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Characterization of Fully Developed Pressure-Driven, Shear-Driven and Combined Pressure and Shear Driven Flow of Sisko Fluids Through Rectangular Channels

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Abstract

Pressure-driven, shear-driven and combined pressure and shear driven flow of a non-Newtonian Sisko fluid through rectangular channels is investigated. Inclusion of the aspect ratio in the formulation yields a highly nonlinear partial differential equation, which is not reported in the existing literature. Thus, neither analytical nor numerical solution to this equation is available in the open literature. In the present study, the partial differential equation, describing the flow, is solved employing the finite difference method. Explicit method is adopted, and the solution for the non-dimensional velocity and wall shear stress is obtained. An exact solution for the flow of a Sisko fluid, for a special case (for non-Newtonian index 2), through large parallel plates (aspect ratio to be zero) is obtained. Expression for the friction factor, including the effect of the aspect ratio, is given. The effects of the aspect ratio, Sisko fluid parameter, non-Newtonian index on the non-dimensional velocity distribution and shear-stress distribution are analyzed both for shear-thinning and shear-thickening fluids. The results indicate that for pressure-driven flow, the effect of the aspect ratio on the velocity is negligible when it is less than 0.1. In case of shear-driven flow and combined pressure and shear driven flow also, the characteristics of flow through large parallel plates exist in nearly 50% of the channel for the aspect ratio of 0.1 or less, which means that for up to 50% of the channel, near the core, the parallel plates assumption will generate reasonably accurate results.

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Abbreviations

\( \tilde{A}_{1} \) :

1st Rivlin–Erickson tensor

\( \vec{B} \) :

Body force per unit volume (N/m3)

a :

Material constant (Ns/m2)

b :

Material constants (Ns/mn−1)

\( b^{*} \) :

Sisko fluid parameter

D e :

Hydraulic diameter (m)

f :

Friction factor

f fr :

Friction factor

g :

Acceleration due to gravity (m/s2)

\( \tilde{I} \) :

Identity matrix

k1, k2, k3, k4 :

Constant

\( \tilde{L} \) :

Velocity gradient matrix (1/s)

2L1 :

Depth of the channel (m)

2L2 :

Width of the channel (m)

n :

Material constant

p :

Pressure (N/m2)

r :

The locator where the maximum velocity occurs (m)

Re:

Reynold number

Rem :

Modified Reynolds number

ReN :

Reynolds number for Newtonian fluid

Rep :

Reynolds number for power law fluid

\( \tilde{S} \) :

Extra stress tensor (N/m2)

\( \tilde{T} \) :

Canely’s stress tensor (N/m2)

t :

Time (s)

u :

Dimensional coordinate along x direction (m/s)

\( u^{*} \) :

Non-dimensional coordinate along axial direction

u p :

Velocity of upper plate (m/s)

u avg :

Average velocity (m/s)

u 1 :

Velocity in region 1

u 2 :

Velocity in region 2

\( \vec{V} \) :

Velocity vector (m/s)

x, y, z :

Dimensional coordinates along axial, vertical and lateral direction (m)

x*, y*, z*:

Non-dimensional coordinates along axial, vertical and lateral directions

ρ :

Density (kg/m3)

τ xy :

Shear stress along axial direction (N/m)

τ xz :

Shear stress along lateral direction (N/m)

μ e :

Effective viscosity (Ns/m2)

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Acknowledgement

The work is supported by Indian Institute of Technology (ISM), Dhanbad (FRS/110/2017-18/MECH. ENGG.).

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Authors and Affiliations

  1. School of Mechanical Engineering, KIIT Deemed to be University, Bhubaneswar, 751024, India

    Sumanta Chaudhuri

  2. Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, India

    Satyabrata Sahoo

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  1. Sumanta Chaudhuri
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Correspondence to Satyabrata Sahoo.

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Chaudhuri, S., Sahoo, S. Characterization of Fully Developed Pressure-Driven, Shear-Driven and Combined Pressure and Shear Driven Flow of Sisko Fluids Through Rectangular Channels. Arab J Sci Eng 45, 5925–5947 (2020). https://doi.org/10.1007/s13369-020-04621-4

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