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This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number

The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $\\epsilon \\leq c_0\\mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \\rightarrow \\infty $. For times $t \\gtrsim \\mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of \"2.5 dimensional\" streamwise-independent solutions referred to as streaks.

A relatively complete continuous relaxation spectrum H ( τ ) of poly(styrene- b -isoprene- b -styrene) (SIS) two-phase system was divided into five regions based on the variation of H ( τ ) strength, which corresponded to: (1) glass transition of PI phase, (2) high elastic state of PI phase, (3) glass transition of PS phase, (4) high elastic state of PS phase, and (5) viscous flow state of the entire SIS molecular chain. Five regions only appeared in SIS1105, because the molecular of SIS1105 experienced glass transition of PI block, PS block, and viscous flow of the whole molecular chain. The stress relaxation of SIS was influenced by the S/I ratios, because the PS microdomains ultimately determined the relaxation characteristics. The stress relaxation of SIS system was also closely related to the viscous flow transition temperature ( T f ). When the S/I ratio was low (15/85), forced stress relaxation occurred; when the S/I ratio was high (29/71 and 45/55), the SIS system did not show stress relaxation below the T f . When the temperature was higher than the T f , the S/I ratio did not affect the stress relaxation. The relaxation information obtained from the Cole–Cole diagram further verified the analysis of the continuous relaxation spectrum H ( τ ) and stress relaxation. Graphical abstract

In this note we investigate the initial-boundary value problem for a Stokes system arising in a free surface viscous flow of a horizontally periodic fluid with fractional boundary operators. We derive an integral representation of solutions by making use of the multiple Fourier series. Moreover, we demonstrate a unique solvability in the framework of the Sobolev space of L 2 -type.

In this study, a novel meshless collocation method based on the Gaussian–cubic hybrid kernel function in conjunction with the ghost-points method and the general Newton–Raphson method is proposed for solving the four-order stream function formulation of Navier–Stokes equations. The decrease of variables and equations results in better computational efficiency of stream function formulation than primitive variable formulations of 2D incompressible viscous flow. The original Naiver–Stokes equations can be transformed into a four-order stream function partial differential equation by introducing the vorticity and stream function. The new proposed Gaussian–cubic backward substitution method is used to solve the corresponding system of equations where the nonlinear stream function formulation is linearized by the general Newton–Raphson method. The ghost-points method is applied as the layout scheme of center points, which can effectively improve the accuracy without requiring more computational resources. Several examples are provided and the results demonstrate the feasibility of the proposed novel approach.

In this paper, the rheological properties of bio-based semi-aromatic high temperature polyamide PA5T/56 were studied by melt index meter and rotational rheometer, and the effect of PA5T content on the rheological properties was discussed. The results showed that the melt index of PA5T/56 decreased with the increase in PA5T content; The bio-based semi-aromatic high temperature polyamide PA5T/56 with different ratios was pseudoplastic fluids, and its non-Newtonian index increased with the increase in temperature, the increase in PA5T content can improve the sensitivity of apparent viscosity to shear stress and shear rate; The apparent viscosity reduced with the increase in shear stress or shear rate; With the increase in shear rate, the viscous flow activation energy decreased, and the apparent viscosity was less sensitive to temperature; With the increase in PA5T content, the viscous flow activation energy of PA5T/56 increased gradually, indicated that the increase in PA5T content can improve the temperature sensitivity of PA5T/56 apparent viscosity; Therefore, in the synthesis and post-treatment of PA5T/56 material, in order to improve its processing properties, make the operation easier and reduce the economic cost, it can be improved by adjusting the processes such as shear stress, shear rate and the ratio of PA5T/56 components.

Variational formulations for viscous flows which lead to the Navier–Stokes equation are examined. Since viscosity leads to dissipation and, therefore, to the irreversible transfer of mechanical energy to heat, thermal degrees of freedom have been included in the construction of viscous dissipative Lagrangians, by embedding of thermodynamics aspects of the flow, such as thermasy and flow exergy. Another approach is based on the presumption that the pressure gradient force is a constrained force, whose sole role is to maintain the continuity constraint, with a magnitude that is minimum at every instant. From these considerations, Lagrangians based on the minimal energy dissipation principal have been constructed from which the application of the Euler–Lagrange equation leads to the standard form of the Navier–Stokes equation directly, or at least they are capable of generating the same equations of motion for simple steady and unsteady one-dimensional viscous flows. These efforts show that there is equivalence between Lagrangian, Hamiltonian, and Newtonian mechanics as far as the derivation of the Navier–Stokes equation is concerned. However, one of the conclusions is that the attractiveness of the variational approach in more complex situations is still an open question for the applied fluid mechanician.

Ion irradiation can cause burrowing of nanoparticles in substrates, strongly depending on the material properties and irradiation parameters. In this study, it is demonstrated that the sinking process can be accomplished with ion irradiation of cube‐shaped Ag nanoparticles on top of silicon; how ion channeling affects the sinking rate; and underline the importance of the amorphous state of the substrate upon ion irradiation. Based on these experimental findings, the sinking process is described as being driven by capillary forces enabled by ion‐induced plastic flow of the substrate. Ion irradiation causes nanoparticles to sink into substrates, influenced by material properties and irradiation specifics. This study reveals nanoparticle shape doesn't dictate sinking. Ion channeling and substrate's amorphous state play crucial roles. Sinking is driven by capillary forces, facilitated by ion‐induced substrate plasticity.

The time dependence for densification of titanium nitride nanopowder during nonisothermal spark plasma sintering at an external pressure of 79.2 MPa in a nitrogen atmosphere was experimentally studied under controlled heating at a constant rate of 0.833 K/s. The densification kinetics was analyzed within the continuum theory of bulk viscous flow of a porous body using computational modeling. In general, the sintering process is characterized by a decrease in the root-mean-square stress within the porous body matrix to the limiting zero value as it approaches the nonporous state and by an increase in the root-mean-square strain rate following a curve with a maximum. Prior to the onset of densification, when thermodynamic temperature reaches 783 K, a stage involving annealing of the strain-hardened matrix forming the porous titanium nitride is observed. In the temperature range of 950–1040 K, weak densification occurs, governed by plastic flow, with a linear dependence of the strain rate on stress and low apparent activation energy (35.1 kJ/mol). At higher temperatures, dislocation climb becomes the acting mechanism, characterized by a power-law dependence (n = 2) of the root-mean-square strain rate on the root-mean-square stress, with an activation energy of 280.8 kJ/mol. The activation of this mechanism at relatively low temperatures, along with the nanosized structure, is attributed to the influence of the electric field. Titanium nitride samples produced by spark plasma sintering exhibit a nanosized structure with an average grain size of 60 nm, which ensures its enhanced mechanical properties.

This article is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier–Stokes–Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory.