Abstract
The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.
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Abbreviations
- h :
-
step length in radial direction
- I 0 :
-
modified Bessel function of first kind order zero
- I l :
-
modified Bessel function of first kind of order one
- k(r) :
-
permeability function =\(\bar k(\bar r)/R^2 = k_0 f(r)\)
- k 0 :
-
permeability factor
- f :
-
radial variation of permeability (=1 or (1−r)/r)
- N :
-
number of sub-interval in radial direction
- p :
-
dimensionless pressure
- P 0 :
-
characteristic pressure gradient
- Q :
-
dimensionless flow rate
- R :
-
radius of the tube
- r :
-
dimensionless radial distance =\(\bar r/R\)
- r j :
-
jth radial nodal point
- r p :
-
dimensionless plug flow radius
- u :
-
dimensionless axial velocity =\(\bar u/u_c \)
- u c :
-
characteristic velocity = −R 2 P 0/2μ∞
- u p :
-
dimensionless plug flow velocity
- ɛ1 :
-
error of tolerance for τ
- ɛ2 :
-
error of tolerance forr p
- μ∞ :
-
Newtonian viscosity (viscosity at high shear rate)
- τ:
-
dimensionless shear stress = τ/τɛc
- τ c :
-
characteristic shear stress = μ∞ u c /R
- τ y :
-
dimensionless yield stress
- +:
-
shear flow region,r p ≤r≤1
- −:
-
plug flow region, 0≤r≤r p
- −:
-
‘overbar’; represents all dimensional quantities
References
Bird, R.B., Stewart, W.E. and Lightfoot, E.N.,Transport Phenomena. John Wiley and Sons, New York (1960) chapter 2.
Fung, Y.C.,Biomechanics, Mechanical Properties of Living Tissues. Springer-Verlag, New York (1981) chapter 3.
Casson, N., A flow equation for pigment oil suspensions of printing, ink type. In: Mill, C.C. (ed.),Rheology of Dispersed System. Pergamon Press, Oxford (1959) pp. 84–102.
Scott Blair, G.W., An equation for flow of blood serum through glass tubes.Nature, London 183 (1959) 613–614.
Charm, S.E. and Kurland, G.S., Viscometry of human blood for shear rates of 0–100,000 sec−1.Nature, London 206 (1965) 617–618.
Merrill, E.W., Benis, A.M., Gilliland, E.R., Sherwood, T.K. and Salzman, E.W., Pressure flow relations of human blood in hollow fibre at low shear rates.Jour. Appl. Physiol. 20 (1965) 954–967.
MacDonald, D.A.,Blood Flows in Arteries, 2nd edition. Arnold, London (1974) chapter 2.
Tammamasi, B., Consideration of certain haemorheological phenomenon from the stand-point of surface chemistry. In Copley, A.L. (ed.),Haemorheology. Pergamon Press, London (1968) p. 89.
Walwander, W.P., Chen, T.Y. and Cala, D.F., An approximate Casson fluid model for tube flow of blood.Biorheology 12 (1975) 111–119.
Vinogradov, G.V. and Malkin, A.Y.,Rheology of Polymers. Mir Publisher, Moscow (1979).
Jayaraman, G., Lautier, A., Jarry, G., Bui-Mong-Huang and Laurent, D., Numerical scheme for modelling oxygen transfer in tubular oxygenators.Med. & Biol. Engng. & Compt. 19 (1981) 524–534.
Dash, R.K., Jayaraman, G. and Mehta, K.N., Estimation of increased flow resistance in a narrow catheterized artery—A theoretical model.Jour. Biomech. 29 (1996) 917–930.
Dash, R.K., Mehta, K.N. and Jayaraman, G., Casson fluid flow in a pipe filled with a homogeneous porous medium.Int. Jour. Engng. Sci. 34 (1996) 1145–1156.
Das, B. and Batra, R.L., Flow of a Casson fluid in the entrance region of a porous tube. In: Nicholas, P. and Cheremisninoff, P. (eds),Encyclopedia of Fluid Mechanics, Supplement 3, Advances in Fluid Dynamics. Gulf Publishing Company, Texas (1994) pp. 123–137.
Das, B. and Batra, R.L., Non-Newtonian flow of a blood in an arteriosclerotic blood vessels with rigid permeable walls.Jour. Theor. Biol. 175 (1995) 1–11.
Bear, J.,Dynamics of Fluids in Porous Media. Elsevier Science, New York (1972).
Vradis, G.C. and Protopapas, A.L., Macroscopic conductivity for flow of Bingham Plastics in porous media.Jour. Hydraul. Engng. 119 (1993) 95–108.
Brinkman, H.K., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles.Appl. Sci. Res. A1 (1947) 27–34.
Merrill, E.W., Rheology of blood.Physiol. Rev. 49 (1969) 863–888.
Chow, C.Y.,An Introduction to Computational Fluid Dynamics. John Wiley and Sons, New York (1979) chapter 4.
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Dash, R.K., Mehta, K.N. & Jayaraman, G. Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube. Appl. Sci. Res. 57, 133–149 (1996). https://doi.org/10.1007/BF02529440
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DOI: https://doi.org/10.1007/BF02529440