Answered: Problem 6: Consider the incompressible…

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Consider the incompressible Newtonian pipe flow (Fig. 6). Assume the flow is essentially
axial, vz ≠ 0 but vr = v = 0 and ∂/∂ = 0. The flow is fully developed and steady in a horizontal pipe with
the constant pressure gradient. Apply the Navier-Stokes equations and:

$### Problem 6: Incompressible Newtonian Pipe Flow Consider the incompressible Newtonian pipe flow (Fig. 6). Assume the flow is essentially axial, $ v_z \neq 0 $ but $ v_r = v_\theta = 0 $ and $\partial/\partial \theta = 0$. The flow is fully developed and steady in a horizontal pipe with the constant pressure gradient. Apply the Navier-Stokes equations and: a) Develop simplified governing equations (continuity and momentum) for this flow; b) Apply the boundary conditions and determine the velocity profile; and c) Develop expressions for the flow rate and mean velocity from the velocity profile. ### Diagram Explanation **Fig. 6: Flow in Circular Pipe** The diagram illustrates a circular pipe with flow moving in the axial direction. The arrows indicate the direction and relative magnitude of the velocity, $ v_z $, which varies across the radius of the pipe. The centerline indicates the direction of maximum velocity, while the velocity decreases towards the pipe walls. The pipe is depicted in cross-section, showing the symmetrical nature of the flow profile.$

Transcribed Image Text:### Problem 6: Incompressible Newtonian Pipe Flow Consider the incompressible Newtonian pipe flow (Fig. 6). Assume the flow is essentially axial, $ v_z \neq 0 $ but $ v_r = v_\theta = 0 $ and $\partial/\partial \theta = 0$. The flow is fully developed and steady in a horizontal pipe with the constant pressure gradient. Apply the Navier-Stokes equations and: a) Develop simplified governing equations (continuity and momentum) for this flow; b) Apply the boundary conditions and determine the velocity profile; and c) Develop expressions for the flow rate and mean velocity from the velocity profile. ### Diagram Explanation **Fig. 6: Flow in Circular Pipe** The diagram illustrates a circular pipe with flow moving in the axial direction. The arrows indicate the direction and relative magnitude of the velocity, $ v_z $, which varies across the radius of the pipe. The centerline indicates the direction of maximum velocity, while the velocity decreases towards the pipe walls. The pipe is depicted in cross-section, showing the symmetrical nature of the flow profile.

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