Search

The linear instability characteristics of a core-annular pipe flow of a Newtonian fluid and a Bingham fluid separated by a mixed region is investigated. The main objective of this study is to investigate the effect of the yield stress of the annular fluid on the stability behaviour. The present linear stability analysis reveals that the corkscrew mode is the most dangerous mode for the parameters considered. The Bingham number seems to destabilise the flow via modifying the velocity profile. The mechanism of instability is found to be inviscid in nature. An energy budget analysis shows that the energy due to the gradient of viscosity perturbation in the radial direction, and the transfer of energy from the basic flow to the perturbation via the “Reynolds stress” term are responsible for the instability observed. The positive contributions of these terms towards instability increases with increasing the yield stress. The critical Reynolds number also decreases with increasing the Bingham number for the corkscrew mode, which indicates the destabilising influence of the yield stress.

Dense granular materials are universal in nature and some common examples of this kind of materials in our daily lives are sand, rice, sugar etc. Research in this area is motivated by numerous applications encountered in industrial processes, such as hopper discharge, chute flow, moving beds, sandpipe flow, etc. and also in geophysics for the description and prediction of natural hazards like landslide and rock avalanches. Here, a continuum description of these granular flows is appropriate. Granular materials in liquid-like state show strong non-Newtonian behavior, which is typically described by phenomenological constitutive laws with local rheology. Very often, fluidization and flows of granular materials are localized in thin shear bands, whereas the granular material remains in solid-like state outside these bands. In this paper we establish a benchmark problem for granular materials, namely as Couette flow which is a very common example for granular materials in industrial application. We formulate the rheology for the continuum approach in the framework of the regularized version of a Bingham fluid and use advanced numerical methods regarding discretization as well as solution aspects, so that we can provide mesh independent results with two different software packages, Featflow and Openfoam, which can be used for validation and evaluation of the different methods and approaches. The authors thank the German Research Foundation DFG for funding under the grants TU 102/44-1 and SCHW 1168/6-1 within the pilot for transnational cooperation of the Dutch Technology Foundation STW and the DFG. This work was also supported by DAAD.

Buoyant miscible displacement flow of a Bingham fluid by a Newtonian fluid along a vertical plane-channel is studied, in the high Peclet number regime. The displacing fluid is lighter than the displaced fluid and the flow direction is upwards, hence density-unstable. The flow is effectively governed by 4 dimensionless parameters: the Newtonian Bingham number (BN), the viscosity ratio (m), the Reynolds number (Re), and modified Froude number (Fr). This is a simple model for micro-annulus formation in the primary cementing of oil and gas wells. The micro-annulus layers correspond to residual drilling mud that remains on the walls after displacement. Here we show that in density unstable situations static residual wall layers can exist for yield stresses below the minimum for density stable regimes. The layers are partially static and may also be thicker than the fully static layers encountered in density stable flows. Adverse buoyancy gradients (represented here by large χ = 2 R e F r 2 ), also seems to promote instability of the flow. These instabilities are various, ranging from Rayleigh–Taylor type frontal instabilities, through to Kelvin–Helmholz type inertial waves, and more viscous-controlled inverse bamboo and mushroom morphologies. We present an overview of flow regimes within our parameter space and approximate onset criteria. It is unclear whether or not the instabilities observed result in improved fluid displacement efficiency.

In this study, applicability of peristaltic waves as an efficient means for bio-particle manipulation is numerically investigated using the lattice Boltzmann method (LBM). The analysis is carried out in a flow configuration which comprises a two-dimensional, closed-ended, vertical cavity filled with a Bingham fluid and equipped with two flexible side walls and two rigid upper and lower plates. It is shown that by judicially adjusting the working parameters of the peristaltic waves (e.g., wavelength, amplitude, speed, and/or phase lag) it is possible to control the dynamics/trajectory of a large, circular bio-particle immersed in the fluid. It is also shown that a suspending circular particle moves faster if the surrounding fluid possess a yield-stress. Based on our obtained numerical results, for a given cross-sectional area, circular particles move faster than elliptical particles when excited peristaltically.

The evolution equations for the particle distribution functions are written in a divergence form applicable in three dimensions. From this set, it is shown that the continuity equation and the equations of motion are satisfied in Cartesian, cylindrical and spherical coordinates for all fluids when additional source terms are added to the equations of evolution in the latter two coordinate systems. If the body forces are present, a new set of source functions is required in each coordinate system and these are described as well. Next, the energy equation is derived by using a separate set of particle distribution functions. Modifications of the relevant equations to be applicable to incompressible fluids is described. The incorporation of boundary conditions and the description of the numerical scheme for the simulation of the flows employing the new approach is given. Validation results obtained through the modelling of a mixed convection flow of a Bingham fluid in a lid-driven square cavity, and the steady flow of a Bingham fluid in a pipe of square cross-section are presented. Next, using the cylindrical coordinate version of the evolution equations, numerical modelling of the steady flow of a Bingham fluid and the Herschel–Bulkley fluid in a pipe of circular cross-section have been performed and compared with the simulation results using the augmented Lagrangian method as well as the analytical solutions for the velocity field and the flow rate. Finally, some comments on the theoretical differences between the present approach and the existing formulations regarding Lattice Boltzmann Equations are offered.

The unclassified tailings paste (UTP) is a typical non-Newtonian fluid with a mass fraction over 70%, and the UTP disposal technology is of great significance to solve the disaster of tailings ponds and underground voids. The widely used Buckingham equation for predicting the pressure gradient of UTP in pipe flow does not consider the effect of wall slip drag reduction and the prediction accuracy is relatively low. To this end, a theoretical derivation based on the apparent wall slip equations established by previous scholars, sedimentation tests, loop tests and rheological tests were carried out. When the UTP is a Bingham fluid, the proposed new pressure gradient model which considers the wall slip effect is composed of the Buckingham equation and a dimensionless wall slip correction coefficient. The correction coefficient decreases with increases in either slip layer thickness and plastic viscosity of UTP, and increases with increases in either pipe diameter and slip layer viscosity. In this study, the dimensionless correction coefficient is 0.83∼0.85. When using the new pressure gradient model, the ratio of solids volume fraction to the maximum packing volume fraction of UTP (the filling degree) is at least 0.76. Compared with the measured pressure gradient from the loop tests in this study and the literature, the average error of the new pressure gradient model is 11.3%, whereas the average error of the Buckingham equation is 29.8%. The advantages of the new pressure gradient model are that the required parameters are easy to obtain, the model is easy to solve, and the prediction accuracy is better than the Buckingham equation. When the UTP is a Herschel-Bulkley fluid, the reliability of the pipe pressure gradient model which considers the wall slip effect still needs to be verified.

The plane Poiseuille flow of a Bingham plastic with pressure-dependent material parameters is analysed. Both the plastic viscosity and the yield stress are assumed to vary linearly with pressure and analytical solutions are derived for the two-dimensional pressure and the one-dimensional velocity. The effects of the plastic-viscosity and yield-stress growth parameters on the thickness of the unyielded plug and the conditions for the occurrence of flow are discussed.

We develop an asymptotic solution for the axisymmetric squeeze flow of a viscoplastic medium. The standard lubrication-style expansions of the problem predict a plug speed which varies slowly in the principal flow direction. This variation implies that the plug region cannot be truly unyielded. Our solution shows that this region is a pseudo-plug region in which the leading order equation predicts a plug, but really it is weakly yielded at a higher order. We follow the asymptotic technique suggested earlier by Balmforth and Craster (1999) and Frigaard and Ryan (2004). This method involves no relaxation of the exact Bingham model. The analytical expression for the squeeze force is in good agreement with previous numerical results. In the second part of the paper, the axisymmetric squeeze flow of a Bingham fluid with slip yield boundary condition at the wall is considered. We provide an asymptotic solution for this type of flow. Depending on the ratio of two dimensionless parameters partial slip (stick-slip) or full slip at the wall (slip) are possible.

The effect of modifying yield stress on turbulent pipe flow of generalised Newtonian fluids at a friction Reynolds number of 323 is investigated using direct numerical simulations. Simulations are carried out for Bingham and Herschel–Bulkley fluids with the yield stress varying from 0% to 20% of the mean wall shear stress. Results show that the effect of increasing yield stress is mostly similar to shear thinning in power-law fluids. The turbulent viscous stress which arises due to viscosity fluctuations is negative for a yield stress fluid and is higher in magnitude for higher yield stress. An analysis of the turbulent kinetic energy budget showed that the effect of yield stress is mainly significant near the wall for y + ≲ 60 which was also seen for shear-thinning power-law fluids at similar Re τ . Additional shear thinning enhances the yield stress effect. The main difference between shear thinning and yield stress is that the effect of yield stress is maximum outside the viscous sublayer whereas shear thinning has a more significant effect inside the viscous sublayer.

We study numerically isoviscous displacement flows of a Bingham fluid by a Newtonian fluid in a horizontal plane channel. We consider a heavy-light displacement flow with a laminar imposed flow. We analyze the flow from various perspectives, using the three dimensionless parameters that largely describe the flow: the Bingham number (Bn), the densimetric Froude number (Fr) and the Reynolds number (Re). In particular, we are able to categorize three classes for the main flow regimes, viz., center-type and slump-type, no-back-flow and temporary-back-flow, and stable and unstable displacements, for which we also quantify their transition boundaries versus the aforementioned dimensionless numbers. We show that the appearance of unstable flows can eventually lead to a number of different flow patterns, such as sinusoidal shaped wave and periodic-detachment forms. In addition, we study in depth an interesting feature exclusively associated to viscoplastic fluid displacements, i.e., the formation of static residual layers on the upper/lower walls of the channel. We also study the evolution of the key displacement flow features, i.e., the trailing and leading fronts, and classify certain secondary flow features related to the leading front, such as plug-like, inertial tip, semi-detached and fully-detached fronts. Finally, we quantify the velocity of the leading front at long times.

Buoyant miscible displacement flow of a Bingham fluid by a Newtonian fluid along a vertical plane-channel is studied, in the high Peclet number regime. The displacing fluid is denser than the displaced fluid and the flow direction is density-stable (upwards). The flow is effectively governed by 4 dimensionless parameters: the Newtonian Bingham number (BN), the viscosity ratio (m), the Reynolds number (Re), and modified Froude number (Fr). This is a simple model for micro-annulus formation in the primary cementing of oil and gas wells. We show that the residual layer thickness is largely determined by two parameters: (BN/m, χ*/m), where χ * = 2 R e F r 2 . Residual wall layers may be either static or mobile, and mobile layers either evolve to become static or are washed from the channel at long times. We show that the different behaviours of the residual layers are linked to different characteristic behaviours of the displacement front, and we show how the latter behaviours can be predicted using a lubrication/thin-film approximation.

We present an assessment of viscosity regularization (VR) in predicting transient flows of viscoplastic fluids (VPF). Although VR has been the most popular approach for computational studies of VPF in the last decade, the understanding of its effects on the predicted flow behaviour has remained largely qualitative. To reveal the effects of VR, we present a systematic quantitative comparison of the solutions of the regularized and exact (i.e. not regularized) problems. We consider free convection of a Bingham fluid in a two-dimensional square cavity with differentially heated sidewalls. We use bi-viscosity as a representative VR model and use augmented Lagrangian (AL) method to solve the exact problem. We show that common methods of assessing the solution error can hugely overestimate the accuracy of the results of VR. In particular, complete loss of accuracy is observed when flow conditions approach the motionless state, rendering the lower computational cost and algorithmic simplicity of VR irrelevant. Furthermore, we quantify the accuracy of bi-viscosity regularization in predicting the unyielded regions. This analysis disproves by counterexample that using yield stress contours produces convergent numerical estimates of the unyielded regions.

In the present paper we analyze the dimensionless numbers that concern the flow of viscoplastic materials. The Bingham material is used to conduct the main discussion but the ideas are generalized to more complex viscoplastic models at the end of the article. Although one can explore the space of solutions with a set of dimensionless numbers where only one of them takes into account the yield stress, like the Bingham number for example, we recommend that the characteristic stress should be defined as the extra-stress intensity evaluated at a characteristic (maximum) deformation rate. Such a definition includes the yield stress in every dimensionless number that is related to viscous effects like the Reynolds number, the viscosity ratio, and the Rayleigh number. This procedure was shown to be more effective on collapsing data into master curves and to provide a fairer comparison with the Newtonian case. This happens because a more representative viscous effect is taken into account, concentrating the plastic effects into a single parameter. The plastic number, the ratio of the yield stress to the maximum stress of the domain, is shown to better capture plastic effects than the usual Bingham number. The analysis of problems where a characteristic stress, but not a characteristic velocity, is provided, indicates that a more representative characteristic velocity should be defined with respect to the driving potential for the motion, i.e., the difference between the characteristic and yield stresses. This method is in contrast to the majority of the literature, where for Bingham materials, the dimensionless numbers are maintained in the same form as the original Newtonian ones, replacing the Newtonian viscosity by the viscous parameter of the Bingham model.

Shear-induced particle migration in monodisperse and bi-disperse suspension of spherical particles in a Bingham fluid is considered. Previous models of particle migration in Newtonian suspensions are extended to account for the existence of un-yielded regions in the visco-plastic domain when forced to flow in a tube and in a concentric Couette device. In suspension with a monodisperse particle phase, it is shown that particle concentration is continuously augmented in low shear rate regions. The yield boundary is monotonically shifting, affecting the velocity profiles, and the power to maintain the flow is monotonically reduced. When the suspension size distribution is bi-modal, the migration, eventually, results in a separation of the species. Larger particles migrate to the low shear rate zones and the smaller phase is pushed away from there. The velocity profiles, yield boundary and power do not change monotonically, and several stages in this dynamics can be identified.

Starting from the basic thermodynamic principles, we derive equations of micropolar Bingham fluids allowing for the variable concentration of polar particles. On the one hand, such fluids exhibit couple stresses, a non-symmetrical stress tensor, microrotations, and microinertia. On the other hand, the fluids support a yield stress. It was proved earlier that a micropolar fluid model can reproduce the Segre–Silberberg effect. Here, we study how the yield stress impacts this effect.

The flow of a Bingham fluid in a tilted channel of non uniform width is considered. The upper wall of the channel is assumed to be flat but not parallel to the bottom one and the flow is driven only by the gravity. A lubrication approximation is considered and an analytical solution is determined. We find conditions ensuring the appearance of the plug at a certain distance from the channel inlet. We also give an explicit formula for the pressure drop along the channel. Some numerical simulations are worked out for different values of the Bingham number and for different slopes of the upper wall.

Peristaltic transport of circular rigid particle(s) suspended in a viscoplastic fluid is numerically investigated in a planar two-dimensional channel using the lattice Boltzmann method (LBM) coupled with the smoothed profile method (SPM). Numerical results could be obtained at non-zero Reynolds numbers for a wide range of the propagating wave's parameters. Our numerical results reveal that a fluid's yield stress has a dramatic effect on the peristaltic transport of solid particles. For a single particle suspended in a viscoplastic fluid, which obeys the bi-viscosity (Bingham) model as its constitutive equation, it is found that there exists a threshold wave number below which the yield stress slows down the particle's transport but above which it has an accelerating effect. A fluid's yield stress is also predicted to lower the average velocity of an initially-centered single particle at any given confinement ratio. For a given Bingham number, particle's velocity is predicted to increase by an increase in the confinement ratio provided that it is larger than a critical value. An increase in the Reynolds number may increase or decrease the particle's average velocity depending on the Bingham number. For the two-particle and four-particle scenarios, our numerical results show that the initial positioning of the particles in the channel has a decisive role in their eventual fate.

In this work, two-dimensional, steady flow of incompressible Bingham plastic fluids past a rotating circular cylinder has been studied. The continuity and momentum equations have been solved numerically over the range of conditions as follows: 0.1 ≤ Re ≤ 40; Bingham number, 0 ≤ Bn ≤ 103 and non-dimensional rotational velocity 0 ≤ α ≤ 5. The objective of the detailed numerical study is to elucidate the effects of Reynolds number, Bingham number and rotational velocity on the yield surfaces, the drag and lift coefficients, and torque acting on the rotating cylinder. The present drag results have been correlated with the modified Reynolds number and rotational velocity to interpolate the intermediate values of Bingham number. In additional, shear rate and yielded/unyielded contours, and streamlines in the vicinity of the cylinder are also presented for detailed visualization of the flow field. The critical Bingham number, denoting the cessation of the flow detachment from the surface of the rotating cylinder, has been plotted as a function of the Reynolds number, and the rotational velocity is shown to suppress the propensity for flow separation.

We present a lattice Boltzmann model to simulate multiphase viscoplastic fluid flow. It is an extended model based on the free-energy-based lattice Boltzmann method (LBM) for multiphase fluids with possible high-density and high-viscosity ratios by applying the Herschel-Bulkley constitutive relationship to account for the variable viscosity. The model shows good agreements between the simulation results and the corresponding theoretical solutions for different cases. Furthermore, the capability and effectivity of this model is tested by examples, including droplet(s) falling and interaction in Bingham fluid, and sessile viscoplastic droplet motion. The results illustrate that our model is able to catch the yield behavior well, and to distinguish various kinds of viscoplastic fluids effectively.

Peristaltic flow of a viscoplastic fluid which obeys the bi-viscous (Bingham) model as its constitutive equation is numerically studied in a planar two-dimensional channel using the multiple-relaxation-time lattice Boltzmann method (MRT-LBM). Numerical results could be obtained at large Reynolds numbers for arbitrary set of wavelength and amplitude ratios of the peristaltic wave. It is shown that depending on the Reynolds number, a fluid's yield stress may increase or decrease the time-mean flow rate of peristaltic pumps. Our numerical results also show that for yield-stress fluids there exists a threshold wave number above which the yield stress can accelerate fluid transport whereas below which it can have a decelerating effect. The yield stress is also predicted to diminish the size of the fluid bolus in the “trap” phenomenon for certain set of parameters.

The flow classification developed by Thompson and Souza Mendes [Int. J. Engng. Sci. 43 (2005) 79-105] is applied to radial and squeeze flows between two parallel disks. Based on the lubrication approximation, explicit expressions of the flow classification parameter, namely ℛ , could be obtained for those flows, and the results for power-law and Bingham fluids are presented. It is shown that 0 ≤ ℛ ≤ 1 in the radial flow of both power-law and Bingham fluids, which reflects mixed shear and extensional flow. In the squeeze flow of a power-law fluid, the overall distribution of ℛ is similar to the radial flow case. In the squeeze flow of a Bingham fluid, on the other hand, it is noted that there can be a particular region where ℛ > 1 , which is due to the rapid rotation of the principal axes of the rate-of-strain tensor.

In this paper we study the flow of a Bingham fluid on a curved (rectangular) pipe. Flows in this kind of geometries present secondary flows due to presence of centripetal forces in the radial direction. This so called Dean Flow has been extensively studied for Newtonian fluids. In considering a yield stress fluid in similar geometries the picture is less clear. Unfortunately, there is not an analytical solution, so the flow has been barely studied. Thus the purpose of this work. We consider only steady flow and develop a solution as a perturbation series in terms of the Dean number, D n . The leading order solution is an axisymmetric azimuthal flow, driven by the pressure gradient. We compute this flow numerically using an augmented Lagrangian method on a regular rectangular mesh. We have also studied the limiting case of zero flow and we give a general expression for the critical Bingham number, B c , for this to happen. The first order velocity perturbation produces 2 recirculating vortices, at top and bottom walls, analogous to those of the Newtonian flow. The size of the vortices appears to decrease slightly as B → B c − , and the magnitude of the secondary flow decays to zero significantly faster than that of the leading order flow. Finally we discuss the short comings of the perturbation series approach and propose how to overcome them.

We extend a method developed by Fusi and Farina (Appl. Math. Comp. 320, 1–15, 2018) to obtain semi-analytical lubrication-approximation solutions for the flow of a Bingham-plastic in a tube of variable radius. The proposed method is applicable provided that the unyielded core extends continuously from the inlet to the outlet. It turns out that the variable radius of the latter core obeys a stiff integral-algebraic equation which is solved both numerically and asymptotically. The pressure distribution is then obtained integrating a 1st-order ODE and the velocity components are computed using analytical expressions. Converging or diverging either linearly or exponentially, undulating and stenosed tubes are considered. The effects of the shape of the wall on the yield surface which separates the yielded region from the unyielded core and on the pressure difference required to drive the flow are investigated and discussed. The results show that the effect of the wall variation amplitude is greatly amplified as the volumetric flow rate (or, equivalently, the imposed pressure difference driving the flow) increases. It is also demonstrated that the pressure difference needed to achieve a given volumetric flow rate in a converging pipe is always higher than that for a diverging one.

In the present paper, the effects of a channel symmetric wall slip are investigated on the stability of plane Poiseuille flow of a Bingham fluid, based on the modal and non-modal linear stability analysis approaches. Both streamwise and spanwise slip conditions are considered and their results are compared with those of the no-slip condition, in terms of the flow stability. The results are obtained for different dimensionless groups that govern the stability picture, for example the Bingham number, the Reynolds number and the streamwise/spanwise slip numbers. The linearized perturbation equations are obtained and the corresponding eigenvalue problem is solved based on the Chebyshev collocation method at the Gauss-Lobatto grid points, using the QZ algorithm. In the modal analysis, the focus is on the changes made by the channel wall slip on the eigenvalue spectra. The flow is shown to be linearly stable for no-slip and streamwise slip conditions. However, for the spanwise slip condition, an unstable eigenvalue is found for large values of the spanwise slip number. The non-modal stability analysis is based on the transient energy growth of the perturbations, which is calculated using the eigenfunctions of the eigenvalue problem representing the amplitude of the perturbations. Increasing the streamwise slip decreases the maximum energy growth, eventually leading to a flow for which the transient energy only decays in time. Thus, two distinct flows showing the energy growth and energy decay regimes can be identified, for a range of Bingham, Reynolds and streamwise slip numbers.

Impinging liquid jets are widely used to clean unwanted soil layers from the walls of structures and vessels. When the soil is a thin layer of an immiscible viscoplastic material, removal involves the growth of a cleared area (which is circular for a jet impinging normally) bounded by a berm of displaced material. Glover et al. [1] presented a semi-empirical model relating the rate of removal to the momentum flow rate in the liquid film. We present a first-order model for cleaning thin layers of these materials based on the rate of rate of viscous dissipation in a shallow wedge of material at the cleaning front. This yields a result of the form of the model in [1] , with expressions linking the kinetic parameters to measurable quantities including the rheology of the soil. The fully coupled problem is not solved: the wedge angle and residual layer thickness need to be specified and were obtained here by fitting to the data. The model is compared with experimental results obtained for three soft solids – two petroleum jellies and a soft paraffin – which exhibited Bingham plastic behaviour and creep, for jet Reynolds numbers between 10,000-37,000, and 0.1 h ¯ / δ o 1.5 , where h ¯ is the average film depth and δo the layer thickness. The data show reasonable agreement with the model.

We derive an expression for the velocity profile of a pressure-driven yield-stress fluid flow-ing around a two-dimensional concentric annulus. This result allows the prediction of the effects of channel curvature on the pressure gradient required to initiate flow for given yield stress, and for the width of the plug region and the flux through the channel at different curvatures. We use it to validate numerical simulations of the flow from a straight channel into a curved channel which show how the fluid first yields everywhere before reaching the predicted velocity profile., arXiv admin note: substantial text overlap with arXiv:1909.06173

We study steady inertial flows of a Bingham fluid through 2D “washout” geometries, that represent longitudinal sections of an oil and gas well under construction, thus extending our previous work (Roustaei et al., 2015) into the inertial regime. The washout geometry is characterised by dimensionless depth h and length δ − 1 . The other dimensionless parameters of problem are the Reynolds number Reand Bingham number B. The effects of increasing Re were studied for fixed B and for fixed Hedstrom number, H e = R e B . The latter represents an increase in flow rate for fixed geometry and fluid properties. In both cases we observed that the variation in flowing area of the washout (i.e. that area that is mobilised) was non-monotone. Increasing Reresulted in a straightening of the streamlines passing through the washout region, in the main part of the channel. For fixed He, beyond a first critical value of Re the flowing area was observed to decrease, i.e. increasing the flow rate results in larger parts of the washout being static, as is quite counterintuitive and contrary to the industrial perception that pumping faster will circulate/condition the mud better. This trend persists for a significant range of Re. On passing a second critical value of Re, we observe the onset of zones of recirculation within the washout. The flowing area thus increases but the area of washout from which fluid is displaced during conditioning continues to decrease. Finally, we considered self-selection of the flow geometry, as observed in Roustaei et al. (2015) for R e = 0 . We conclude that the same phenomenon holds in inertial flows for a given geometry, i.e. at large hthe flowing areas of washouts with static regions become similar in shape.

In this paper we study the planar squeeze flow of a Bingham plastic in the lubrication approximation. We assume that the domain occupied by the fluid is closed at one end and open at the other (planar geometry). We consider two cases: (i) planar walls approaching each other in a prescribed way; (ii) parallel walls whose shape depends on both time and longitudinal coordinate. The dynamics of the unyielded region is determined exploiting the integral formulation of the linear momentum balance. We prove that in proximity of the closed end the material is always yielded, so that the rigid part is always detached from it. When dealing with case (ii), we show that the dynamics of the rigid domain is governed by a very complex integral equation, whose qualitative analysis is beyond the aims of this paper. Conversely, in case (i) we obtain an almost explicit solution.

The analytical solution of the start-up flow of a Bingham fluid between two co-axial cylinders is presented in this study. We focus on that all fluid is at rest initially, then a constant shear stress is exerted by the inner cylinder while the outer remains stationary. The problem is solved using methods of Laplace transform and numerical integration. The unsteady solutions of shear stress and angular velocity as well as the motion of yielding surface are given. We also conduct the limiting analysis at infinitesimal values of the time after initiation.