Bifurcation Analysis for Peristaltic Transport of a Power-Law Fluid

2019 ◽  
Vol 74 (3) ◽  
pp. 213-225 ◽  
Author(s):  
Nasir Ali ◽  
Kaleem Ullah
Keyword(s):  
Power Law ◽  
Global Bifurcation ◽  
Shear Thickening ◽  
Peristaltic Transport ◽  
Trapping Region ◽  
Shear Thickening Fluids ◽  
The Stability ◽  
Autonomous Differential Equations ◽  
Fluid Behaviour ◽  
Behaviour Index

AbstractIn this work, the streamline topologies and their bifurcations for peristaltic transport of shear-thinning and shear-thickening fluids characterised by power-law model are analysed. The flow is assumed in a two-dimensional symmetric channel. The analytical solution is obtained in a wave frame of reference under low Reynolds number and long wavelength approximations. To study the streamline topologies, a system of non-linear autonomous differential equations is formed and the method of dynamical system is employed to investigate the bifurcations and their changes. Three different types of flow situations occur: backward flow, trapping and augmented flow. The conversions of backward flow to trapping and then trapping to augmented flow correspond to bifurcations. The stability and nature of bifurcations and their topological changes are explained graphically. For this purpose, a global bifurcation diagram is constructed. The backward flow and trapping regions are significantly affected by fluid behaviour index. In fact, the trapping region expands and the backward region shrinks by increasing the fluid behaviour index. Theoretical results are verified by comparing them with the experimental data, which is available in the literature.

Bifurcation and stability analysis of stagnation points for an asymmetric peristaltic transport

2020 ◽  
Vol 98 (2) ◽  
pp. 172-182 ◽  
Author(s):  
Kaleem Ullah ◽  
Nasir Ali
Keyword(s):  
Flow Field ◽  
Amplitude Ratio ◽  
Nonlinear Differential Equations ◽  
Global Bifurcation ◽  
Flow Region ◽  
Peristaltic Transport ◽  
Trapping Region ◽  
Long Wavelength ◽  
Stagnation Points ◽  
The Stability

This paper investigates the streamline topologies and stability of stagnation points and their bifurcations for an asymmetric peristaltic flow. The asymmetry of channel is due to the propagation of peristaltic waves with different phases and amplitudes on the flexible channel walls. An exact analytic solution of the flow problem subject to the constraints of low Reynolds number and long wavelength is obtained in wave frame of reference moving with wave velocity. A system of nonlinear differential equations is established to locate and classify the stagnation points in the flow domain. Different flow situations, manifested in the flow field, are categorized as: backward flow, trapping, and augmented flow. The transition from one situation to the other corresponds to bifurcation, which is explored graphically through local and global bifurcation diagrams. This analysis discloses the stability status of stagnation points and ranges of involved parameters in which various flow conditions appear in the flow field. It is concluded that the trapping in an asymmetric peristaltic transport can be reduced by increasing the phase difference of the channel walls. It is also found that the augmented flow region shrinks and the trapping region expands by increasing the amplitude ratio of the channel walls.

Drops of power-law fluids falling on a coated vertical fibre

2014 ◽  
Vol 751 ◽  
pp. 184-215
Author(s):  
Liyan Yu ◽  
John Hinch
Keyword(s):  
Solitary Waves ◽  
Power Law ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Bond Number ◽  
Power Law Fluid ◽  
Shear Thinning Fluids ◽  
Power Law Fluids ◽  
Shear Thickening Fluids ◽  
The Stability

AbstractWe study the solitary wave solutions in a thin film of a power-law fluid coating a vertical fibre. Different behaviours are observed for shear-thickening and shear-thinning fluids. For shear-thickening fluids, the solitary waves are larger and faster when the reduced Bond number is smaller. For shear-thinning fluids, two branches of solutions exist for a certain range of the Bond number, where the solitary waves are larger and faster on one and smaller and slower on the other as the Bond number decreases. We carry out an asymptotic analysis for the large and fast-travelling solitary waves to explain how their speeds and amplitudes change with the Bond number. The analysis is then extended to examine the stability of the two branches of solutions for the shear-thinning fluids.

Stability of Shear-Thickening Liquids Between Rotating Cylinders

2010 ◽  
Author(s):  
Nariman Ashrafi ◽  
Habib Karimi Haghighi
Keyword(s):  
Couette Flow ◽  
Dynamical Behavior ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Taylor Vortex ◽  
Shear Dependent Viscosity ◽  
Shear Thinning Fluids ◽  
Shear Thickening Fluids ◽  
The Stability ◽  
Dependent Viscosity

The effects of nonlinearities on the stability are explored for shear thickening fluids in the narrow-gap limit of the Taylor-Couette flow. It is assumed that shear-thickening fluids behave exactly as opposite of shear thinning ones. A dynamical system is obtained from the conservation of mass and momentum equations which include nonlinear terms in velocity components due to the shear-dependent viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of Couette flow becomes higher as the shear-thickening effects increases. Similar to the shear thinning case, the Taylor vortex structure emerges in the shear thickening flow, however they quickly disappear thus bringing the flow back to the purely azimuthal flow. Naturally, one expects shear thickening fluids to result in inverse dynamical behavior of shear thinning fluids. This study proves that this is not the case for every point on the bifurcation diagram.

Fully Developed Flow of Power-Law Fluid Through a Cylindrical Microfluidic Pipe: Pressure Drop and Electroviscous Effects

2008 ◽  
Author(s):  
Ram P. Bharti ◽  
Dalton J. E. Harvie ◽  
Malcolm R. Davidson
Keyword(s):  
Pressure Drop ◽  
Field Strength ◽  
Charge Density ◽  
Power Law ◽  
Maximum Velocity ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Electrical Field ◽  
Electrical Field Strength ◽  
Fluid Behaviour

Pressure drop and electroviscous effects in the axisymmetric, steady, fully developed, pressure-driven flow of incompressible power-law fluids through a cylindrical microchannel at low Reynolds number (Re = 0.01) have been investigated. The Poisson-Boltzmann equation (describing the electrical potential) and the momentum equations in conjunction with electrical force and power-law fluid rheology have been solved numerically using the finite difference method. The pipe wall is considered to have uniform surface charge density (S = 4) and the liquid is assumed to be a symmetric electrolyte solution. In particular, the influence of the dimensionless inverse Debye length (K = 2, 20) and power-law flow behaviour index (n = 0.2, 1, 1.8) on the EDL potential, ion concentrations and charge density profiles, induced electrical field strength, velocity and viscosity profiles and pressure drop have been studied. As expected, the local EDL potential, local charge density and electrical field strength increases with decreasing K and/or increasing S. The velocity profiles cross-over away from the charged pipe wall with increasing K and/or decreasing n. The maximum velocity at the center of the pipe increases with increasing n and/or increasing S and/or decreasing K. The shear-thinning fluid viscosity is strongly dependent on K and S, whereas the shear-thickening viscosity is very weakly dependent on K and S. For fixed K, as the fluid behaviour changes from Newtonian (n = 1) to shear-thinning (n < 1), the induced electrical field strength increases and maximum velocity reduces. On the other hand, the change in fluid behaviour from Newtonian (n = 1) to shear-thickening (n > 1) decreases the electrical field strength and increases the maximum velocity. The non-Newtonian effects on maximum velocity and pressure drop are stronger in shear-thinning fluids at small K and large S, the shear-thickening fluids show opposite influence. Electroviscous effects enhance with decreasing K and/or increasing S. The electroviscous effects show complex dependence on the non-Newtonian tendency of the fluids. The shear-thickening (n > 1) fluids and/or smaller K show stronger influence on the pressure drop and thus, enhance the electroviscous effects than that in shear-thinning (n < 1) fluids and/or large K where EDL is very thin.

Shear-Thickening Flow Between Coaxial Cylinders

2011 ◽  
Author(s):  
Nariman Ashrafi ◽  
Habib Karimi Haghighi
Keyword(s):  
Couette Flow ◽  
Dynamical Behavior ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Taylor Vortex ◽  
Shear Dependent Viscosity ◽  
Shear Thinning Fluids ◽  
Shear Thickening Fluids ◽  
The Stability ◽  
Dependent Viscosity

The effects of nonlinearities on the stability are explored for shear thickening fluids in the narrow-gap limit of the Taylor-Couette flow. A dynamical system is obtained from the conservation of mass and momentum equations which include nonlinear terms in velocity components due to the shear-dependent viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of Couette flow becomes higher as the shear-thickening effects increases. Similar to the shear thinning case, the Taylor vortex structure emerges in the shear thickening flow; however they quickly disappear thus bringing the flow back to the purely azimuthal flow. Naturally, one expects shear thickening fluids to result in inverse dynamical behavior of shear thinning fluids. This study proves that this is not the case for every point on the bifurcation diagram.

Power-Law Fluid Flow Passing Two Square Cylinders in Tandem Arrangement

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Izadpanah Ehsan ◽  
Sefid Mohammad ◽  
Nazari Mohammad Reza ◽  
Jafarizade Ali ◽  
Ebrahim Sharifi Tashnizi
Keyword(s):  
Reynolds Number ◽  
Power Law ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Newtonian Fluids ◽  
Power Law Fluid ◽  
Tandem Arrangement ◽  
Square Cylinders ◽  
Shear Thickening Fluids ◽  
Power Law Index

Two-dimensional laminar flow of a power-law fluid passing two square cylinders in a tandem arrangement is numerically investigated in the ranges of 1< Re< 200 and 1 ≤ G ≤ 9. The fluid viscosity power-law index lies in the range 0.5 ≤ n ≤ 1.8, which covers shear-thinning, Newtonian and shear-thickening fluids. A finite volume code based on the SIMPLEC algorithm with nonstaggered grid is used. In order to discretize the convective and diffusive terms, the third order QUICK and the second-order central difference scheme are used, respectively. The influence of the power-law index, Reynolds number and gap ratio on the drag coefficient, Strouhal number and streamlines are investigated, and the results are compared with other studies in the literature to validate the methodology. The effect of the time integration scheme on accuracy and computational time is also analyzed. In the ranges of Reynolds number and power-law index studied here, vortex shedding is known to occur for square cylinders in tandem. This study represents the first systematic investigation of this phenomenon for non-Newtonian fluids in the open literature. In comparison to Newtonian fluids, it is found that the onset of leading edge separation occurs at lower Reynolds number for shear-thinning fluids and is delayed to larger values for shear-thickening fluids.

Two Dimensional Unsteady Laminar Flow of Power Law Fluids Past a Square Cylinder: A Numerical Study

2008 ◽  
Author(s):  
Akhilesh K. Sahu ◽  
Raj P. Chhabra ◽  
V. Eswaran
Keyword(s):  
Reynolds Number ◽  
Power Law ◽  
Shear Thickening ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Shear Thinning ◽  
Shear Thinning Fluids ◽  
Power Law Fluids ◽  
Shear Thickening Fluids ◽  
Edge Separation

The two-dimensional and unsteady flow of power-law fluids past a long square cylinder has been investigated numerically in the range of conditions 60 ≤ Re ≤ 160 and 0.5 ≤ n ≤ 2.0. Over this range of Reynolds numbers, the flow is periodic in time for Newtonian fluids. However, no such information is available for power law fluids. A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. The macroscopic quantities such as drag coefficients, Strouhal number, lift coefficient as well as the detailed kinematic variables like stream function, vorticity and so on, have been calculated as functions of the pertinent dimension-less groups. In particular, the effects of Reynolds number and of the power-law index have been investigated in the unsteady laminar flow regime. The leading edge separation in shear-thinning fluids produces an increase in drag values with the increasing Reynolds number, while shear-thickening behaviour delays the leading edge separation. So, the drag coefficient in the above-mentioned range of Reynolds number, Re, in shear-thinning fluids (n &lt; 1) initially decreases but at high values of the Reynolds number, it increases. As expected, on the other hand, in case of shear-thickening fluids (n &gt; 1) drag coefficient reduces with Reynolds number, Re. Furthermore, the present results also suggest the transition from steady to unsteady flow conditions to occur at lower Reynolds numbers in shear-thickening fluids than that in Newtonian fluids. Also, the spectra of lift signal for shear-thickening fluids show that the flow is truly periodic in nature with a single dominant frequency in the above range of Reynolds number. In shear-thinning fluids at higher Re, quasi-periodicity sets in with additional frequencies, which indicate the transition from the 2-D to 3-D flows.

Flow of power-law fluids in fixed beds of cylinders or spheres

2012 ◽  
Vol 713 ◽  
pp. 491-527 ◽  
Author(s):  
John P. Singh ◽  
Sourav Padhy ◽  
Eric S. G. Shaqfeh ◽  
Donald L. Koch
Keyword(s):  
Drag Force ◽  
Newtonian Fluid ◽  
Power Law ◽  
Test Particle ◽  
Body Force ◽  
Equations Of Motion ◽  
Fixed Bed ◽  
Shear Thickening ◽  
Length Scale ◽  
Shear Thickening Fluids

AbstractAn ensemble average of the equations of motion for a Newtonian fluid over particle configurations in a dilute fixed bed of spheres or cylinders yields Brinkman’s equations of motion, where the disturbance velocity produced by a test particle is influenced by the Newtonian fluid stress and a body force representing the linear drag on the surrounding particles. We consider a similar analysis for a power-law fluid where the stress $\boldsymbol{\tau} $ is related to the rate of strain $ \mathbisf{e} $ by $\boldsymbol{\tau} = 2m \mathop{ \vert \mathbisf{e} \vert }\nolimits ^{n\ensuremath{-} 1} \mathbisf{e} $, where $m$ and $n$ are constants. In this case, the ensemble-averaged momentum equation includes a body force resulting from the nonlinear drag exerted on the surrounding particles, a power-law stress associated with the disturbance velocity of the test particle, and a stress term that is linear with respect to the test particle’s disturbance velocity. The latter term results from the interaction of the test particle’s velocity disturbance with the random straining motions produced by the neighbouring particles and is important only in shear-thickening fluids where the velocity disturbances of the particles are long-ranged. The solutions to these equations using scaling analyses for dilute beds and numerical simulations using the finite element method are presented. We show that the drag force acting on a particle in a fixed bed can be written as a function of a particle-concentration-dependent length scale at which the fluid velocity disturbance produced by a particle is modified by hydrodynamic interactions with its neighbours. This is also true of the drag on a particle in a periodic array where the length scale is the lattice spacing. The effects of particle interactions on the drag in dilute arrays (periodic or random) of cylinders and spheres in shear-thickening fluids is dramatic, where it arrests the algebraic growth of the disturbance velocity with radial position when $n\geq 1$ for cylinders and $n\geq 2$ for spheres. For concentrated random arrays of particles, we adopt an effective medium theory in which the drag force per unit volume in the medium surrounding a test particle is assumed to be proportional to the local volume fraction of the neighbouring particles, which is derived from the hard-particle packing. The predictions of the averaged equations of motion are validated by comparison with simulations of randomly distributed hydrodynamically interacting cylinders.

Pattern selection in the thermal convection of non-Newtonian fluids

2011 ◽  
Vol 668 ◽  
pp. 500-550 ◽  
Author(s):  
BASHAR ALBAALBAKI ◽  
ROGER E. KHAYAT
Keyword(s):  
Fluid Layer ◽  
Shear Thickening ◽  
Newtonian Fluids ◽  
Critical Threshold ◽  
Physical Parameters ◽  
Pattern Selection ◽  
Nonlinear Analyses ◽  
Weakly Nonlinear ◽  
Shear Thickening Fluids ◽  
The Stability

The thermogravitational instability in a fluid layer of a non-Newtonian medium heated from below is investigated. Linear and weakly nonlinear analyses are successively presented. The fluid is assumed to obey the Carreau–Bird model. Although the critical threshold is the same as for a Newtonian fluid, it is found that non-Newtonian fluids can convect in the form of rolls, squares or hexagons, depending on the shear-thinning level. Similar to Newtonian fluids, shear-thickening fluids convect only in the form of rolls. The stability of the convective steady branches is carried out to determine under which specific conditions a pattern is preferred. The influence of the rheological and physical parameters is examined and discussed in detail.

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