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Performance of one- and two-dimensional models for a slow flow system in a long, permeable tubule

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Abstract

A radially symmetric slow flow system in a long cylinder with moderate wall leakage is considered, with physical parameters taken from the renal proximal tubule. Dimensional analysis yields a simplified system which, under certain assumptions, is well-posed, and possesses a unique solution if a solution exists. A lumped parameter analysis generates a one-dimensional model identical to a typical one-dimensional model in a special case. The one- and two-dimensional models agree well in the situations examined. When the wall fluxes are of a certain class, the concentration to which the solution tends as the fluid proceeds down the tubule can be computed algebraically.

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Abbreviations

a, b:

terms in example inequalities

A, B, C, D:

real-valued functions of a scalar variable on [0, 1]

c:

concentration

c0,c1,c2 :

constants

C0,C1, C2 :

constants

D:

diffusion coefficient

d1, d2 :

scalar domains

f(r):

factor of nondimensional concentration

F:

flux

g(y):

factor of nondimensional concentration

G:

gas constant

h(r):

function arising from a solution of the PDM

I:

interval of integration

Js :

solute flux averaged over axial variable

K:

constant

kj :

constant associated with wall solute pump: j=1, constant pump; j=2, pump ∝ concentration

L:

length of PT

Lp :

hydraulic conductivity of wall of PT

M:

volume flow into PT from glomeralus

p:

pressure

P:

permeability of PT to solute

Q:

flow resistance

r:

radial variable

R:

radius of PT

Re:

Reynolds number

S:

nondimensional constant

Sc:

Schmidt number

u:

axial velocity averaged over radial variable

v:

velocity

x:

limit of integration

y:

axial variable

Y:

number in the interval [0, 1]

z:

dummy variable

α:

constant

γ:

dimensionless concentration-1

δ:

ratio of tubule radius to length

Δ:

difference between value inside and outside PT at wall

ɛ:

small positive parameter

μ:

viscosity

ϱ:

density

φ:

decimal percentage of fluid leaving the PT through the walls

σ:

reflection coefficient

ω:

dummy variable

f:

fixed value

i:

value at glomerular end of PT

0:

value at distal end of PT

r:

radial

s:

solute; serosa

v:

volume

y:

axial

∞:

limiting value

-:

log mean

~:

used to distinguish between the function ci(r) and the constant ci

boldface:

dimensional

normal typeface:

dimensionless

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Authors and Affiliations

  1. Mathematical Research Branch, National Institute of Arthritis, Diabetes and Digestive and Kidney Diseases, National Institutes of Health, 20892, Bethesda, MD, USA

    Kathleen Morrish

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  1. Kathleen Morrish
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Morrish, K. Performance of one- and two-dimensional models for a slow flow system in a long, permeable tubule. J. Math. Biol. 24, 237–258 (1986). https://doi.org/10.1007/BF00275636

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  • DOI: https://doi.org/10.1007/BF00275636

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