Numerical solutions laminar pipe flow

Because the velocity field is fully developed, the variations of U and E with R are known. The solution to Eq. (7.93) can therefore be obtained using a similar procedure to that used in Chapter 4 to solve for thermally developing laminar pipe flow, i.e., using separation of variables. Here, however, a numerical finite-difference solution procedure will be used because it is more easily adapted to the situation where the wall temperature is varying with Z. 

Larrain and Bonilla conducted theoretical analysis of pressure drop in laminar flow of fluid in a coiled pipe [97]. They extended the series to 14th order and solved by means of computer. Austin and Seader came up with a comprehensive review of previous work and gave a detailed numerical solution in the whole laminar range [98]. Their solution, based on the vorticity field, gave excellent agreement with experiments but it did not yield any understanding of the complex interactions between the different forces. 

Roscoe, D. F., Numerical Solution of the Navier-Stokes Equations for a Three-Dimensional Laminar Flow in Curved Pipes using Finite Difference Methods, J. Engg. Math., 12, 303-323 (1978). 

Flow through resistive porous elements has been studied by many in the particle filtration community to determine basic relations and empirical correlations (Ergun, 1952 Jones and Krier, 1982 Laws and Livesey, 1978 Munson, 1988 Brundrett, 1993 Olbricht, 1996 Sodre and Parise, 1997 Wakeland and Keolian, 2003 Wu et al 2005 Valli et al 2009). A detailed and rigorous review of previous analytical and numerical solutions in porous pipe, annulus, and channel flow is reserved in Appendix F only highlights are presented here. Porous channel flow is classified by the size of flow within the channel (laminar or turbulent), the number of porous walls (one or two), the size (small, large, arbitrary), and nature (uniform or variable) of injection into the porous element, the type of transverse and axial boundary conditions at the porous surface (suction or injection), and whether or not there is heat transfer and/or electrical or magnetic component, where the injection Reynolds number is defined as ... 

---