Peristaltic propulsion of generalized Burgers' fluids through a non-uniform porous medium: a study of chyme dynamics through the diseased intestine.

A mathematical study of the peristaltic flow of complex rheological viscoelastic fluids using the generalized fractional Burgers' model through a non-uniform channel is presented. This model is designed to study the movement of chyme and undigested chyme (biophysical waste materials) through the small intestine to the large intestine. To simulate blockages and impedance of debris generated by cell shedding, infections, adhesions on the wall and undigested material, a drag force porous media model is utilized. This effectively mimicks resistance to chyme percolation generated by solid matrix particles in the regime. The conduit geometry is mathematically simulated as a sinusoidal propagation with linear increment in shape of the bolus along the length of channel. A modified Darcy-Brinkman model is employed to simulate the generalized flows through isotropic, homogenous porous media, a simplified but physically robust approximation to actual clinical situations. To model the rheological properties of chyme, a viscoelastic Burgers' fluid formulation is adopted. The governing equations are simplified by assuming long wavelength and low Reynolds number approximations. Numerical and approximate analytical solutions are obtained with two semi-numerical techniques, namely the homotopy perturbation method and the variational iteration method. Visualization of the results is achieved with Mathematica software. The influence of the dominant hydromechanical and geometric parameters such as fractional viscoelastic parameters, wave number, non-uniformity constant, permeability parameter, and material constants on the peristaltic flow characteristics are depicted graphically.

[1]  H. Niroomand-Oscuii,et al.  NEW APPROACH IN MODELING PERISTALTIC TRANSPORT OF NON-NEWTONIAN FLUID , 2013 .

[2]  M. K. Chaube,et al.  Peristaltic transport of multilayered power-law fluids with distinct viscosities: a mathematical model for intestinal flows. , 2011, Journal of theoretical biology.

[3]  T. Hayat,et al.  On the MHD flow of fractional generalized Burgers’ fluid with modified Darcy’s law , 2007 .

[4]  Wenchang Tan,et al.  Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary , 2005 .

[5]  Haitao Qi,et al.  On accelerated flows of a viscoelastic fluid with the fractional Burgers’ model , 2009 .

[6]  N. Akbar,et al.  Peristaltic flow of Walter’s B fluid in a uniform inclined tube , 2010 .

[7]  G. W. Scott-Blair.  A survey of general and applied rheology , 1951 .

[8]  Tasawar Hayat,et al.  Exact solution for MHD flow of a generalized Oldroyd-B fluid with modified Darcy’s law , 2006 .

[9]  V. Seshadri,et al.  Peristaltic pumping in non-uniform tubes. , 1976, Journal of biomechanics.

[10]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[11]  N. Balmforth,et al.  Peristaltic pumping of rigid objects in an elastic tube , 2011, Journal of Fluid Mechanics.

[12]  F. Kelvin,et al.  Obstruction of the small intestine: accuracy and role of CT in diagnosis. , 1993, Radiology.

[14]  F. Mainardi,et al.  Creep, relaxation and viscosity properties for basic fractional models in rheology , 2011, 1110.3400.

[15]  Wenchang Tan,et al.  STOKES FIRST PROBLEM FOR AN OLDROYD-B FLUID IN A POROUS HALF SPACE , 2005 .

[16]  Constantin Fetecau,et al.  Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate , 2008, Appl. Math. Comput..

[17]  Dharmendra Tripathi,et al.  Mathematica simulation of peristaltic pumping with double-diffusive convection in nanofluids: a bio-nano-engineering model , 2011 .

[18]  Pierre M. Adler,et al.  Real porous media: local geometry and transports , 2002, Poromechanics II.

[19]  S. Weinberg,et al.  Peristaltic pumping with long wavelengths at low Reynolds number , 1968, Journal of Fluid Mechanics.

[20]  H. Brenner,et al.  Dispersion resulting from flow through spatially periodic porous media , 1980, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[21]  V. Harrison.  Survey of general and applied rheology , 1950 .

[22]  Self-sustained peristaltic waves: explicit asymptotic solutions. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  James G. Brasseur,et al.  Non-steady peristaltic transport in finite-length tubes , 1993, Journal of Fluid Mechanics.

[24]  D. Tripathi.  Peristaltic Hemodynamic Flow of Couple-Stress Fluids Through a Porous Medium with Slip Effect , 2012, Transport in Porous Media.

[25]  L. M. Srivastava,et al.  Peristaltic transport of a power-law fluid: Application to the ductus efferentes of the reproductive tract , 1988 .

[26]  D. Tripathi.  NUMERICAL STUDY ON PERISTALTIC TRANSPORT OF FRACTIONAL BIO-FLUIDS , 2011 .

[27]  Taha Sochi,et al.  Pore-Scale Modeling of Non-Newtonian Flow in Porous Media , 2010, 1011.0760.

[28]  Dharmendra Tripathi,et al.  Transient magneto-peristaltic flow of couple stress biofluids: a magneto-hydro-dynamical study on digestive transport phenomena. , 2013, Mathematical biosciences.

[29]  Kambiz Vafai,et al.  The role of porous media in modeling flow and heat transfer in biological tissues , 2003 .

[30]  Dharmendra Tripathi,et al.  A mathematical model for swallowing of food bolus through the oesophagus under the influence of heat transfer , 2012 .

[31]  D. Tripathi.  Numerical study on peristaltic flow of generalized burgers' fluids in uniform tubes in the presence of an endoscope , 2011 .

[32]  H. S. Lew,et al.  Peristaltic carrying and mixing of chyme in the small intestine (an analysis of a mathematical model of peristalsis of the small intestine). , 1971, Journal of biomechanics.

[33]  D. Tripathi.  Numerical Study on Creeping Flow of Burgers' Fluids through a Peristaltic Tube , 2011 .

[34]  C. Ng,et al.  Lagrangian transport induced by peristaltic pumping in a closed channel. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Neil J. Balmforth,et al.  Peristaltic pumping of viscous fluid in an elastic tube , 2011, Journal of Fluid Mechanics.

[36]  Dharmendra Tripathi,et al.  A mathematical model for the peristaltic flow of chyme movement in small intestine. , 2011, Mathematical biosciences.

[37]  Dharmendra Tripathi,et al.  Peristaltic transport of fractional Maxwell fluids in uniform tubes: Applications in endoscopy , 2011, Comput. Math. Appl..

[38]  O. Bég,et al.  A Numerical Study of Oscillating Peristaltic Flow of Generalized Maxwell Viscoelastic Fluids Through a Porous Medium , 2012, Transport in Porous Media.

[39]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[40]  Wenchang Tan,et al.  Stability analysis of a Maxwell fluid in a porous medium heated from below , 2007 .

[41]  J. Wood.  Neurophysiology of Auerbach's plexus and control of intestinal motility. , 1975, Physiological reviews.

[42]  C. L. Tien,et al.  Boundary and inertia effects on flow and heat transfer in porous media , 1981 .

[43]  R. Doerr,et al.  Small Bowel Obstruction , 2001 .

[44]  J. Christensen,et al.  Finite-element analysis of flow induced by contractions like those of the intestine. , 1981, Journal of biomechanics.

[45]  S. Salinari,et al.  An analysis of the peristaltic reflex , 1979, Biological Cybernetics.

[46]  D. Tripathi.  A Mathematical Study on Three Layered Oscillatory Blood Flow Through Stenosed Arteries , 2012 .

[47]  Tasawar Hayat,et al.  Some exact solutions for fractional generalized Burgers’ fluid in a porous space , 2008 .

[48]  I. Podlubny.  Fractional differential equations , 1998 .

[49]  K. Takeuchi,et al.  Measurement of small intestinal damage. , 2010, Current protocols in toxicology.

[50]  S Salinari,et al.  A mathematical model of intestinal motor activity. , 1978, Journal of biomechanics.

[51]  A. Tharakan.  Modelling of physical and chemical processes in the small intestine , 2009 .

[52]  Sohail Nadeem,et al.  Influence of heat transfer on a peristaltic flow of Johnson Segalman fluid in a non uniform tube , 2009 .

[53]  K. Mekheimer,et al.  Peristaltic transport of a couple stress fluid in a uniform and non-uniform channels. , 2002, Biorheology.

[54]  Tan Wen-chang,et al.  A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates , 2003 .

[55]  Youbing Yin,et al.  Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model , 2006, Appl. Math. Comput..

[56]  W M Bayliss,et al.  The movements and innervation of the small intestine , 1899, The Journal of physiology.