Abstract
We derive analytical expressions for the flow of Newtonian and power law fluids in elastic circularly symmetric tubes based on a lubrication approximation where the flow velocity profile at each cross section is assumed to have its axially dependent characteristic shape for the given rheology and cross sectional size. Two pressure–area constitutive elastic relations for the tube elastic response are used in these derivations. We demonstrate the validity of the derived equations by observing qualitatively correct trends in general and quantitatively valid asymptotic convergence to limiting cases. The Newtonian formulae are compared to similar formulae derived previously from a one-dimensional version of the Navier–Stokes equations.
Highlights
The flow of Newtonian and non-Newtonian fluids in distensible conduits of elastic or viscoelastic nature is common in many biological and industrial systems
The 1D Navier-Stokes distensible flow model is widely used in the flow simulations in single distensible tubes and networks of interconnected distensible tubes especially in the hemodynamic studies, it has a number of limitations
The method is based on a lubrication approximation where the flow velocity profile is assumed to be determined locally by the fluid rheology and the size of the local cross sectional area as in the case of the flow in a tube with a cross sectional area that, under flow state, is constant in shape and size over the whole tube length
Summary
The flow of Newtonian and non-Newtonian fluids in distensible conduits of elastic or viscoelastic nature is common in many biological and industrial systems. The 1D Navier-Stokes distensible flow model is widely used in the flow simulations in single distensible tubes and networks of interconnected distensible tubes especially in the hemodynamic studies, it has a number of limitations One of these limitations is its restriction to the Newtonian flow, on which the NavierStokes momentum equation is based, and to accommodate non-Newtonian rheologies, the use of approximations or employing different models, if they exist, is required. Another limitation is the relatively large number of parameters (e.g. α, κ, ρ, etc.) that define the 1D model; which make it difficult to choose proper numerical values for these parameters in practical situations without considerable amount of experimental and observational work, and some ambiguity or arbitrariness may be attached to the assumptions and results of this model. Broadening the proposed method to include conduit geometries other than the regular cylindrical tube with a constant cross sectional area in the axial direction (e.g. conduits of elliptically-shaped cross sections or of convergingdiverging nature) may be possible; we will not consider any of these extensions in the present paper
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