (as anyone who has used a very, very long garden hose knows). The terms πa 4 and μ in (3.2) need a little more explanation. One portion of the πa 4 term (namely πa 2 ) can be seen simply from the fact that the tube’s cross-sectional area is given by πa 2 and volume flow is proportional to cross-sectional area for a given pressure drop. The remaining a 2 dependence requires a look at the fluid flow velocity inside the tube. For Poiseuille’s Law to hold, the flow profile is assumed to be laminar (or “layered”), as diagrammed in Fig. 3.5. A laminar profile is characterized by a “no-slip” condition at the walls; that is, the fluid velocity is zero where the fluid touches the walls, and then the velocity increases parabolically toward the peak velocity at the centerline of the tube.

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  McDonald’s Blood Flow in Arteries

  Theoretical, Experimental and Clinical Principles

  • Wilmer W. Nichols, Michael O'Rourke, Elazer R. Edelman, Charalambos Vlachopoulos, Wilmer W. Nichols, Michael O'Rourke, Elazer R. Edelman, Charalambos Vlachopoulos(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  Poiseuille used rigid uniform capillary tubes varying in internal diameter from 0.03 to 0.14 mm because of his conclusion that tubes of such small size provided the greatest resistance to blood flow through the circulation. His results were expressed by the equation

  Q =

  K

  D 4

  P 1

  −

  P 2

  L

  ,

  3.2

  where, as previously defined, Q, (P1 − P2 ) and L are the volume flow, pressure drop, and length of the tube, respectively; D is the tube internal diameter, and K is a constant.

  The value of K was determined under various conditions and shown to fall with decreasing temperature. This constant is clearly a measure of the viscosity, but by purely experimental work it is not possible to define it other than empirically.

  The form of Poiseuille’s equation with which we are familiar is, in fact, dependent on the theoretic solution of the problem. Navier’s early work on the equations of motion for viscous liquids were amplified and corrected by Stokes in the 1840s, and the Navier–Stokes equations are the general solution of this problem. Stokes, however, did not tackle the particular case of liquid flow in a tube. The solution of this case was made independently by Wiedemann in 1856 and Hagenbach in 1860, who both produced the result that

  Q =

  π

  R 4

  P 1

  −

  P 2

  8 μ L

  ,

  3.3

  where µ is the dynamic viscosity of the liquid, and R the internal radius of the tube. It can thus be seen that Poiseuille’s constant K is π/128µ (as R4 = D4 /16), or µ = π/128K.

  Eduard Hagenbach (1833–1910) calculated µ from Poiseuille’s data and obtained the result in modern units of 0.013084 poise at 10°C (the modern value is 0.013077 poise) (see Bingham and Jackson, 1918; Barr, 1931; Sutera and Skalak, 1993).

  The method of derivation of the solution used by Hagenbach was a simple one, and it seems odd that it should have taken so long to produce. Compared with the advances made in pure mathematics by the middle of the nineteenth century, this is a very elementary problem (in applied mathematics) and furthermore a problem that had been subjected to experimental investigation for over 40 years.

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  Fluid Mechanics for Petroleum Engineers

  • E. Bobok(Author)
  • 1993(Publication Date)
  • Elsevier Science

    (Publisher)

  For a laminar flow the Hagen-Poiseuille equation gives the head loss as (8.39) For practical applications this equation has to be somewhat modified. In en- gineering practice the diameter of a pipe is given rather than the radius. We also wish to obtain an equation for the head loss which depends explicitly on the Reynolds number. Expressing Eq. (8.39) in terms of the diameter D and rearrang- ing the terms, we have (8.40) which may be written as (8.41) L c2 D 29 h;-2=A- -, where A is called the friction factor. It can be calculated from the equation 64 Re A= -, (8.42) This expression is valid for the range of Re < 2300, i.e. laminar flow only. If the Reynolds number of the flow exceeds the critical value of 2300, the flow becomes turbulent, and the friction factor increases abruptly. Thus the smallest value of the friction factor is obtained for laminar flow immediately before the transition. For Re = 2300 the friction factor is A=-.- -0.0218 2300 If the laminar-turbulent transition could be retarted the friction factor, and thus the head loss, would be much smaller. This is a frequently applied technique in the petroleum industry. Long-chain polymer additives can reduce the friction factor of a “solvent”, thus retarding the laminar-turbulent transition. This phenomenon will be discussed in detail in the next chapter. Turbulent flow is a more complex phenomenon compared to laminar flow. Thus the determination of the head loss for such flows is a rather difficult problem. For turbulent flow in a pipe two cases can be distinguished. If the laminar sublayer covers the surface roughness of the pipe wall the turbulent flow is not affected by the roughness, as the pipe can be considered to be absolutely smooth. This case is referred to as that of a turbulent flow in a hydraulically smooth pipe. On the other hand there are flows, where the laminar sublayer cannot cover the surface rough- ness of the pipe wall, or there is no laminar sublayer at the pipe wall.

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  Transport by Advection and Diffusion

  • Ted Bennett(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 31 Fully Developed Turbulent Flow 31.1 Turbulent Poiseuille Flow between Smooth Parallel Plates 31.2 Turbulent Couette Flow between Smooth Parallel Plates 31.3 Turbulent Poiseuille Flow in a Smooth-Wall Pipe 31.4 Utility of the Hydraulic Diameter 31.5 Turbulent Poiseuille Flow in a Smooth Annular Pipe 31.6 Reichardt’s Formula for Turbulent Diffusivity 31.7 Poiseuille Flow with Blowing between Walls 31.8 Problems This chapter treats turbulent momentum transport in fully developed internal flows using the mixing length model developed in the context of boundary layers in Chapter 30. There would be little hope of quantitative success in applying the same model if similar turbulent behavior were not exhibited for these two types of flows. Fortunately, experi- mental measurements of turbulent flows through smooth pipes have demonstrated the same near wall velocity profile as for turbulent boundary layers. Nikuradse [1] found that the velocity profile near the wall is given by u þ ¼ 2:5 ln y þ þ 5:5 ð31-1Þ where the wall coordinates are defined the same as in Chapter 30, u þ ¼ υ x ffiffiffiffiffiffiffiffi t w = ρ p y þ ¼ y ffiffiffiffiffiffiffiffiffi t w = ρ p ν , ð31-2Þ and y ¼ R  r is the distance from the wall of the pipe of radius R. Nikuradse’s equation implies mixing length constants of κ ¼ 0:40 and E þ ν ¼ 11:6. In contrast, the values used for boundary layers in Chapter 30 were κ ¼ 0:41 and E þ ν ¼ 10:8. Other experimental inves- tigations into κ and E þ ν show some scatter around the values given by the Nikuradse equation. (A review of some of this literature can be found in reference [2].) A value of E þ ν ¼ 11:6 corresponds to A þ ν ¼ 27 in the Van Driest damping function (although a value of A þ ν ¼ 26 is used more often in the literature). For boundary layers, the dimension of the momentum boundary layer thickness limits the largest scale of the mixing length. In contrast, the dimension of the pipe or channel for internal flows limits the largest scale of the mixing length.

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