Newtonian flow problems circular tube

In this section, we instead consider two well-known examples of heat transfer in the fully developed, laminar, and unidirectional flow of a Newtonian fluid in a straight circular tube. We begin with a problem in which there is a prescribed heat flux into the fluid at the walls of the tube, so that there is a steady-state temperature distribution in the tube. At the end of the section, we consider the transient evolution of the temperature distribution beginning with an initially sharp temperature jump within the fluid at a fixed position (say z = 0), which illustrates an important phenomenon that is known as Taylor dispersion. 

Example 5.4 Flow of a Non-Newtonian Fluid. Write a general MATLAB function for solution of a boundary value problem by the shooting method using the Newton s technique. Apply this function to find the velocity profile of a non-Newtonian fluid that is flowing through a circular tube as shown in Fig. E5.4a. Also calculate the volumetric flow rate of the fluid. The viscosity of this fluid can be described by the Carreau model [5] ... 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

A circular, horizontal tube contains asphalt. Its viscosity is 100,000 cP( = 1000 P), and its density is 701bm/ftl The tube radius is 1 in. Asphalt may be considered a newtonian fluid for the purposes of this problem, although it is not always one. We now apply a pressure gradient of 1.0 (lbf/in )/ft. What is the steady-state flow rate I... 

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