Flow of a Second Order Fluid through Constricted Tube with Slip Velocity at Wall Using Integral Method
The steady flow of a second order fluid through
constricted tube with slip velocity at wall is modeled and analyzed
theoretically. The governing equations are simplified by implying no
slip in radial direction. Based on Karman Pohlhausen procedure
polynomial solution for axial velocity profile is presented.
Expressions for pressure gradient, shear stress, separation and
reattachment points, and radial velocity are also calculated. The
effect of slip and no slip velocity on magnitude velocity, shear stress,
and pressure gradient are discussed and depicted graphically. It is
noted that when Reynolds number increases magnitude velocity of
the fluid decreases in both slip and no slip conditions. It is also found
that the wall shear stress, separation, and reattachment points are
strongly affected by Reynolds number.
[1] M. R. Roach, "An experimental study of the production and time course
of post-stenotic dialation in the femoral and carotid arteries of adult
dogs". Circulation Res, vol 13, 1963, pp. 537-551.
[2] S. Rodbard, "Dynamics of blood flow in stenotic vascular lesions". Am.
Heart J. vol 72, 1966, pp. 698-704.
[3] J. H. Forrester, "Flow through a converging-diverging tube and its
implications in occlusive vascular disease". PhD Thesis; Ames, Iowa,
library, Iowa State University, 1968.
[4] J. A. Fox and A.E. Hugh, "Localization of atheroma a theory based on
boundary layer separation". British heart Journal, vol 28, 1966, pp. 388-
399.
[5] D. F. Young, "Effect of a Time-Dependent stenosis on flow through a
tube". J. Engg. Int., Trans. Am. Soc. Mech. Engrs., vol 90, 1968, pp.
248-254.
[6] J. H. Forrester and D. F. Young, "Flow through a converging-diverging
tube and its implications in occlusive disease". J. Biomech., vol 3, 1970,
pp.297-316.
[7] J. S. Lee and Y. C. Fung, "Flow in locally constricted tubes at low
Reynolds number". J. Appl. Mech.,vol 37, 1970, pp. 513-524.
[8] E. W. Merill, "Rheolgy of human blood and some speculations on its
role in vascular homeostasis. In Biophysical mechanisms in vascular
homeostasis and intravascular thrombosis", Sawyer, P. N., ed. New York:
Appleton-Century-Croft. 1965.
[9] D. L. Fry, "Acute vascular Endothelial changes Associated with
Increased Blood velocity Gradients". Circulation Res, vol 22, 1968, pp.
165-197.
[10] B. E. Morgan, and D. F. Young, "An Integral method for the analysis of
flow in arterial stenosis". Bulletin of Math Bio. vol 36, 1974, pp. 39-46.
[11] K. Haldar, "Analysis of separation of blood flow in constricted arteries".
Archives of Mechanics, vol 43, 1991, pp. 107-113.
[12] J. C. F. Chow and K. Soda, "Laminar flow in tubes with constriction".
Physics of fluids, vol 15, issue 10, 1972, pp. 1700-1706.
[13] J. C. F. Chow, K. Soda and C. Dean, "On laminar flow in wavy
channe"l. Development in Mech, Vol 6, Proceeding of the 12 th
Midwestern Mechanics Conference, 1971.
[14] A. Mirza, A. R. Ansari, A. M. Siddiqui and T. Haroon, "On steady twodimensional
flow with heat transfer in the presence of a stenosis". WEAS
Transactions on fluids mechanics, vol 8, issue 4, 2013,pp. 149-158.
[15] L. Bennett," Red cell slip at wall in vitro". Sci, vol 15, 1967, pp. 1554-
1556.
[16] Y. Nubar, “Blood flow, slip and viscometer”, Biophysics J, vol 11, 1971,
pp. 252-264.
[17] P. Brumm, “The velocity slip for polar fluids”, Rheol. Acta, vol 14,
1975,1039-1054.
[18] M. Sarma, “Analysis of blood flow through stenosed vessel under effect
of magnetic field.”Int. J. for Basic Sci and Social sci, vol 1, issue 3,
2012, pp. 78-88.
[19] J. C. Misra and B. K. Bar, “Momentum Integral method for studying
flow characteristics of blood through a stenosed vessel”, Biorheology,
vol 26, issue 1, 1989, pp. 23-35.
[20] B. D. Coleman, & W. Noll, “An approximation theorem for functionals,
with applications in continuum mechanics”. Arch. Rational Mech. Anal.
Vol 6, 1960, pp. 355-370
[21] Schlichting, H. Boundary Layer Theory, 6th Edn.,McGraw-Hill,
NewYork, 1968. Pp. 122-192.
[22] L. Prandtl, "Motion of fluids with very little viscosity." ntrs.nasa.gov.
1928.
[1] M. R. Roach, "An experimental study of the production and time course
of post-stenotic dialation in the femoral and carotid arteries of adult
dogs". Circulation Res, vol 13, 1963, pp. 537-551.
[2] S. Rodbard, "Dynamics of blood flow in stenotic vascular lesions". Am.
Heart J. vol 72, 1966, pp. 698-704.
[3] J. H. Forrester, "Flow through a converging-diverging tube and its
implications in occlusive vascular disease". PhD Thesis; Ames, Iowa,
library, Iowa State University, 1968.
[4] J. A. Fox and A.E. Hugh, "Localization of atheroma a theory based on
boundary layer separation". British heart Journal, vol 28, 1966, pp. 388-
399.
[5] D. F. Young, "Effect of a Time-Dependent stenosis on flow through a
tube". J. Engg. Int., Trans. Am. Soc. Mech. Engrs., vol 90, 1968, pp.
248-254.
[6] J. H. Forrester and D. F. Young, "Flow through a converging-diverging
tube and its implications in occlusive disease". J. Biomech., vol 3, 1970,
pp.297-316.
[7] J. S. Lee and Y. C. Fung, "Flow in locally constricted tubes at low
Reynolds number". J. Appl. Mech.,vol 37, 1970, pp. 513-524.
[8] E. W. Merill, "Rheolgy of human blood and some speculations on its
role in vascular homeostasis. In Biophysical mechanisms in vascular
homeostasis and intravascular thrombosis", Sawyer, P. N., ed. New York:
Appleton-Century-Croft. 1965.
[9] D. L. Fry, "Acute vascular Endothelial changes Associated with
Increased Blood velocity Gradients". Circulation Res, vol 22, 1968, pp.
165-197.
[10] B. E. Morgan, and D. F. Young, "An Integral method for the analysis of
flow in arterial stenosis". Bulletin of Math Bio. vol 36, 1974, pp. 39-46.
[11] K. Haldar, "Analysis of separation of blood flow in constricted arteries".
Archives of Mechanics, vol 43, 1991, pp. 107-113.
[12] J. C. F. Chow and K. Soda, "Laminar flow in tubes with constriction".
Physics of fluids, vol 15, issue 10, 1972, pp. 1700-1706.
[13] J. C. F. Chow, K. Soda and C. Dean, "On laminar flow in wavy
channe"l. Development in Mech, Vol 6, Proceeding of the 12 th
Midwestern Mechanics Conference, 1971.
[14] A. Mirza, A. R. Ansari, A. M. Siddiqui and T. Haroon, "On steady twodimensional
flow with heat transfer in the presence of a stenosis". WEAS
Transactions on fluids mechanics, vol 8, issue 4, 2013,pp. 149-158.
[15] L. Bennett," Red cell slip at wall in vitro". Sci, vol 15, 1967, pp. 1554-
1556.
[16] Y. Nubar, “Blood flow, slip and viscometer”, Biophysics J, vol 11, 1971,
pp. 252-264.
[17] P. Brumm, “The velocity slip for polar fluids”, Rheol. Acta, vol 14,
1975,1039-1054.
[18] M. Sarma, “Analysis of blood flow through stenosed vessel under effect
of magnetic field.”Int. J. for Basic Sci and Social sci, vol 1, issue 3,
2012, pp. 78-88.
[19] J. C. Misra and B. K. Bar, “Momentum Integral method for studying
flow characteristics of blood through a stenosed vessel”, Biorheology,
vol 26, issue 1, 1989, pp. 23-35.
[20] B. D. Coleman, & W. Noll, “An approximation theorem for functionals,
with applications in continuum mechanics”. Arch. Rational Mech. Anal.
Vol 6, 1960, pp. 355-370
[21] Schlichting, H. Boundary Layer Theory, 6th Edn.,McGraw-Hill,
NewYork, 1968. Pp. 122-192.
[22] L. Prandtl, "Motion of fluids with very little viscosity." ntrs.nasa.gov.
1928.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71318", author = "Nosheen Zareen Khan and Abdul Majeed Siddiqui and Muhammad Afzal Rana", title = "Flow of a Second Order Fluid through Constricted Tube with Slip Velocity at Wall Using Integral Method", abstract = "The steady flow of a second order fluid through
constricted tube with slip velocity at wall is modeled and analyzed
theoretically. The governing equations are simplified by implying no
slip in radial direction. Based on Karman Pohlhausen procedure
polynomial solution for axial velocity profile is presented.
Expressions for pressure gradient, shear stress, separation and
reattachment points, and radial velocity are also calculated. The
effect of slip and no slip velocity on magnitude velocity, shear stress,
and pressure gradient are discussed and depicted graphically. It is
noted that when Reynolds number increases magnitude velocity of
the fluid decreases in both slip and no slip conditions. It is also found
that the wall shear stress, separation, and reattachment points are
strongly affected by Reynolds number.", keywords = "Approximate solution, constricted tube, non-Newtonian fluids, Reynolds number.", volume = "9", number = "10", pages = "625-7", }
