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Laminar non-Newtonian flow in a porous pipe

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The problem of flow of certain non-Newtonian liquids in a porous pipe is discussed by introducing second order terms in the stress-strain velocity relations of classical hydrodynamics. The Navier-Stokes equations resulting therefrom for the system have been solved exactly to obtain a complete description of the non-Newtonian flow. The analysis has been limited to two-dimensional steady-state laminar flow. The solution of the flow equations leads to detailed expressions for the dependence of the velocity components and the pressure on position coordinates, dimensions of the pipe and fluid properties.

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a :

radius of the porous pipe

p(x, λ) :

pressure in the pipe at the point (x, λ)

u(x, λ) :

velocity component in the x-direction at the point (x, λ) in the channel

ū 0 :

x-component of the velocity, averaged over the pipe cross-section at the entrance of the pipe x=0

v(x, λ) :

λ component of velocity at the point (x, λ) in the pipe

v w :

cross-flow velocity of the fluid at the wall

R :

Reynolds number for the flow through the pipe wall, R=av w /ν

N Re :

Reynolds number for flow entering the pipe through the inlet, N Re= 0/ν

λ :

dimensionless distance parameter, r/a

μ :

fluid viscosity

ν :

kinematic viscosity (μ/ρ)

μ 1 :


ν 1 :

kinematic cross-viscosity (μ 1/ρ)

K=ν 1/a 2 :

dimensionless parameter

ψ :

Stokes stream function

Ψ :

dimensionless stream function


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Narasimhan, M.N.L. Laminar non-Newtonian flow in a porous pipe. Appl. sci. Res. 10, 393 (1961).

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