Poiseuille Flow in Tubes and Capillaries

FIGURE 8.6 Pressure-driven flow in a capillary tube. 

Furthermore, based on the axial symmetry of the tube, only derivatives of the stress tensor with respect to r are nonzero (Appendix 8.A). The differential momentum balance equations in cylindrical coordinates (Appendix 8.B) can be simplified as follows  

The dependencies on r arise from the structure of the velocity field, v = v (f)e. Since is only a function of r, Equation 8.9 can be readily rearranged to give 

The total volumetric flow rate in the capillary can be found  

The final integral in Equation 8.12 requires a priori knowledge of y(r) to determine Q. However, to determine y(r), we need to know v/r), which in turn requires knowledge of the relationship between shear stress and shear rate for the material the latter is precisely the objective of rheological measnrements. This dilemma can be avoided by directly transforming Equation 8.12 to obtain an explicit relationship between Q and Xg. To do this, we utilize the expression for x and rewrite Equation 8.10 as r = (R/xR)Xrz and use the result to rewrite Equation 8.12 in terms of x.  

---