On the dispersion of a solute in oscillating flow through a channel

The paper presents an exact analysis of the dispersion of a solute in an incompressible viscous fluid flowing slowly in a parallel plate channel under the influence of a periodic pressure gradient. Using a generalised dispersion model which is valid for all times after the solute injection, the diffusion coefficientsK _i (τ)(i=1,2,3,…) are determined as functions of timeτ when the initial distribution of the solute is in the form of a slug of finite extent. The second coefficientK ₂(τ) gives a measure of the longitudinal dispersion of the solute due to the combined influence of molecular diffusion and nonuniform velocity across the channel cross-section. The analysis leads to the novel result thatK ₂(τ) consists of a steady partS and a fluctuating partD ₂(τ) due to the pulsatility of the flow. It is shown thatS increases with increase inλ (the amplitude of pressure pulsation) for small values ofω (the frequency of the pulsation). But for largeω, S decreases with increase inλ. It is also found that for fixedλ, there is very little fluctuation inD ₂(τ) forω=1, butD ₂(τ) shows fluctuation with large amplitude whenω slightly exceeds unity. The amplitude ofD ₂(τ) then decreases with further increase inω. Thus the variation of bothS andD ₂(τ) withω is non-monotonic. Finally,? _m , the average concentration of the solute over the channel cross-section is determined for various values ofλ andω.     

On the dispersion of soluble matter in a channel flow of non-Newtonian fluid

Summary An analysis of the dispersion of a solute in Rivlin-Ericksen third-order fluid in a parallel plate channel is carried out. It is seen that the solute which is dispersed relative to a plane moving with the mean speed of the flow has its effective Taylor diffusion coefficient which decreases with increasing the non-Newtonian parameter With 1 table     

On the dispersion of a solute in Poiseuille flow of a third-grade fluid in a channel

Gill and Sankarasubramanian's analysis of the dispersion of Newtonian fluids in laminar flow between two parallel walls are extended to the flow of non-Newtonian viscoelastic fluid (known as third-grade fluid) using a generalized dispersion model which is valid for all times after the solute injection. The exact expression is obtained for longitudinal convective coefficient K₁(Γ), which shows the effect of the added viscosity coefficient Γ on the convective coefficient. It is seen that the value of the K₁(Γ) for Γ≠0 is always smaller than the corresponding value for a Newtonian fluid. Also, the effect of the added viscosity coefficient on the K₂(t,Γ) (which is a measure of the longitudinal dispersion coefficient of the solute) is explored numerically. Finally, the axial distribution of the average concentration C_m of the solute over the channel cross-section is determined at a fixed instant after the solute injection for several values of the added viscosity coefficient.     

On solute dispersion in pulsatile flow through a channel with absorbing walls

The paper presents the longitudinal dispersion of passive tracer molecules released in an incompressible viscous fluid flowing through a channel with reactive walls under the action of a periodic pressure gradient. A finite-difference implicit scheme is adopted to solve the unsteady advection-diffusion equation based on the Aris-Barton method of moments for all time period. Here it is shown how the spreading of tracers is influenced by the shear flow, lateral diffusion about its mean position due to the action of absorption at both the walls. The analysis has been performed for three different cases: steady, periodic and the combined effect of steady and periodic currents, separately. The results show that for all cases the dispersion coefficient asymptotically reaches a stationary state after a certain critical time and it achieves a stationary state at earlier instant of time, when absorption at the walls increases. The axial distributions of mean concentration are determined from the first four central moments by using Hermite polynomial representation for all three different flow velocities.     

Nonisothermal flow of non-Newtonian fluids in a channel

A study is made of the nonisothermal motion of a rheologically complex fluid in a plane-parallel channel in the case of boundary conditions of the third kind on the outer surfaces of the channel walls and with allowance for the dissipation of mechanical energy and temperature dependence of the coefficients. A qualitative investigation of the problem is made for arbitrary temperature dependence of the yield. Special cases are considered: flow of linear viscoplastic medium and a power-law medium with abrupt change with the temperature of the yield point and consistency. It is shown that under certain conditions several different flow regimes can be realized simultaneously in the channel and the change in the flow rate of the medium in a channel with varying temperature of the surrounding medium can exhibit hysteresis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–10, May–June, 1980.     

Finite element analysis of non-Newtonian fluid flow in 2-d branching channel

This paper presents finite element analysis of non-Newtonian fluid flow in 2-d branching channel. The Galerkin method and mixed finite element method are used. Here the fluid is considered as incompressible, non-Newtonian fluid with Oldyord differential-type constitutive equation. The non-linear algebraic equation system which is formulated with finite element method is solved by means of continuous differential method. The results show that finite element method is suitable for the analysis of non-Newtonian fluid flow with complex geometry.     

Impulsive motion of a non-Newtonian fluid between two oscillating parallel plates

An exact solution for the flow of an incompressible viscoelastic fluid between two infinitely extended parallel plates, due to the harmonic oscillations of the upper plate and the impulsively started harmonic oscillations of the lower plate from rest, in the respective planes of the plates, has been obtained. The momentum transfer towards the central region and the skin friction of the lower plate are found to be greater for the viscoelastic fluid than that for viscous fluid. The effect of out-of-phase oscillations of the plates with different amplitudes on the flow characteristics has also been investigated.     

Free-surface non-Newtonian fluid flow in a round pipe

Free-surface pseudoplastic and viscoplastic fluid flows in a round pipe were studied for the case where the direction of motion coincides with the direction of gravity. Numerical modeling was performed using a technique based on a combination of the SIMPLE algorithm and the method of invariants. Three characteristic filling regimes were found to exist: a complete filling regime, a regime characterized by air-cavity formation on the solid wall, and a jet regime. Critical parameter values separating the regions of existence of these regimes were calculated. The evolution of quasisolid cores was studied for flow of a fluid with an yield point.     

Exact analysis of laminar dispersion in non-Newtonian flow

An exact analysis is presented for dispersion of a solute in an Eyring model fluid flowing between two parallel plates under uniform pressure gradient. Using a generalised dispersion model, which is valid for all time after injection of the solute, expressions have been developed for the time-dependent longitudinal dispersion coefficient as well as of the area-mean concentration. It is observed that the dispersion coefficient markedly decreases as the fluid parameterG increases, which is attributable to the gradual flattening of the velocity profile. Consequently the concentration distribution becomes steeper at largerG. It is also established that forG<1, the extent of dispersion is practically same as in the case of Newtonian flow.

---

Exakte Analyse einer laminaren Dispersion in einer Nicht-Newtonschen Strömung

Zusammenfassung Es wird eine exakte Analyse für eine Dispersion in einer Eyring-Modellflüssigkeit vorgestellt, die zwischen zwei parallelen Platten unter gleichmäßigem Druckgefälle fließt. Es wurden sowohl Formeln für den zeitabhängigen Longitudinaldispersionskoeffizienten, als auch für die mittlere Umgebungskonzentration entwickelt, wobei ein allgemeines Dispersionsmodell benutzt wurde, welches für den ganzen Zeitraum nach dem Einleiten der Lösung gültig ist. Es wird beobachtet, daß der Dispersionskoeffizient deutlich abnimmt, wenn der FlüssigkeitsparameterG sich erhöht, was bezeichnend für die allmähliche Abflachung des Geschwindigkeitsprofils ist. Folglich wird die Konzentrationsverteilung mit steigendemG steiler. Es wurde bewiesen, daß für G 1 die Ausdehnung der Dispersion praktisch mit der Newtonschen Strömung identisch ist.

---

Nomenclature a half of channel width - A, B Eyring model parameters - C local concentration of the solute - C ₀ concentration of uniform slug - ¯ C dimensionless local concentration - ¯ C _m area-average concentration defined by (8) - D molecular diffusivity - G dimensionless Eyring model parameter - K ₁() time-dependent dispersion coefficient - Pe Peclet number - t time - u axial velocity - U average axial velocity - x axial coordinate - x _s length of uniform slug - X dimensionless axial coordinate - X _s dimensionless slug length - y transverse coordinate - Y dimensionless transverse coordinate Greek symbols dimensionless Eyring model parameter - coefficient of viscosity - dimensionless time     

MHD non-Newtonian micropolar fluid flow and heat transfer in channel with stretching walls

A study is presented for magnetohydrodynamics （MHD） flow and heat transfer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with stretching walls. The micropolar model introduced by Eringen is used to describe the working fluid. The transformed self similar ordinary differential equations together with the associated boundary conditions are solved numerically by an algorithm based on quasi-linearization and multilevel discretization. The effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. The present investigations may be beneficial in the flow and thermal control of polymeric processing.     

Slow flow of a non-Newtonian fluid past a micro-capsule

The streaming motion past a spherical microcapsule is studied. The particle consists of a thin elastic membrane enclosing an incompressible fluid. Since the problem is highly nonlinear, a perturbation solution is sought in the limiting case where the deviation from sphericity is small. Obviously, the capsule remains nearly spherical when λ, the ratio of viscous forces to elastic (shape-restoring) membrane forces is small. In this limit, the rheology of the inside fluid is immaterial and the problem is essentially characterized by three parameters: λ, the Reynolds number Re (interia effect), and the Weissenberg number We (non-newtonian effect). The deformation is obtained explicitly under the restriction We<1, Re<1. It is shown that to leading order, the capsule deforms exactly into a spheroid which can be either oblate or prolate, depending mainly upon the elasticity number We/Re: for We/Re<0.57 the spheroid is oblate, while for We/Re>0.81 a prolate spheroid results. For 0.57<We/Re<0.81 additional details of the rheology of the membrane and of the suspending fluid are needed. The degree of the deformation is governed by the parameters λ Re. All parameters of the problem enter into the expression of the drag force. On a qualitative basis, these results are similar to those for droplets although major differences exist quantitatively.     

On the time development of dispersion in electroosmotic flow through a rectangular channel

This is an analytical study on the time development of hydrodynamic dispersion of an inert species in electroosmotic flow through a rectangular channel.The objective is to determine how the channel side walls may affectthe dispersion coefficient at different instants of time.Tothis end,the generalized dispersion model,which is valid forshort and long times,is employed in the present study.Analytical expressions are derived for the convection and dispersion coefficients as functions of time,the aspect ratio ofthe channel,and the Debye-Hu¨ckel parameter representingthe thickness of the electric double layer.For transport ina channel of large aspect ratio,the dispersion may undergoseveral stages of transience.The initial,fast time development is controlled by molecular diffusion across the narrowchannel height,while the later,slower time development isgoverned by diffusion across the wider channel breadth.Fora sufficiently large aspect ratio,there can be an interludebetween these two periods during which the coefficient isnearly steady,signifying the resemblance of the transportto that in a parallel-plate channel.Given a sufficiently longtime,the dispersion coefficient will reach a fully-developedsteady value that may be several times higher than that without the side wall effects.The time scales for these periods oftransience are identified in this paper.     

On the reversed flow and oscillating wake in an asymmetrically heated channel

The detailed processes of flow reversal in a buoyancy-induced flow through a one-side-heated vertical channel of finite height were simulated numerically. It is of interest to note that the wake above the heated plate is oscillatory at high Rayleigh number and there exists a minimum in the transient variation of the average Nusselt number. Additionally, the predicted steady average Nusselt number and induced flow rate are correlated by empirical equations.     

The stability of the plane Poiseuille flow of a non-Newtonian fluid

Rheologica Acta - This paper considers the flow and the stability of the flow of a fluid whose viscosity depends on the shear in the form $$\nu = {\nu _0}\left\{ {r - s{{\left( {\frac{{d\bar...     

Heat removal from oscillating flow in a vertical annular channel

In this study, heat removal from a surface, which is located into the reciprocating flow in a vertical annular liquid column, is investigated experimentally. The experiments are carried out for four different oscillation frequencies and three heat fluxes while the amplitude remains constant for all cases. Instantaneous and time-averaged surface and bulk temperature variations are presented. The cycle-averaged values are considered in the calculation of heat transfer using the experimental measurements. Heat removal from the cold surface due to the oscillating liquid column is determined in terms of Nusselt number. Based on the experimental data, an empirical equation is obtained for the cycle averaged Nusselt number as a function of kinetic Reynolds number.     

Stability of steady-state non-Newtonian fluid layer flow

The stability of the steady-state plane-parallel flow of a non-Newtonian fluid layer in the gravity field along an inclined rigid surface is investigated. It is shown that the most dangerous are the long-wave perturbations propagating over the free surface. The stability maps are plotted for such perturbations in the Reynolds number — gravity parameter plane. With increase in the gravity number the layer flow becomes less stable. The layer deviation from the vertical lines stabilizes the flow.     

Secondary flow in the neighborhood of a cylinder oscillating in a viscoelastic fluid

Measurements were made of the steady secondary flow which is caused by a cylinder executing small-amplitude oscillations perpendicular to the cylinder generators. Test fluids were a water-glycerine mixture and two non-Newtonian fluids: aqueous solutions of Separan AP-30 and PolyHall 295 polyacrylamide polymers. The fluids were contained in a cylinder whose center coincides with the rest position of the oscillating cylinder. Experimental results are consistent with a theoretical analysis for a three-constant Oldroyd fluid model in its simplified convected-Maxwell form. Both experiment and theory show that at an oscillation frequency of 40 Hz the secondary flow of the dilute polymer solutions is, essentially, in a sense opposite from that of a Newtonian fluid.     

Hydrodynamic stability of Poiseuille flow of a viscoplastic non-Newtonian fluid

The hydrodynamic stability of Poiseuille flow of a viscoplastic fluid is investigated. The flow is shown to be stable for infinitesimal disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–154, November–December, 1974.In conclusion the authors thank S. A. Regirer for critical remarks.