A numerical solution of the Navier-Stokes equations using the finite element technique

Laminar flow through a porous channel bounded by isothermal parallel plates

: The authors discuss the creation of a finite element program suitable for solving incompressible, viscous free surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. The authors conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related stick-slip problem and the tube inlet problem, both of which also contain stress singularities, are also given. (Modified author abstract)

The solution of viscous incompressible jet and free-surface flows using finite-element methods

The authors’ research work into fully developed pulsating and oscillating laminar pipe and channel flows raised questions regarding the development length of the corresponding steady flow. For this development length, i.e., the distance from the entrance of the pipe to the axial position where the flow reaches the parabolic velocity profile of the Hagen-Poiseuille flow, a wide range of contradictory data exists. This is shown through a short review of the existing literature. Superimposed diffusion and convection, together with order of magnitude considerations, suggest that the normalized development length can be expressed as L∕D=C0+C1Re and for Re→0 one obtains C0=0.619, whereas for Re→∞ one obtains C1=0.0567. This relationship is given only once in the literature and it is presumed to be valid for all Reynolds numbers. Numerical studies show that it is only valid for Re→0 and Re→∞. The development length of laminar, plane channel flow was also investigated. The authors obtained similar results to those for the pipe flow: L∕D=C0′+C1′; Re, where C0′=0.631 and C1′=0.044. Finally, correlations are given to express L∕D analytically for the entire Re range for both laminar pipe and channel flows.

The Development Lengths of Laminar Pipe and Channel Flows

Shear stress is an important biomechanical parameter in regulating human mesenchymal stem cell (hMSC) construct development. In this study, the biomechanical characteristics of hMSCs within highly porous 3-D poly (ethylene terephthalate) (PET) matrices in a perfusion bioreactor system were analyzed for two flow rates of 0.1 and 1.5 mL/min, respectively over a 20-day culture period. A 1.4 times higher proliferation rate, higher CFU-F formation, and more fibronectin and HSP-47 secretion at day 20 were observed at the flow rate of 0.1 mL/min compared to those at the flow rate of 1.5 mL/min. The higher flow rate of 1.5 mL/min upregulated osteogenic differentiation potential at day 20 as measured by the expression of alkaline phosphatase activity and calcium deposition in the matrix after 14 days osteogenic induction, consistent with those reported in literatures. Mathematical modeling indicated that shear stress existed in the range of 1 x 10(-5) to 1 x 10(-4) Pa in the constructs up to a depth of 70 microm due to flow penetration in the porous constructs. Analysis of oxygen transport in the constructs for the two flow rates yielded oxygen levels significantly higher than those at which cell growth and metabolism are affected (Jiang et al., 1996). This indicates that differences in convective transport have no significant influence on cell growth and metabolism for the range of flow rates studied. These results demonstrate that shear stress is an important microenvironment parameter that regulates hMSC construct development at a range significantly lower than those reported previously in the perfusion system.

Effects of shear stress on 3-D human mesenchymal stem cell construct development in a perfusion bioreactor system: Experiments and hydrodynamic modeling

Equations are given that relate the entrance length to Reynolds number for pipe and channel geometries with a flat velocity profile as the initial condition. These equations are linear combinations of the creeping flow and boundary layer solutions. The former is obtained by minimization of the viscous dissipation using the finite element method. The equation for the pipe entrance is shown to be in good agreement with experimental data.

Low Reynolds number developing flows

An optical technique for the quantitative determination of point velocities in rapidly developing laminar flows of transparent liquids is useful as an experimental tool in problems of complex geometries and fluids. The accuracy of the technique has been confirmed by comparison of experimental data with the numerical solution of the equation of motion for the entrance region problem. A distributor capable of generating a flat velocity profile has been developed for this purpose.

J. M. Smith

Papers

Low Reynolds number developing flows

Measurements of velocity profile in developing liquid flows