Non-Newtonian Fluids in Tubes and Channels

Let us consider a steady-state axisymmetric flow of a non-Newtonian fluid in a straight horizontal circular tube of radius a. The coordinate Z is measured along the tube axis and is directed downstream. We restrict our consideration to the hydrodynamically stabilized flow far from the input cross-section, where the streamlines are parallel to the tube axis. In this case, the pressure increment decreases with increasing Z, and the pressure gradient is negative and constant, 

In this problem, all derivatives of the velocity with respect to t, Z, and ip, as well as the velocity components and Vn, are zero. Thus, the equation of motion (see Supplement 6) is 

One can see that the absolute value of the friction stress linearly increases from zero on the tube axis to its maximum value rs = aAP/L on the tube wall irrespective of the type of the non-Newtonian fluid. 

General formulas. The shear rate in the tube is negative, that is, 7 = dV/ dlZ 0, where V = Vz- In the general case of nonlinearly viscous fluids, we represent the dependence of the shear rate on the stress as follows  

By substituting the expression (6.4.2) into (6.4.3), we obtain an equation for the fluid velocity V - Vz- The solution of this equation satisfying the no-slip condition on the tube wall (V - 0 at H = a) has the form 

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