Thermal entry flow problem for Giesekus fluid inside an axis-symmetric tube through isothermal wall condition: a comparative numerical study between exact and approximate solution

The thermal entry flow problem also known as the Graetz problem is investigated for a Giesekus fluid model. Both analytical (exact) and approximate solutions for velocity are obtained. The nondimensional pressure gradient is numerically obtained via the mean flow rate relation. The energy equation along with the Giesekus fluid velocity is analytically solved for the constant wall temperature case by using the classical separation of variable method. This method transforms the energy equation into a Sturm–Liouville (SL) boundary value problem. The MATLAB solver bvp5c is employed to compute the eigenvalues and the related eigenfunctions numerically. The impact of mobility parameter and Weissenberg number on local Nusselt number, mean temperature, and average Nusselt number is discussed and displayed graphically. It is also found that the presence of the Weissenberg number elevates the Nusselt numbers. Further, the presence of the mobility parameter of the Giesekus fluid model delays the prevalence fully developed conditions in both entrance and fully developed regions. The comparison between approximate and exact solution is also presented. It reveals that both solutions have an exact match with each other for smaller values of mobility parameter and Weissenberg number. However, there is a deviation for larger values of both parameters.

Migration and Alignment of Three Interacting Particles in Poiseuille Flow of Giesekus Fluids

Effect of rheological property on the migration and alignment of three interacting particles in Poiseuille flow of Giesekus fluids is studied with the direct-forcing fictitious domain method for the Weissenberg number (Wi) ranging from 0.1 to 1.5, the mobility parameter ranging from 0.1 to 0.7, the ratio of particle diameter to channel height ranging from 0.2 to 0.4, the ratio of the solvent viscosity to the total viscosity being 0.3 and the initial distance (y0) of particles from the centerline ranging from 0 to 0.2. The results showed that the effect of y0 on the migration and alignment of particles is significant. The variation of off-centerline (y0 ≠ 0) particle spacing is completely different from that of on-centerline (y0 = 0) particle spacing. As the initial vertical distance y0 increased, the various types of particle spacing are more diversified. For the off-centerline particle, the change of particle spacing is mainly concentrated in the process of cross-flow migration. Additionally, the polymer extension is proportional to both the Weissenberg number and confinement ratio. The bigger the Wi and confinement ratio is, the bigger the increment of spacing is. The memory of shear-thinning is responsible for the reduction of d1. Furthermore, the particles migrate abnormally due to the interparticle interaction.

On the stationary nonlocal Cahn–Hilliard–Navier–Stokes system: Existence, uniqueness and exponential stability

The Cahn–Hilliard–Navier–Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn–Hilliard–Navier–Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder’s theorem. Further, we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn–Hilliard–Navier–Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable (for convex singular potentials) under suitable conditions on mobility parameter and viscosity.

Calibration of material parameters based on 180$$^\circ $$ and 90$$^\circ $$ ferroelectric domain wall properties in Ginzburg–Landau–Devonshire phase field models

Ferroelectric phase field models based on the Ginzburg–Landau–Devonshire theory are characterized by a large number of material parameters with problematic physical interpretation. In this study, we systematically address the relationship between these parameters and the main properties of ferroelectric domain walls. A variational approach is used to derive closed form solutions for the polarization fields at the phase transition regions as well as for the propagation velocities of the domain walls. Introducing a modified set of material parameters, which appropriately scales different contributions to the free energy, we are able to accurately calibrate these parameters based on domain wall thickness and energy of both 180$$^\circ $$ ∘ and 90$$^\circ $$ ∘ domain walls. Moreover, the mobility parameter appearing in the Ginzburg–Landau evolution equation can be accurately calibrated based on the propagation velocity of the domain walls.

Near-critical swirling flow of a viscoelastic fluid in a circular pipe

The interaction between flow inertia and elasticity in high-Reynolds-number, axisymmetric and near-critical swirling flows of an incompressible and viscoelastic fluid in an open finite-length straight circular pipe is studied at the limit of low elasticity. The stresses of the viscoelastic fluid are described by the generalized Giesekus constitutive model. This model helps to focus the analysis on low fluid elastic effects with shear thinning of the viscosity. The application of the Giesekus model to columnar streamwise vortices is first investigated. Then, a nonlinear small-disturbance analysis is developed from the governing equations of motion. It reveals the complicated interactions between flow inertia, swirl and fluid rheology. An effective Reynolds number that links between steady states of swirling flows of a viscoelastic fluid and those of a Newtonian fluid is revealed. The effects of the fluid viscosity, relaxation time, retardation time and mobility parameter on the flow development in the pipe and on the critical swirl for the appearance of vortex breakdown are explored. It is found that in vortex flows with either an axial jet or an axial wake profile, increasing the shear thinning by decreasing the ratio of the viscoelastic characteristic times from one (with fixed values of the Weissenberg number and the mobility parameter) increases the critical swirl ratio for breakdown. Increasing the fluid elasticity by increasing the Weissenberg number from zero (with a fixed ratio of the viscoelastic characteristic times and a fixed value of the mobility parameter) or increasing the fluid mobility parameter from zero (with fixed values of the Weissenberg number and the ratio of viscoelastic times) causes a similar effect. The results may explain the trend of changes in the appearance of breakdown zones as a function of swirl level that were observed in the experiments by Stokes et al. (J. Fluid Mech., vol. 429, 2001, pp. 67–115), where Boger fluids were used. This work extends for the first time the theory of vortex breakdown to include effects of non-Newtonian fluids.

Hydrodynamics of pedestrians' instability in floodwaters

People's safety is the first objective to be fulfilled by flood risk mitigation measures, and according to existing reports on the causes of casualties, most of the fatalities are due to inappropriate behaviour such as walking or driving in floodwaters. Currently available experimental data on people instability in floodwaters suffer from a large dispersion primarily depending on the large variability of the physical characteristics of the subjects. This paper introduces a dimensionless mobility parameter θP for people partly immersed in flood flows, which accounts for both flood and subject characteristics. The parameter θP is capable of identifying a unique threshold of instability depending on a Froude number, thus reducing the scatter of existing experimental data. Moreover, a three-dimensional (3-D) numerical model describing the detailed geometry of a human body and reproducing a selection of critical pairs of water depth and velocity is presented. The numerical results in terms of hydrodynamic forces and force coefficients are analysed and discussed. Both the mobility parameter θP and the numerical results hint at the crucial role of the Froude number and relative submergence as the most relevant dimensionless numbers to interpret the loss of stability. Finally, the mobility parameter θP is compared with an analogous dimensionless parameter for vehicles' instability in floodwaters, providing a new contribution to support flood risk management and educating people.