Poiseuille Flow

12 Key excerpts on "Poiseuille Flow"

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  Introduction to Biomedical Engineering

  Biomechanics and Bioelectricity - Part I

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

    (Publisher)

  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. Note that the velocity must build in a parabolic manner from zero (at the walls) to a peak value (at the center). So the smaller the radius of the tube, the less distance the velocity has to build to a peak value, thus reducing this peak value and the volumetric flow rate. When the 32 CHAPTER 3. POISEUILLE’S LAW: PRESSURE-DRIVEN FLOW THROUGH TUBES two-dimensional nature of the cross-section is considered, the additional a 2 term is found, leading to the overall πa 4 dependence. fluid velocity u y z Figure 3.5: Velocity profile for laminar flow in a tube. This fourth power dependence on radius a is dramatic. It means that if the radius of a tube is reduced by 1/2, the flow rate will be reduced to 1/16th of its original flow! This would severely limit the amount of blood flowing through the tiny capillaries were it not for the fact that there are many, many capillaries in parallel in the vascular network. The presence of the viscosity term μ in the denominator of (3.2) also needs some discussion. Viscosity is a measure of the resistance of the layers of the fluid to flowing past one another—“sliding friction” as it were. We first saw the term in Chapter 2 relating to flow through membranes. The laminar flow profile in Fig. 3.5 requires that neighboring layers of fluid must slide past each other going from the walls to the centerline, imposing a shearing nature to the flow, as shown for a local region of the fluid in Fig. 3.6. velocity u distance y force F Figure 3.6: Layers of fluid sliding past each other, giving a shear rate u/ y , which requires a force F .

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  Mechanics of Fluids

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

    (Publisher)

  6.2 STEADY LAMINAR FLOW IN CIRCULAR PIPES: THE HAGEN–POISEUILLE LAW The law governing laminar flow in circular pipes was one of the first examples to be studied. About 1840, experimental investigations of flow in straight pipes of circular cross-section were carried out independently by two men. The first results, published in 1839, were the work of the German engineer G. H. L. Hagen (1797–1884). He had experimented with the flow of water through small brass tubes, and his figures showed that the loss of head experienced by the water as it flowed through a given length of the tube 192 Laminar flow between solid boundaries was directly proportional to the rate of flow, and inversely proportional to the fourth power of the diameter of the tube. At the same time a French physician, J. L. M. Poiseuille (1799–1869) was engaged in a long series of careful and accurate experiments with the object of studying the flow of blood through veins. In his experiments he used water in fine glass capillary tubes and arrived at the same conclusions as Hagen (although the reason for an apparent discrepancy found with very short tubes escaped him). His first results were made public in 1840. Subsequently the law governing laminar flow in circular pipes was derived theoretically. It is of interest to note that, invaluable though Poiseuille’s results have been in pointing the way to the theory of laminar flow in circular tubes, they are not really applicable to the flow of blood in veins; for one thing the walls of veins are not rigid, and also blood is not a Newtonian fluid, that is, it does not have a constant viscosity, even at a fixed temperature. Figure 6.1 shows, on the left, a side view of a straight pipe of constant internal radius R . On the right is shown the circular cross-section. We assume that the part of the pipe considered is far enough from the inlet for condi-tions to have become settled, or, to use the technical term, fully developed .

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  Micro- and Nanoscale Fluid Mechanics

  Transport in Microfluidic Devices

  • Brian J. Kirby(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  To solve for the Hagen–Poiseuille Flow, we assume that all flow is in the z direc-tion ( u = 0, u = 0) and that velocity gradients in the and z directions are zero. The Navier–Stokes equations are ∂ u ∂ t + u · ∇ u = −∇ p + ∇ 2 u . (2.26) Fig. 2.2 Poiseuille Flow in a circular tube. 47 2.1 Steady Pressure-and Boundary-Driven Flow Through Long Channels Simplified equation. Because of the geometric simplifications, the convection and un-steady terms are all zero, leaving: ∇ p = ∇ 2 u . (2.27) For radially symmetric flow, we write this as: ∂ p ∂ z = 1 ∂ ∂ ∂ u z ∂ . (2.28) Solution. To solve, we assume that ∂ p ∂ z is uniform, and integrate the terms: ∂ p ∂ z 1 2 2 + C 1 = ∂ u z ∂ , (2.29) rearrange, ∂ p ∂ z 2 + C 1 = ∂ u z ∂ (2.30) and integrate again to find u z = ∂ p ∂ z 2 4 + C 1 ln + C 2 . (2.31) By requiring that (a) u z be bounded at = 0 and (b) u z = 0 at = R , we solve and find u z = − 1 4 ∂ p ∂ z ( R 2 − 2 ) . (2.32) Poiseuille solution for flow in a tube of radius R We can integrate this distribution spatially to find the volumetric flow rate Q : Q = − R 4 8 ∂ p ∂ z , (2.33) flow rate in a tube of radius R caused by a pressure gradient and normalize by the area to generate the area-averaged velocity u z : u z = − ∂ p ∂ z R 2 8 . (2.34) mean velocity for flow in a tube of radius R caused by a pressure gradient Physical interpretation. The steady Poiseuille Flow has no convective transport of mo-mentum, nor acceleration. The governing equation highlights the balance between the net pressure forces and the net viscous force. Because the pressure gradient (and thus the net pressure force) is uniform, the derivative of the shear stress is uniform, and thus the concavity (i.e., 2nd derivative with respect to y ) of the distribution is uniform. If we draw a control volume around the fluid in its entirety, we can see that the net pressure

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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)

  3 Nature of Flow of a Liquid

  Wilmer Nichols

  DOI: 10.1201/9781351253765-3

  It is convenient in mechanics to classify matter into solids and fluids. A solid substance has a definite size and shape and can be either perfectly elastic or plastic (see Chapter 4 ). A fluid is a substance that can flow and includes both liquids and gases. Gases are easily compressed, whereas liquids are almost incompressible. This text deals almost exclusively with liquids and the physical laws that govern their flow.

  In considering the vascular system with its contained blood, we are naturally concerned with the physical laws governing the flow of liquids through solid tubes. The simplest example is that of a straight, uniform, rigid tube with a steady (or constant) rate of nonturbulent liquid flow through it. To maintain such a steady flow, there must be a constant head of pressure applied to the liquid because of its viscosity or “internal friction”. Steady flow in a rigid cylindrical tube is described by the well-known Poiseuille equation, which states that the pressure drop (or gradient) along the tube is directly proportional to the length of the tube, the rate of flow through the tube and the viscosity of the liquid and is inversely proportional to the fourth power of the internal radius. If dye is injected into the tube, the liquid in the axis of the tube moves much faster than that near the wall, and the front of the dye assumes a paraboloid shape. This occurs because the particles of liquid are flowing in a series of laminae parallel to the sides of the tube, the liquid in contact with the wall is stationary, and each successive lamina is slipping against the viscous friction of the lamina outside it. When flow occurs in such parallel laminae, it is called “laminar” (Kaufmann, 1963). If two tubes are joined to form a trunk and dye is injected into only one of them, there is no mixing transversely across the tube, and the two streams remain distinct and are seen to flow side by side in the main trunk; this phenomenon is called “streamlining” (McDonald, 1974; Nichols and O’Rourke, 1990, 1998). Hence laminar flow is often called “streamline flow”. Alternatively, it may be called “Poiseuille-type flow” because it obeys the Poiseuille equation (Caro et al., 1978; Milnor, 1989; Fung, 1997; Wood, 1999; Nakayama and Bouncher, 1999; Wang et al., 2015; Zhang et al

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  Introduction to Fluid Dynamics

  Understanding Fundamental Physics

  • Young J. Moon(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  1 Equation (7.18) shows that the averaged velocity of the Poiseuille Flow is found to be twice the centerline velocity: u = Q 𝜋R 2 = R 2 8 𝜇 |Δp| L = 1 2 u(0) = 1 2 u max (7.19) and that it is proportional to the pressure gradient of the pipe, as well as to the radius to the power of 2. Also, it is inversely proportional to the fluid viscosity. From Eq. (7.19), the pressure drop (or loss) can be written as follows: |Δp| = 64 Re D ⋅ L D ⋅ 𝜌 u 2 2 (7.20) 1 A hydrodynamic resistance R h can be defined as follows: R h = |Δp| Q = 8 𝜇 L 𝜋R 4 = f (𝜇, L, R) which is the ratio of the pressure difference to the volumetric flow rate. 236 7 Internal Viscous Flows Laminar Fully turbulent Transitional Critical 0.02 10 3 10 4 10 5 10 6 10 7 Re = UD/𝜐 0.04 0.06 0.08 Smooth Relative roughness 𝜀/D 0.0001 0.0004 0.001 0.003 0.006 inar Transitional i Critical Critical Darcy friction factor r Figure 7.4 Moody’s chart (pipe flow) If we express the pressure loss in the pipe as a head loss gh f = |Δp| 𝜌 (7.21) the head loss coefficient 𝜅 can be defined as follows: 𝜅 = gh f u 2 ∕2 = 64 Re ⋅ L D (7.22) where Darcy–Weisbach’s friction factor f is defined as follows: f = 𝜅 L∕D = 64 Re D (7.23) This relation of f vs. Re D , called Moody’s chart, is presented in Figure 7.4. In the laminar flow regime (Re D < 2000), this follows exactly Eq. (7.23) derived from the Poiseuille Flow solution. It is interesting to note that the loss coefficient 𝜅 is only a function of L∕D and Re D . 7.2.2.1 A Fully Developed Turbulent Flow If the flow in pipe transits from laminar to turbulent, the field of Poiseuille Flow drastically changes, reflecting turbulent characteristics such as random fluctuations of the physical quantities (e.g. momentum, heat), three-dimensional nature of vortex dynamics, and a turbulent energy cascade process. Figure 7.5 shows turbulent pipe flow computed by an incompressible large-eddy simula- tion at Re D = 5000.

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  Introductory Fluid Mechanics

  • Joseph Katz(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Viscous Incompressible Flow: Exact Solutions (Leading to Some Practical Engineering Solutions) 5.1 Introduction In Chapter 3 the effects of pressure in a fluid were isolated (because the fluid was not moving), and in Chapter 4 the inertia terms were added. The inclusion of viscosity, its effects, and the resulting velocity distribution are discussed here. For example, the velocity distribution for the laminar flow inside a pipe is formulated and the average velocity is calculated. This provides the relation between the simple 1D average velocity model (of Chapter 4) and the more complex (and realistic) 2D or 3D flows. The solutions presented early in this chapter are often called exact solutions . This means that, for a few limited cases, a set of logical assumptions leads to simplifi-cation of the fluid dynamic equations, which allows their solution (for laminar flow)! Also, the cases presented in this chapter (e.g., the flow in pipes) is often termed as internal flows. The discussion on external flows is delayed to the following three chapters. The second part of this chapter demonstrates the approach that evolved during the past 200 years for solving fluid dynamic problems (because there is no closed-form analytic solution to the complete fluid dynamic equations). According to this approach, to develop a practical engineering solution, we must start with a simple but exact solution that determines the major parameters and the basic trends of the problem (e.g., the pressure drop in a circular pipe versus the Reynolds number). Based on these parameters, an empirical database can be developed for treating a wider range of engineering problems. As an example, the viscous laminar flow model in circular pipes is extended into the high-Reynolds-number range and the effects of turbulent flow are discussed.

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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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  Essentials of Micro- and Nanofluidics

  With Applications to the Biological and Chemical Sciences

  • A. Terrence Conlisk(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  In this chapter, pressure-driven flow is considered exclusively, and only steady flows are presented. Once the fundamentals of electrostatics and electrochemistry are described, electrokinetically driven flow of an electrolyte solution, which also involves principles of mass transfer and electrostatics, will be combined with the results of this chapter. In the next section, the structure of steady flow in a pipe or channel is described, followed by the presentation of Poiseuille Flow in a channel and pipe. This is followed by a discussion of the slip boundary condition and the velocity profile for both gases and liquids. Next a series of other viscous flow problems are discussed, including thin film flow, the Blasius boundary layer, fully developed and developing suction flows, lubrication, flow around a bubble, Stokes flow past a sphere, and a simple model for blood flow. 4.2 The structure of flow in a pipe or channel The structure of flow in a pipe or a channel is depicted in Figure 4.2 for a pipe. For a channel, replace r with y and D with h or 2 h . Suppose that there is a large reservoir upstream of the entrance to the pipe or channel, as is the case 142 Essentials of Micro-and Nanofluidics U 0 Inviscid Developing Boundary Layers Entrance Length r z Laminar D Viscous Fully Developed Figure 4.2 Flow through a pipe, showing the entrance region followed by the fully developed region. in many of the microdevices already described. 2 Then the velocity profile at the inlet will be relatively uniform. To adjust to the presence of the solid walls, the fluid at the wall must assume the wall velocity and the no-slip condition, which, in the case of a stationary wall, is zero. This adjustment region, is the entrance region , and its length, the entrance length , is a function of the Reynolds number. For a pipe, the entrance length for laminar flow is conventionally taken to be ( Langhaar , 1942 ) L e D = 0 .

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  Monitoring of Respiration and Circulation

  • J. A. Blom(Author)
  • 2003(Publication Date)
  • CRC Press

    (Publisher)

  48 Monitoring of Respiration and Circulation ρ = density of the fluid v = fluid velocity as a function of x, y, z ∇ = nabla operator (spatial derivative) g = gravitation p = pressure η = viscosity of the fluid 1 ∆ = ∇ 2 Solutions of the Navier-Stokes equation nicely show the onset of turbulence, the interaction of shear layers, and most other interesting fluid dynamics phenomena. The Navier-Stokes equation, however, still has no general solution, although it can be solved for some practical situations where appropriate boundary conditions exist. It is also a starting point for finite element methods, even though computations may take hundreds of hours on the fastest computers available. 2.3. LAMINAR FLOW AND THE POISEUILLE EQUATION Since the Navier-Stokes equation is far too complex, we start by introducing a large number of simplifications. Although none of these is true in the respiratory and circulatory systems, they provide a good starting point for understanding. Some of these simplifications introduce only minor differences from what actually happens; other simplifications will be dropped in subsequent sections. We can derive Poiseuille’s formula from the Navier-Stokes equation when we assume a steady , laminar flow of a homogeneously viscous fluid (the viscosity is in-dependent of the shear rate) in a straight , horizontal , rigid tube with a circular cross section , where the friction of the fluid with the wall is much larger than the fluid’s internal friction; the latter results in a zero velocity at the wall. Converting to cylinder coordinates — because we have rotational symmetry — we obtain x p r v r 1 r v 2 2 ∂ ∂ =         ∂ ∂ + ∂ ∂ η (2.8) where v is the velocity component in the direction of the tube (all others are zero), r is the radial distance from the center of the tube, and x is the distance along the tube. The boundary condition is zero velocity at the wall: v (r 0 ) = 0.

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  Microfluidics: Modeling, Mechanics and Mathematics

  • Bastian E. Rapp(Author)
  • 2016(Publication Date)
  • Elsevier

    (Publisher)

  Fig. 16.4 Velocity and shear force profile in a channel of 10 µm height.

  Fig. 16.5 Velocity and shear force profile in a channel of 50 µ m height.

  16.3 Flows in Circular Cross-Sections: Hagen-Poiseuille Flow

  16.3.1 Introduction

  Obviously, circular cross-sections are a very commonly occurring in technical flow systems. Macroscopic examples include pipes and the tap; microfluidic examples include capillaries and circular tubes. The fluid mechanics of circular tubes or channels is commonly referred to as Hagen-Poiseuille 1 flow . As we will see, the flow profile is parabolic in shape and the shear stress is linear (see Fig. 16.6 ). Calculating the flow in circular cross-sections is not difficult as it exploits the fact that the Navier-Stokes equation in cylindrical coordinates is one-dimensional. There are two approaches deriving the same equation. We will first look at the direct solution of the Navier-Stokes equation in cylindrical coordinates which is the Euler equation.

  Fig. 16.6 Hagen-Poiseuille Flow in a circular tube. Flow profile (a) and shear stress profile (b).

  16.3.2 Derivation Using the Euler Equation

  Derivation. The first method for approaching the fluid mechanics of circular tube can be directly derived from the Euler equation (see Eq. 14.8 ), which we derived for the circular flow tube. As such, a circular tube or pipe is nothing else than a circular flow tube, which makes this approach very intuitive. Again we only consider the fully-developed and stationary case and neglect gravity. In this case the Euler equation simplifies to

  τ r

  +

  ∂ τ

  ∂ r

  =

  d p

  d z

    (Eq. 16.17)

  where we assume a pressure drop along the x -axis, which is independent of r ; this allows us to treat

  d p

  d x

  as a constant. If we reduce the fluids to incompressible Newtonian fluids we can use Eq. 9.5 to obtain a dependency between the shear forces τ and the velocity v . Please note that, as we will be working with cylindrical coordinates, we use z as the coordinate axis along the channel, in which case Eq. 16.17

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  Physics of Continuous Matter

  Exotic and Everyday Phenomena in the Macroscopic World

  • B. Lautrup(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  1 /L=a : (16.52) Torricelli’s result is only obtained for sufficiently short pipes where the denominator becomes unity. For sufficiently long pipes the terminal velocity decreases as 1= p L and this will even-tually bring the Reynolds number down into the smooth pipe region and finally into laminar one where it according to (16.43) decreases as 1=L . Example 16.8 [Barrel of wine, continued]: For the barrel of wine (Example 13.3 on page 215), which empties through a wooden spout of length L D 20 cm and diameter 2a D 5 cm, the Reynolds number is about 300,000, well into the rough pipe region. Disregarding entry cor-rections and assuming a wood roughness of k 0:5 mm, corresponding to k=2a D 0:01 , we find C f . 1 / 9 10 3 and 2C f . 1 /L=a 0:14 . The Toricelli value for the exit velocity is thus only reduced by about 7% by turbulence. 16.6 Laminar cylindric flow Maurice Marie Alfred Cou-ette (1858–1943). French physi-cist from the provincial university of Angers. Published only seven papers, all from 1888 to 1900. Invented what is now called the Couette viscometer. A layer of fluid lubricating the space between rotating coaxial cylinders is used in nearly all machines. Since the fluid must hang on to the cylindrical surfaces, they will exert a drag on each other that is similar to the drag created by the fluid between moving planar surfaces (page 247). The flow solution is attributed to Couette, but as always in science the true history is tortuous [DMS05]. Today, Couette flow is used to denote any kind of motion-driven laminar flow, including laminar channel flow between moving parallel plates. Maximally symmetric flow Infinitely extended coaxial cylinders are both translationally symmetric along the rotation axis and rotationally symmetric around it. In cylindrical coordinates a maximally symmetric circulating flow takes the form v D v .r/ O e ; (16.53) where O e is the tangential unit vector and r the distance from the axis (see Appendix D).

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  Munson, Young and Okiishi's Fundamentals of Fluid Mechanics

  • Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  For such conditions, the left side of the Poiseuille’s law can be obtained from the Navier– Stokes equations. 8.2 Fully Developed Laminar Flow 413 Eq. 8.14 is zero. This is equivalent to saying that the fluid experiences no acceleration as it flows along. The same constraint was used in the previous section when considering F = m a for the fluid cylinder. Thus, with g = − g k ˆ the Navier–Stokes equations become 𝛁 ∙ V = 0 𝛁 p + ρg k ˆ = μ 𝛁 2 V (8.15) The flow is governed by a balance of pressure, weight, and viscous forces in the flow direction, similar to that shown in Fig. 8.10 and Eq. 8.10. If the flow were not fully developed (as in an entrance region, for example), it would not be possible to simplify the Navier–Stokes equations to that form given in Eq. 8.15 (the nonlinear term ( V ∙ 𝛁 ) V would not be zero), and the solution would be very difficult to obtain. Because of the assumption that V = u ( r ) i ˆ , the continuity equation, Eq. 8.13, is automati-cally satisfied. This conservation of mass condition was also automatically satisfied by the incom-pressible flow assumption in the derivation in the previous section. The fluid flows across one section of the pipe at the same rate that it flows across any other section. When it is written in terms of polar coordinates (as was done in Section 6.9.3), the component of Eq. 8.15 along the pipe becomes ∂p ∂x + ρg sin θ = μ 1 r ∂ ∂r ( r ∂u ∂r ) (8.16) Since the flow is fully developed, u = u ( r ) and the right side is a function of, at most, only r . The left side is a function of, at most, only x . It was shown that this leads to the condition that the pres-sure gradient in the x direction is a constant— ∂p ∂x = −Δ p ℓ . The same condition was used in the derivation of the previous section (Eq. 8.3). It is seen from Eq. 8.16 that the effect of a nonhorizontal pipe enters into the Navier–Stokes equations in the same manner as was discussed in the previous section.

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