cross sections various fittings and changes in pipe diameter through which the

# Cross sections various fittings and changes in pipe

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cross sections, various fittings and changes in pipe diameter (through which the flow is no longer fully-developed), and turbulent flows (including parametrizations associated with different levels of surface roughness). We begin with a statement of the governing equations, the Navier-Stokes equations in polar coordinates. We then provide a detailed treatment of the solution procedure, and we conclude the section with some discussions of the physics of the solution, including the derivation of mean and maximum velocities that will be of later use. Governing Equations The equations governing Hagen–Poiseuille flow are the steady, incompressible N.–S. equations in cylindrical-polar coordinates, in the absence of body-force terms. We list these here as ∂u ∂z + 1 r ∂r ( rv ) = 0 , (continuity) (4.34a) ρ parenleftbigg u ∂u ∂z + v ∂u ∂r parenrightbigg = ∂p ∂z + μ bracketleftbigg 2 u ∂z 2 + 1 r ∂r parenleftbigg r ∂u ∂r parenrightbiggbracketrightbigg , ( z momentum) (4.34b) ρ parenleftbigg u ∂v ∂z + v ∂v ∂r parenrightbigg = ∂p ∂r + μ bracketleftbigg 2 v ∂z 2 + 1 r ∂r parenleftbigg r ∂v ∂r parenrightbiggbracketrightbigg . ( r momentum) (4.34c)
130 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS Note that there are no θ -direction derivative terms due to the axisymmetric assumption imposed earlier for right-circular pipes. If we invoke the fully-developed flow assumption so that u z = 0 and v = 0, these equations can be readily reduced to ∂u ∂z = 0 , (4.35a) μ r ∂r parenleftbigg r ∂u ∂r parenrightbigg = ∂p ∂z , (4.35b) ∂p ∂r = 0 . (4.35c) This system of equations holds for a physical situation similar to that depicted in Fig. 4.11, but only beyond the x -direction point where the boundary layers have merged, i.e. , beyond the entrance length L e (corresponding to the fully-developed assumption). Figure 4.13 displays the current situation. R p 1 p 2 1 2 r L z Figure 4.13: Steady, fully-developed pipe flow. Solution Derivation We begin by noting that there is no new information in the continuity equation (4.35a) since this merely expresses one of the requirements for fully-developed flow. Next, we observe from Eq. (4.35c) that pressure does not depend on the radial coordinate, or more formally, p = C ( z ) . In particular, pressure can depend only on the z direction. (The reader should recall that we came to an analogous conclusion when studying plane Poiseuille flow.) Differentiation of the above with respect to z gives ∂p ∂z = ∂C ∂z , indicating that the z component of the pressure gradient can depend only on z . Now from the fact that the flow is fully developed it follows that u can be a function only of r , implying that the left-hand side of Eq. (4.35b) can depend only on r . But we have just seen that the right-hand side, ∂p/∂z , depends only on z . Thus, as was the case in plane Poiseuille flow, this pressure gradient must be a constant, i.e. , a trivial function of z , and we set ∂p ∂z = p L , (4.36)
4.5. PIPE FLOW 131 just as we did in the planar case treated earlier. Here, ∆ p p 1 p 2 , with the minus sign being used for later notational convenience. (Note that account of the minus sign shows that this is precisely the same pressure gradient treatment used in the plane Poiseuille case, Eq. (4.28).) Substitution

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