Recent zbMATH articles in MSC 76A https://www.zbmath.org/atom/cc/76A 2021-11-25T18:46:10.358925Z Werkzeug Rigorous derivation of a mean field model for the Ostwald ripening of thin films https://www.zbmath.org/1472.35028 2021-11-25T18:46:10.358925Z "Dai, Shibin" https://www.zbmath.org/authors/?q=ai:dai.shibin During the late stages of the evolution of thin liquid films on a solid substrate, liquid droplets are connected by an ultra-thin residual film. Their number decreases due to migration and collision on the one hand, and exchange of matter through a diffusive field in the residual film on the other hand. Supposing that, at time $$t>0$$, there are $$N(t)\ge 1$$ droplets $$\{B(x_i,R_i(t))\ :\ 1\le i \le N(t)\}$$ in the square $$\Omega_{\mathcal{L}} =(0,\mathcal{L})^2$$ and neglecting the motion of the centers $$x_i$$ due to the no-slip boundary condition for the fluid at the substrate, the dynamics of the radii $$(R_i)$$ and the diffusive field $$u$$ may be described by \begin{align*} - \Delta u(t,x) &= 0, \qquad x\in \Omega_{\mathcal{L}}\setminus \bigcup_{i=1}^{N(t)} \bar{B}(x_i,R_i(t)), \ t>0, \\ u(t,x) &= \frac{1}{R_i(t)}, \qquad x\in B(x_i,R_i(t)), \ t>0, \\ \frac{dR_i}{dt}(t) &= \frac{1}{R_i(t)^2} \int_{\partial B(x_i,R_i(t))} [\nabla u(t,s)\cdot \mathbf{n}(s)]\ ds, \qquad t>0, \end{align*} supplemented with periodic boundary conditions on $$\partial\Omega_{\mathcal{L}}$$. In the above integral term, $$[\nabla u(t,s)\cdot \mathbf{n}(s)]$$ denotes the jump of the normal gradient of $$u$$ across the boundary of $$B(x_i,R_i(t))$$. After introducing a small parameter $$\varepsilon>0$$ and scaling $$\mathcal{L}$$, $$x$$, $$t$$, $$N$$, $$(R_i)$$, and $$u$$ in an appropriate way, homogeneization techniques are used to establish the convergence of the rescaled diffusion fields to a mean field $$u_*$$. The latter is a weak solution to \begin{align*} -\Delta u_*(t,y) + 2\pi\delta \int_0^\infty \left( u_*(t,y) - \frac{1}{r} \right) f(t,y,r)\ dr & = 0, \qquad r\in (0,\infty),\\ \partial_t f(t,y,r) + \partial_r \left( \frac{2\pi}{r^2} \left( u_*(t,y) - \frac{1}{r} \right) f(t,y,r) \right) & = 0, \qquad r\in (0,\infty), \end{align*} for $$t>0$$ and $$y\in \Omega_1$$, supplemented with periodic boundary conditions on $$\Omega_1$$. The parameter $$\delta$$ is prescribed by the scaling, while $$f$$ is in general a measure-valued solution to the above transport equation. Compactness and sharp lower bound for a 2D smectics model https://www.zbmath.org/1472.35143 2021-11-25T18:46:10.358925Z "Novack, Michael" https://www.zbmath.org/authors/?q=ai:novack.michael-r "Yan, Xiaodong" https://www.zbmath.org/authors/?q=ai:yan.xiaodong Summary: We consider a 2D smectics model $E_{\epsilon}(u)=\frac{1}{2}\int_\varOmega\frac{1}{\varepsilon}\left(u_z-\frac{1}{2}u_x^2\right)^2+\varepsilon(u_{xx})^2\text{d}x\,\text{d}z.$ For $$\varepsilon_n\rightarrow 0$$ and a sequence $$\{u_n\}$$ with bounded energies $$E_{\varepsilon_n}(u_n)$$, we prove compactness of $$\{\partial_zu_n\}$$ in $$L^2$$ and $$\{\partial_xu_n\}$$ in $$L^q$$ for any $$1\le q<p$$ under the additional assumption $$\Vert\partial_xu_n\Vert_{L^p}\le C$$ for some $$p>6$$. We also prove a sharp lower bound on $$E_{\varepsilon}$$ when $$\varepsilon\rightarrow 0.$$ The sharp bound corresponds to the energy of a 1D ansatz in the transition region. Kelvin-Voigt equations with anisotropic diffusion, relaxation and damping: blow-up and large time behavior https://www.zbmath.org/1472.35284 2021-11-25T18:46:10.358925Z "Antontsev, S." https://www.zbmath.org/authors/?q=ai:antontsev.stanislav-nikolaevich "De Oliveira, H. B." https://www.zbmath.org/authors/?q=ai:de-oliveira.hermenegildo-borges "Khompysh, Kh." https://www.zbmath.org/authors/?q=ai:khompysh.kh|khompysh.khonatbek Summary: A nonlinear initial and boundary-value problem for the Kelvin-Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established by the first author et al. [J. Math. Anal. Appl. 473, No. 2, 1122--1154 (2019; Zbl 1458.74026)]. In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation. On the long-time behavior of dissipative solutions to models of non-Newtonian compressible fluids https://www.zbmath.org/1472.35294 2021-11-25T18:46:10.358925Z "Feireisl, Eduard" https://www.zbmath.org/authors/?q=ai:feireisl.eduard "Kwon, Young-Sam" https://www.zbmath.org/authors/?q=ai:kwon.young-sam "Novotný, Antonín" https://www.zbmath.org/authors/?q=ai:novotny.antonin Summary: We identify a class \textit{maximal} dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times. Weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation times https://www.zbmath.org/1472.35300 2021-11-25T18:46:10.358925Z "Karazeeva, N. A." https://www.zbmath.org/authors/?q=ai:karazeeva.n-a Summary: A system of equations describing the motion of fluids of Maxwell type is considered: $\frac{\partial }{\partial t}v +v \cdot \nabla v -\underset{0}{\overset{t}{\int }}K\left(t-\tau \right) d\tau +\nabla p=f\left(x,t\right), \quad\mathrm{div}v =0.$ Here $$K(t)$$ is an exponential series $$K(t)=\sum \limits_{s=1}^{\infty }{\beta}_s{e}^{-{\alpha}_st}$$. The existence of a weak solution for the initial boundary value problem $v (x,0)=v_0(x),\quad v \cdot n|_{\partial \Omega }=0,\quad rot v |_{\partial \Omega }=0$ is proved. Free boundary problem in a polymer solution model https://www.zbmath.org/1472.35423 2021-11-25T18:46:10.358925Z "Petrova, A. G." https://www.zbmath.org/authors/?q=ai:petrova.anna-georgevna "Pukhnachev, V. V." https://www.zbmath.org/authors/?q=ai:pukhnachov.vladislav-v The authors consider the integro-differential equation $\frac{\partial w}{\partial t}+\frac{\partial w}{\partial y}\int_{0}^{y}w(z,t)dz-w^{2}=\frac{\partial^{2}w}{\partial y^{2}}+\gamma (\frac{\partial^{3}w}{\partial y^{2}\partial t}+\frac{\partial^{3}w}{\partial y^{3}}\int_{0}^{y}w(z,t)dz- \frac{\partial^{2}w}{\partial y^{2}}),$ posed in the domain $$\Omega_{T} = \{y,t:0 < y < h(t)$$, $$0\leq t\leq T\}$$. This model accounts for the flow of a mixture of water and polymer. This equation is completed with: $$\frac{dh}{dt}=\int_{0}^{h}w(y,t)dy$$. The boundary conditions $$w(0,t)=0$$ and $$\frac{\partial w}{\partial y}+\gamma (\frac{\partial^{2}w}{\partial y\partial t}+ \frac{\partial^{2}w}{\partial y^{2}}\int_{0}^{h}w(y,t)dy-w\frac{\partial w}{\partial y})(h(t),t)=0$$ are added, together with the initial conditions $$w(y,0)=w_{0}(y)$$, $$0\leq y\leq 1$$, $$h(0)=1$$. Here $$\gamma >0$$ is a constant and $$w_{0}$$ is a smooth ($$C^{3}$$) function of $$y$$ satisfying the conditions $$w_{0}(0)=w_{0}^{\prime}(1)=0$$. The first main result proves the existence of a local in time strong solution ($$h\in C^{1}([0,t^{\ast}])$$, $$w\in C^{3,1}([0,h(t)]\times \lbrack 0,t^{\ast}])$$) to this problem. If the initial condition further satisfies $$w_{0}(y)\leq 0$$, $$w_{0}(y)-\gamma w_{0}^{\prime \prime}(y)\leq 0$$, the authors prove the existence of a classical solution $$h\in C^{1}([0,T])$$, $$w\in C^{3,1}([0,h(t)]\times \lbrack 0,T])$$ to the above problem. Both existence results are obtained through appropriate transformations and using Schauder's theorem. The authors then consider the case where $$\gamma$$ tends to 0 and they observe that the problem turns into that of the deformation of a strip of viscous fluid. They here prove that the solution to this problem is destructed in finite time. They finally introduce asymptotic expansions with respect to $$\gamma$$ and they express the second term of this asymptotic expansion. Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations https://www.zbmath.org/1472.35448 2021-11-25T18:46:10.358925Z "Bao, Ngoc Tran" https://www.zbmath.org/authors/?q=ai:bao.ngoc-tran "Hoang, Luc Nguyen" https://www.zbmath.org/authors/?q=ai:hoang.luc-nguyen "Van, Au Vo" https://www.zbmath.org/authors/?q=ai:van.au-vo "Nguyen, Huy Tuan" https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan. "Zhou, Yong" https://www.zbmath.org/authors/?q=ai:zhou.yong Summary: This paper investigates an inverse problem for fractional Rayleigh-Stokes equations with nonlinear source. The fractional derivative in time is taken in the sense of Riemann-Liouville. The proposed problem has many applications in some non-Newtonian fluids. We obtain some results on the existence and regularity of mild solutions. $$L^q$$-solvability for an equation of viscoelasticity in power type material https://www.zbmath.org/1472.45002 2021-11-25T18:46:10.358925Z "de Andrade, Bruno" https://www.zbmath.org/authors/?q=ai:de-andrade.bruno "Silva, Clessius" https://www.zbmath.org/authors/?q=ai:silva.clessius "Viana, Arlúcio" https://www.zbmath.org/authors/?q=ai:viana.arlucio The authors study the existence, uniqueness, regularity, continuous dependence, unique continuation, a blow-up alternative for mild solutions, and global well-posedness of the nonlinear Volterra equation $u_t = \int_0^t dg_\alpha(s) \Delta u(t-s,x)- \nabla p +h - (u\cdot \nabla u),\quad \textrm{div}(u)=0,$ in $$(0,\infty)\times \Omega$$, where $$u=0$$ on $$(0,\infty)\times \partial \Omega$$ and $$u(0,x)=u_0(x)$$ in $$\Omega$$. Here the kernel is taken to be $$g_\alpha (t)= t^{\alpha}/\Gamma(\alpha+1)$$ with $$0\leq \alpha <1$$ and a mild solution is a solution to the equation $u(t)= S_\alpha (tA)u_0 + \int_0^t S_\alpha((t-s)A)(F(u)(s)+Ph(s))\, ds,$ where $$P$$ is the Leray projection on divergence free functions, $$F(u)= P(u\cdot \nabla)u$$, $$A=P\Delta$$ and $S_\alpha(tA)= \frac 1{2\pi i} \int_{Ha}e^{\lambda t}\lambda^\alpha(\lambda^{\alpha+1}I+A)^{-1}\, d\lambda,$ where $$t>0$$ and $$Ha$$ is a suitable path. The existence results show that the mild solutions have more spatial regularity in terms of estimates on norms in fractional power spaces when $$\alpha$$ is closer to $$0$$, the case of the Navier-Stokes equations. The linear estimates needed are stated in an abstract setting for sectorial operators which makes it possible to restate the results for some other equations as well. On the unique solvability of the optimal starting control problem for the linearized equations of motion of a viscoelastic medium https://www.zbmath.org/1472.49068 2021-11-25T18:46:10.358925Z "Artemov, M. A." https://www.zbmath.org/authors/?q=ai:artemov.mikhail-anatolevich Summary: We study an optimization problem for the linearized evolution equations of the Oldroyd model of motion of a viscoelastic medium. The equations are given in a bounded three-dimensional domain. The velocity distribution at the initial time is used as a control function. The objective functional is terminal. The existence of a unique optimal control is proved for a given set of admissible controls. A variational inequality characterizing the optimal control is derived. A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities https://www.zbmath.org/1472.53066 2021-11-25T18:46:10.358925Z "Alexander, Gareth P." https://www.zbmath.org/authors/?q=ai:alexander.gareth-p "Machon, Thomas" https://www.zbmath.org/authors/?q=ai:machon.thomas Summary: We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films. The relationship between viscoelasticity and elasticity https://www.zbmath.org/1472.74038 2021-11-25T18:46:10.358925Z "Snoeijer, J. H." https://www.zbmath.org/authors/?q=ai:snoeijer.jacco-h "Pandey, A." https://www.zbmath.org/authors/?q=ai:pandey.anupam "Herrada, M. A." https://www.zbmath.org/authors/?q=ai:herrada.miguel-angel "Eggers, J." https://www.zbmath.org/authors/?q=ai:eggers.jens-g Summary: Soft materials that are subjected to large deformations exhibit an extremely rich phenomenology, with properties lying in between those of simple fluids and those of elastic solids. In the continuum description of these systems, one typically follows either the route of solid mechanics (Lagrangian description) or the route of fluid mechanics (Eulerian description). The purpose of this review is to highlight the relationship between the theories of viscoelasticity and of elasticity, and to leverage this connection in contemporary soft matter problems. We review the principles governing models for viscoelastic liquids, for example solutions of flexible polymers. Such materials are characterized by a relaxation time $$\lambda$$, over which stresses relax. We recall the kinematics and elastic response of large deformations, and show which polymer models do (and which do not) correspond to a nonlinear elastic solid in the limit $$\lambda \rightarrow \infty$$. With this insight, we split the work done by elastic stresses into reversible and dissipative parts, and establish the general form of the conservation law for the total energy. The elastic correspondence can offer an insightful tool for a broad class of problems; as an illustration, we show how the presence or absence of an elastic limit determines the fate of an elastic thread during capillary instability. Capillary imbibition of non-Newtonian fluids in a microfluidic channel: analysis and experiments https://www.zbmath.org/1472.76001 2021-11-25T18:46:10.358925Z "Gorthi, Srinivas R." https://www.zbmath.org/authors/?q=ai:gorthi.srinivas-r "Meher, Sanjaya Kumar" https://www.zbmath.org/authors/?q=ai:meher.sanjaya-kumar "Biswas, Gautam" https://www.zbmath.org/authors/?q=ai:biswas.gautam "Mondal, Pranab Kumar" https://www.zbmath.org/authors/?q=ai:mondal.pranab-kumar Summary: We have presented an experimental analysis on the investigations of capillary filling dynamics of inelastic non-Newtonian fluids in the regime of surface tension dominated flows. We use the Ostwald-de Waele power-law model to describe the rheology of the non-Newtonian fluids. Our analysis primarily focuses on the experimental observations and revisits the theoretical understanding of the capillary dynamics from the perspective of filling kinematics at the interfacial scale. Notably, theoretical predictions of the filling length into the capillary largely endorse our experimental results. We study the effects of the shear-thinning nature of the fluid on the underlying filling phenomenon in the capillary-driven regime through a quantitative analysis. We further show that the dynamics of contact line motion in this regime plays an essential role in advancing the fluid front in the capillary. Our experimental results on the filling in a horizontal capillary re-establish the applicability of the Washburn analysis in predicting the filling characteristics of non-Newtonian fluids in a vertical capillary during early stage of filling [\textit{R. M. Digilov}, Capillary rise of a non-Newtonian power law liquid: impact of the fluid rheology and dynamic contact angle'', Langmuir 24, 13663--13667 (2008; \url{doi:10.1021/la801807j})]. Finally, through a scaling analysis, we suggest that the late stage of filling by the shear-thinning fluids closely follows the variation $$x \sim \sqrt{t}$$. Such a regime can be called the modified Washburn regime [\textit{E. W. Washburn}, The dynamics of capillary flow'', Phys. Rev.17, No. 3, 273--283 (1921; \url{doi:10.1103/PhysRev.17.273})]. Corrigendum to: Heat and mass mixed convection for MHD viscoelastic fluid past a stretching sheet with ohmic dissipation'' https://www.zbmath.org/1472.76002 2021-11-25T18:46:10.358925Z "Hsiao, Kai-Long" https://www.zbmath.org/authors/?q=ai:hsiao.kai-long Corrigendum to the author's paper [ibid. 15, No. 7, 1803--1812 (2010; Zbl 1222.76013)]. Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations https://www.zbmath.org/1472.76003 2021-11-25T18:46:10.358925Z "Wang, Yong" https://www.zbmath.org/authors/?q=ai:wang.yong.6 "Wu, Wenpei" https://www.zbmath.org/authors/?q=ai:wu.wenpei The paper in question deals with compressible viscoelastic electrical conducting fluids. Their flows are governed by the elastic Navier-Stokes-Poisson system \begin{align*} \partial_t\rho + \nabla\cdot(\rho u) & = 0,\\ \partial_t(\rho u) + \nabla\cdot (\rho u \otimes u) + \nabla P(\rho) & = \mu\Delta u + (\mu + \lambda)\nabla\nabla\cdot u + c^2 \nabla\cdot(\rho \mathbb F\mathbb F^T) + \rho\nabla\Phi,\\ \partial_t\mathbb F + u \cdot \nabla \mathbb F & = \nabla u \mathbb F,\\ \Delta \Phi & = \rho - \overline\rho, \end{align*} which holds true on a certain bounded domain $$\Omega\subset \mathbb R^3$$. Here the unknowns are the density $$\rho$$, the velocity $$u,$$ the deformation gradient $$\mathbb F,$$ and the electrostatic potential $$\Phi$$. The system is endowed with an initial condition $(\rho, u, \mathbb F)(x,0) = (\rho_0,u_0,\mathbb F_0)(x),$ with the homogeneous Dirichlet bouundary condition for $$u$$ and with either $\Phi|_{\partial\Omega} = 0\qquad \mbox{or }\quad \nabla\Phi\cdot \nu|_{\partial\Omega} = 0.$ The main theorem of this paper provides the global-in-time existence of a unique global solution. This result is obtained under additional assumption on the smallness of initial data $$(\rho_0 - \overline\rho,u_0,\mathbb F_0)$$. In order to obtain the necessary estimates allowing multiple usage of the local-in-time existence theorem, the authors work with the deformation $$\varphi$$ rather than with the deformation gradient $$\mathbb F$$. Here the deformation is defined as $$\varphi:= X(x,t) - x$$, where $$X$$ is the Lagrangian coordinate, i.e., an inverse to a function $$x(X,t)$$ defined by the following ordinary differential equation \begin{align*} \frac{\mathrm{d}x(X,t)}{\mathrm{d}t} & = u(x(X,t),t),\\ x(X,0) & = X. \end{align*} A pseudo-anelastic model for stress softening in liquid crystal elastomers https://www.zbmath.org/1472.76004 2021-11-25T18:46:10.358925Z "Mihai, L. Angela" https://www.zbmath.org/authors/?q=ai:mihai.l-angela "Goriely, Alain" https://www.zbmath.org/authors/?q=ai:goriely.alain Summary: Liquid crystal elastomers exhibit stress softening with residual strain under cyclic loads. Here, we model this phenomenon by generalizing the classical pseudo-elastic formulation of the Mullins effect in rubber. Specifically, we modify the neoclassical strain-energy density of liquid crystal elastomers, depending on the deformation and the nematic director, by incorporating two continuous variables that account for stress softening and the associated set strain. As the material behaviour is governed by different forms of the strain-energy density on loading and unloading, the model is referred to as pseudo-anelastic. We then analyse qualitatively the mechanical responses of the material under cyclic uniaxial tension, which is easier to reproduce in practice, and further specialize the model in order to calibrate its parameters to recent experimental data at different temperatures. The excellent agreement between the numerical and experimental results confirms the suitability of our approach. Since the pseudo-energy function is controlled by the strain-energy density for the primary deformation, it is valid also for materials under multiaxial loads. Our study is relevant to mechanical damping applications and serves as a motivation for further experimental tests. Analysis and computations of a non-local thin-film model for two-fluid shear driven flows https://www.zbmath.org/1472.76005 2021-11-25T18:46:10.358925Z "Papageorgiou, D. T." https://www.zbmath.org/authors/?q=ai:papageorgiou.demetrios-t "Tanveer, S." https://www.zbmath.org/authors/?q=ai:tanveer.shakera|tanveer.saleh Summary: This paper is concerned with analysis and computations of a non-local thin-film model developed in Kalogirou \& Papageorgiou (\textit{J. Fluid Mech.}802, 5-36, 2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points are identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a \textit{quasi-solution} analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number $$R$$ and the primary wavelength $$2 \pi \nu^{-1/2}$$ of the disturbance, is determined analytically. A geometric diffuse-interface method for droplet spreading https://www.zbmath.org/1472.76006 2021-11-25T18:46:10.358925Z "Holm, Darryl D." https://www.zbmath.org/authors/?q=ai:holm.darryl-d "Náraigh, Lennon Ó." https://www.zbmath.org/authors/?q=ai:naraigh.lennon-o "Tronci, Cesare" https://www.zbmath.org/authors/?q=ai:tronci.cesare Summary: This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation valid in the case of large-scale droplet spreading---the geometric diffuse-interface method. The method possesses some advantages when compared with the existing models of droplet spreading, namely the slip model, the precursor-film method and the diffuse-interface model. These advantages are discussed and a case is made for using the geometric diffuse-interface method for the purpose of numerical simulations. The mathematical solutions of the geometric diffuse interface method are explored via such numerical simulations for the simple and well-studied case of large-scale droplet spreading for a perfectly wetting fluid---we demonstrate that the new method reproduces Tanner's Law of droplet spreading via a simple and robust computational method, at a low computational cost. We discuss potential avenues for extending the method beyond the simple case of perfectly wetting fluids. Corrigendum to: Hydrodynamic-driven morphogenesis of karst draperies: spatio-temporal analysis of the two-dimensional impulse response'' https://www.zbmath.org/1472.76007 2021-11-25T18:46:10.358925Z "Ledda, P. G." https://www.zbmath.org/authors/?q=ai:ledda.pier-giuseppe "Balestra, G." https://www.zbmath.org/authors/?q=ai:balestra.gioele "Lerisson, G." https://www.zbmath.org/authors/?q=ai:lerisson.gaetan "Scheid, B." https://www.zbmath.org/authors/?q=ai:scheid.benoit "Wyart, M." https://www.zbmath.org/authors/?q=ai:wyart.matthieu "Gallaire, F." https://www.zbmath.org/authors/?q=ai:gallaire.francois From the text: We found an error in (2.4) of our paper [ibid. 910, Paper No. A53, 33 p. (2021; Zbl 1461.76044)]. The correct non-dimensional expression for the curvature of the free surface is $\kappa=\boldsymbol{\nabla}\cdot\left(\frac{\nabla(h+h^0)}{\sqrt{1+\left(\frac{h_N}{l^\ast_c}\right)^2|\nabla(h+h^0)|^2}}\right).$ While this error does not bear any consequence in the linear analysis at the core of this paper, the value $$h_N/l^\ast_c=1$$ has to be specified in figures 15, 16, 17, 21, without altering the discussion. On the dynamics of thin layers of viscous flows inside another viscous fluid https://www.zbmath.org/1472.76008 2021-11-25T18:46:10.358925Z "Pernas Castaño, Tania" https://www.zbmath.org/authors/?q=ai:pernas-castano.tania "Velázquez, Juan J. L." https://www.zbmath.org/authors/?q=ai:velazquez.juan-j-l Summary: In this work we will study the dynamics of a thin layer of a viscous fluid which is embedded in the interior of another viscous fluid. The resulting flow can be approximated by means of the solutions of a free boundary problem for the Stokes equation in which one of the unknowns is the shape of a curve which approximates the geometry of the thin layer of fluid. We also derive the equation yielding the thickness of this fluid. This model, that will be termed as the \textit{Geometric Free Boundary Problem}, will be derived using matched asymptotic expansions. We will prove that the Geometric Free Boundary Problem is well posed and the solutions of the thickness equation are well defined (in particular they do not yield breaking of fluid layers) as long as the solutions of the Geometric Free Boundary Problem exist. Optimization of consistent two-equation models for thin film flows https://www.zbmath.org/1472.76009 2021-11-25T18:46:10.358925Z "Richard, G. L." https://www.zbmath.org/authors/?q=ai:richard.gael-loic "Gisclon, M." https://www.zbmath.org/authors/?q=ai:gisclon.marguerite "Ruyer-Quil, C." https://www.zbmath.org/authors/?q=ai:ruyer-quil.christian "Vila, J. P." https://www.zbmath.org/authors/?q=ai:vila.jean-paul Summary: A general study of consistent two-equation models for thin film flows is presented. In all models derived by the energy integral method or by an equivalent method, the energy of the system, apart from the kinetic energy of the mean flow, depends on the mean velocity. We show that in this case the model does not satisfy the principle of Galilean invariance. All consistent models derived by the momentum integral method are Galilean invariant but they admit an energy equation and a capillary energy only if the Galilean-invariant part of the first-order momentum flux does not depend on the mean velocity. We show that, both for theoretical and numerical reasons, two-equations models should be derived by a momentum integral method admitting an energy equation leading to the structure of the equations of fluids endowed with internal capillarity. Among all models fulfilling these conditions, those having the best properties are selected. The nonlinear properties are tested from the speed of solitary waves at the high Reynolds number limit while the linear properties are studied from the neutral stability curves and from the celerity of the kinematic waves along these curves. The latter criterion gives the best consistent way to write the second-order diffusive terms of the model. Optimized consistent two-equation models are then proposed and numerical results are compared to numerical and experimental results of the literature. Swelling and shrinking in prestressed polymer gels: an incremental stress-diffusion analysis https://www.zbmath.org/1472.76010 2021-11-25T18:46:10.358925Z "Rossi, Marco" https://www.zbmath.org/authors/?q=ai:rossi.marco|rossi.marco.1 "Nardinocchi, Paola" https://www.zbmath.org/authors/?q=ai:nardinocchi.paola "Wallmersperger, Thomas" https://www.zbmath.org/authors/?q=ai:wallmersperger.thomas Summary: Polymer gels are porous fluid-saturated materials which can swell or shrink triggered by various stimuli. The swelling/shrinking-induced deformation can generate large stresses which may lead to the failure of the material. In the present research, a nonlinear stress-diffusion model is employed to investigate the stress and the deformation state arising in hydrated constrained polymer gels when subject to a varying chemical potential. Two different constraint configurations are taken into account: (i) elastic constraint along the thickness direction and (ii) plane elastic constraint. The first step entirely defines a compressed/tensed configuration. From there, an incremental chemo-mechanical analysis is presented. The derived model extends the classical linear poroelastic theory with respect to a prestressed configuration. Finally, the comparison between the analytical results obtained by the proposed model and a particular problem already discussed in literature for a stress-free gel membrane (one-dimensional test case) will highlight the relevance of the derived model. On the problem of resonant incompressible flow in ventilated double glazing https://www.zbmath.org/1472.76042 2021-11-25T18:46:10.358925Z "Akinaga, T." https://www.zbmath.org/authors/?q=ai:akinaga.takeshi "Harvey-Ball, T. M." https://www.zbmath.org/authors/?q=ai:harvey-ball.t-m "Itano, T." https://www.zbmath.org/authors/?q=ai:itano.tomoaki "Generalis, S. C." https://www.zbmath.org/authors/?q=ai:generalis.s-c "Aifantis, E. C." https://www.zbmath.org/authors/?q=ai:aifantis.elias-c Summary: We employ a homotopy method, rather than conventional stability theory, in order to resolve the degeneracy due to resonance, which exists in fluid motion associated with a channel of infinite extent in ventilated double glazing. The introduction of a symmetry breaking perturbation, in the form of a Poiseuille flow component, alters substantially the resonant bifurcation tree of the original flow. Previously unknown resonant higher order nonlinear solutions, i.e. after the removal of the perturbative Poiseuille flow component, are discovered. A possible extension of the methodology to consider non-Newtonian gradient enhanced incompressible viscous fluids is also briefly discussed. Homogenization of a micropolar fluid past a porous media with nonzero spin boundary condition https://www.zbmath.org/1472.76070 2021-11-25T18:46:10.358925Z "Suárez-Grau, Francisco J." https://www.zbmath.org/authors/?q=ai:suarez-grau.francisco-javier After studying the well-posedness of the microscopic problem -- a micropolar fluid past a perforated domain with non vanishing spin boundary conditions, the author uses the periodic unfolding technique to pass to the periodic homogenization limit. The weak formulation of the upscaled limit is provided. Free and circular jets cooled by single phase nanofluids https://www.zbmath.org/1472.76113 2021-11-25T18:46:10.358925Z "Turkyilmazoglu, Mustafa" https://www.zbmath.org/authors/?q=ai:turkyilmazoglu.mustafa Summary: Nanofluids are widely known to enhance the heat transfer rate resulting in a cooled system. In the present paper, we show mathematically that the nanofluids indeed cool the system as the nanoparticles volume fraction is increased. The key role is explained for a two-dimensional laminar free nanofluid jet and for a circular axisymmetric free nanofluid jet issuing into the same nanofluid medium. Exact nanofluid flow results are obtained and, integral flux relations of momentum and thermal layers concerning five most studied nanofluids, respectively Ag, Cu, CuO, Al$$_{2}$$O$$_{3}$$ and TiO$$_{2}$$, are derived. A shape factor is defined controlling the momentum layer thickness. By means of another shape factor representing the thermal layer thickness, the relevant energy equation enables one to identify the regimes of nanoparticle size leading to a coolant jet, without a need to solve the energy equation fully. Two recently popular nanofluid models, resulting in the same conclusion, are examined on the considered free nanofluid jets. Additionally, an exact temperature field associated with the laminar two-dimensional free jet of nanoparticles is obtained offering explicit support to the current approach. Entropy analysis on unsteady MHD flow of Casson nanofluid over a stretching vertical plate with thermal radiation effect https://www.zbmath.org/1472.76127 2021-11-25T18:46:10.358925Z "Shit, G. C." https://www.zbmath.org/authors/?q=ai:shit.g-c "Mandal, S." https://www.zbmath.org/authors/?q=ai:mandal.samir-ch|mandal.soham|mandal.susobhan|mandal.sayanta|mandal.swarnendu|mandal.sudin|mandal.subhendu-bikash|mandal.subhrangsu|mandal.subhayan|mandal.sikta|mandal.sonia|mandal.sanjoy|mandal.swapan|mandal.satyanarayan|mandal.s-p|mandal.satya|mandal.sayan|mandal.shubhadeep|mandal.santosh-kumar|mandal.sudhindu-bikash|mandal.subhro-jyoti|mandal.siddhartha|mandal.sudhansu-s|mandal.sandip|mandal.sayantan|mandal.soumyajit|mandal.samiran|mandal.seikh-hannan|mandal.swagata|mandal.shyamapada|mandal.sekhar|mandal.soma|mandal.sujit|mandal.supriya|mandal.sanjib-kumar|mandal.shyamal-kumar-das|mandal.saumendranath|mandal.somnath|mandal.soumik|mandal.saptarshi|mandal.sanjay-kumar|mandal.sajib|mandal.saumen|mandal.saroj|mandal.swarup|mandal.shobhan|mandal.sudipto|mandal.sourav-k The authors aim at entropy analysis of unsteady MHD flow of Casson nanofluid over a stretching vertical plate with thermal radiation effect using a fourth-order Runge-Kutta-Fehlberg method along with the shooting method. The flow behavior of non-Newtonian Casson nanofluid, heat transfer characteristics and nanoparticle concentration profiles are obtained to understand the impact of the different dimensionless parameters. Furthermore, the expressions for the dimensionless wall shear stress, heat and mass transfer rate over a heated vertical plate are derived. The results are in good agreement with existing results. Magnetohydrodynamics boundary layer flow of micropolar fluid over an exponentially shrinking sheet with thermal radiation: triple solutions and stability analysis https://www.zbmath.org/1472.76128 2021-11-25T18:46:10.358925Z "Yahaya, Rusya Iryanti" https://www.zbmath.org/authors/?q=ai:yahaya.rusya-iryanti "Arifin, Norihan Md" https://www.zbmath.org/authors/?q=ai:arifin.norihan-md "Isa, Siti Suzilliana Putri Mohamed" https://www.zbmath.org/authors/?q=ai:isa.siti-suzilliana-putri-mohamed "Rashidi, Mohammad Mehdi" https://www.zbmath.org/authors/?q=ai:rashidi.mohammad-mehdi Summary: The flow of electrically conducting micropolar fluid past an exponentially permeable shrinking sheet in the presence of a magnetic field and thermal radiation is studied. Similarity transformations are applied to the governing partial differential equations to form ordinary differential equations. The solution for the resultant equations, subject to boundary conditions, is then computed numerically using the bvp4c solver in MATLAB. The effects of several parameters on the local skin friction coefficient, couple stress, Nusselt number, velocity, microrotation and temperature of the fluid are analysed. Because the numerical computations for this problem result in triple solutions, stability analysis is carried out to ascertain the stability and significance of these solutions. The first solution is revealed to be stable, hence more physically meaningful than the other solutions. Meanwhile, it is found that the increase in magnetic and thermal radiation parameters reduces the fluid temperature. Liquid crystals on deformable surfaces https://www.zbmath.org/1472.82040 2021-11-25T18:46:10.358925Z "Nitschke, Ingo" https://www.zbmath.org/authors/?q=ai:nitschke.ingo "Reuther, Sebastian" https://www.zbmath.org/authors/?q=ai:reuther.sebastian "Voigt, Axel" https://www.zbmath.org/authors/?q=ai:voigt.axel Summary: Liquid crystals with molecules constrained to the tangent bundle of a curved surface show interesting phenomena resulting from the tight coupling of the elastic and bulk-free energies of the liquid crystal with geometric properties of the surface. We derive a thermodynamically consistent Landau-de Gennes-Helfrich model which considers the simultaneous relaxation of the $$Q$$-tensor field and the surface. The resulting system of tensor-valued surface partial differential equation and geometric evolution laws is numerically solved to tackle the rich dynamics of this system and to compute the resulting equilibrium shape. The results strongly depend on the intrinsic and extrinsic curvature contributions and lead to unexpected asymmetric shapes. Tear film dynamics with blinking and contact lens motion https://www.zbmath.org/1472.92091 2021-11-25T18:46:10.358925Z "Anderson, Daniel M." https://www.zbmath.org/authors/?q=ai:anderson.daniel-m "Corsaro, Maria" https://www.zbmath.org/authors/?q=ai:corsaro.maria "Horton, Jonathan" https://www.zbmath.org/authors/?q=ai:horton.jonathan "Reid, Tim" https://www.zbmath.org/authors/?q=ai:reid.tim "Seshaiyer, Padmanabhan" https://www.zbmath.org/authors/?q=ai:seshaiyer.padmanabhan Summary: We develop a lubrication theory-based mathematical model that describes the dynamics of a tear film during blinking and contact lens (CL) wear. The model extends previous work on pre-corneal tear film dynamics during blinking by coupling the partial differential equation for tear film thickness to a dynamic model for CL motion. We explore different models for eyelid motion and also account for possible voluntary and involuntary globe (eyeball) rotation that may accompany blinking. Boundary conditions for mass flux at the eyelids are also adapted to account for the presence and motion of the CL. Our predictions for CL motion compare reasonably with existing data. Away from the eyelids the pre-lens tear film (PrLTF) is shifted, relative to its pre-corneal counterpart, in the direction of CL motion. Near the eyelids, the inflow/outflow of fluid under the eyelids also influences the PrLTF profile. We also compare our PrLTF dynamics to existing \textit{in vivo} tear film thickness measurements. Models of bacteria swimming in a nematic liquid crystal https://www.zbmath.org/1472.92142 2021-11-25T18:46:10.358925Z "Duan, Mochong" https://www.zbmath.org/authors/?q=ai:duan.mochong "Walkington, Noel J." https://www.zbmath.org/authors/?q=ai:walkington.noel-j Summary: Models of dilute systems of bacteria swimming in a nematic liquid crystal are developed and analyzed. The motion and orientation of the bacteria are simulated using ordinary differential equations coupled with the partial differential equations modeling the nematic liquid crystal (Ericksen Leslie equations). The analysis and numerical simulations of this system are shown to predict interesting phenomena observed experimentally.