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Experimental measurements of non-Newtonian fluid flow through a stenotic tube

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Abstract

This paper presents an experimental study of the laminar flow of a non-Newtonian fluid through 75% (by area reduction) stenotic tubes. The fluid behaviour was described by the Herschel Bulkey non-Newtonian model. The non-Newtonian fluids were aqueous solutions of 0.1% Carbopol 940. Upstream flow conditions were steady and spanned a range of generalized Reynolds numbers Reg from 0.20 to 13.66. The velocity profiles were measured with a Laser Doppler Anemometry (LDA). This study allows us to see locally the influence of the geometry and the non-Newtonian character of the fluid on the velocity profiles, the pressure drops and flow resistance. From the experimental data, the frictional resistance decreases with increasing generalized Reynolds number Reg and resistance gave a weak value in a stenotic tube as compared to the flow in a simple tube. At the level of stenosis, a correlation relating of the Euler number to the generalized Reynolds number is developed. To compare the upstream and downstream parts of the stenosis, it is preferable to represent the pressure drops by the friction factor f. This factor f in upstream and downstream decreases linearly with the generalized Reynolds number Reg.

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Abbreviations

D :

Tube diameter, m

Eu :

Euler number; \(\left(\frac{\Delta \mathbf{P}}{{\varvec{\uprho}}{\mathbf{U}}_{\mathbf{m}}^{2}}\right)\)

\(f\) :

Friction factor; \(\left(\frac{2{{\varvec{\tau}}}_{{\varvec{p}}}}{{\varvec{\rho}}{{\varvec{U}}}_{{\varvec{m}}}^{2}}\right)\)

k :

Constant in the Herschel–Bulkley model, Pa.sn

L :

Distance between two pressure taps, m

L 0 :

Length of the stenosis, m

n :

Flow behaviour index

p :

Pressure, Pa

\(\Delta \mathrm{P}\) :

Pressure drop difference, Pa

r :

Radial location within tube, m

R:

Tube radius, m

Reg:

Generalized Reynolds number; \(\left(\frac{{\varvec{\rho}}{{\varvec{U}}}_{{\varvec{m}}}^{2-{\varvec{n}}}{R}^{n}}{k}\right)\)

U m :

Average velocity, m/s

U z :

Axial velocity, m/s

U r :

Radial velocity, m/s

Q V :

Flow rate, m3/s

S :

Stenosis severity;\(\left(s=(1-{\delta }^{2}/{D}^{2})*100\%\right)\)

\(\gamma\) :

Shear rate (s−1)

\(\delta\) :

Contraction of stenosis, m

\(\rho\) :

Fluid density, kg/m3

\(\lambda\) :

Resistance, N.s.m−5; \(\left(\frac{\Delta \mathrm{P}}{\mathrm{Qv}}\right)\)

\({\lambda }_{simple}\) :

Resistance in simple tube, N.s.m−5

\({\lambda }_{stenosis}\) :

Resistance in stenotic tube, N.s.m−5

\(\tau\) :

Shear stress, Pa

\({\tau }_{ad}\) :

Shear stress adjusted, Pa

\({\tau }_{m}\) :

Shear stress measured, Pa

\({\tau }_{w}\) :

Wall shear stress, Pa\(\left(\frac{\Delta \mathrm{P D}}{4\mathrm{ L}}\right)\)

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Authors and Affiliations

  1. Department of Energy and Fluid Mechanics, University of Sciences and Technology Houari Boumedienne, USTHB, Algiers, Algeria

    Imane Trea, Mohamed Mahfoud & Fadila Haddad

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  1. Imane Trea
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  2. Mohamed Mahfoud
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  3. Fadila Haddad
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Appendix A: Statistical study

Appendix A: Statistical study

(1) Pearson coefficient

$$R=\frac{N\sum_{i=1}^{N}{\tau }_{m}{\tau }_{ad}-\sum_{i=1}^{N}{\tau }_{m}\sum_{i=1}^{N}{\tau }_{ad}}{\sqrt{\left(N\sum_{i=1}^{N}{{\tau }_{m}}^{2}-{\left(\sum_{i=1}^{N}{\tau }_{m}\right)}^{2}\right)\left(N\sum_{i=1}^{N}{{\tau }_{ad}}^{2}-{\left(\sum_{i=1}^{N}{\tau }_{ad}\right)}^{2}\right)}}$$

\({R}_{Bingham}=\) 0.939, \({R}_{\mathrm{Herschel}-\mathrm{bulkley}}=0.999\), \({R}_{Casson}=0.952\)

A correlation coefficient of 1 means that two variables are perfectly positively linearly related.

(2) Thiel coefficient

$$Te=\frac{\sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left({\tau }_{m}-{\tau }_{ad}\right)}^{2}}}{\sqrt{\frac{1}{N}\sum_{i=1}^{N}{{\tau }_{ad}}^{2}}+\sqrt{\frac{1}{N}\sum_{i=1}^{N}{{\tau }_{m}}^{2}}}$$

\({Te}_{Bingham}=\) 0.159, \({Te}_{\mathrm{Herschel}-\mathrm{bulkley}}=0.018\), \({Te}_{Casson}=0.25\)

With 0 ≤ Te ≤ 1.

If Te reaches 0, the model is favorable, else if go to 1, the model is not appropriate.

(3) Calculation of dispersion

$$D\left(\%\right)=\sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left(\frac{{\tau }_{ad}-{\tau }_{m}}{{\tau }_{m}}\right)}^{2}}.100$$

\({D}_{Bingham}=\) 12.85,\({D}_{\mathrm{Herschel}-\mathrm{bulkley}}=1.55\), \({D}_{Casson}=8.08\)

If the dispersion tends to zero, the model is more adjusted with the experimental values.

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Trea, I., Mahfoud, M. & Haddad, F. Experimental measurements of non-Newtonian fluid flow through a stenotic tube. Korea-Aust. Rheol. J. 34, 317–326 (2022). https://doi.org/10.1007/s13367-022-00038-7

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