A framework for synthetic validation of 3D echocardiographic particle image velocimetry

Particle image velocimetry (PIV) has been significantly advanced since its conception in early 1990s. With the advancement of imaging modalities, applications of 2D PIV have far expanded into biology and medicine. One example is echocardiographic particle image velocimetry that is used for in vivo mapping of the flow inside the heart chambers with opaque boundaries. Velocimetry methods can help better understanding the biomechanical problems. The current trend is to develop three-dimensional velocimetry techniques that take advantage of modern medical imaging tools. This study provides a novel framework for validation of velocimetry methods that are inherently three dimensional such as but not limited to those acquired by 3D echocardiography machines. This framework creates 3D synthetic fields based on a known 3D velocity field \({\mathbf{V}}\) and a given 3D brightness field \({\mathbf{B}}\). The method begins with computing the inverse flow \({\mathbf{V}}^{\varvec{*}} \) based on the velocity field \({\mathbf{V}}\). Then the transformation of \({\mathbf{B}}\), imposed by \({\mathbf{V}}\), is calculated using the computed inverse flow according to \({\mathbf{B}}^{\varvec{*}} \left( {\mathbf{x}} \right) = {\mathbf{B}}\left( {{\mathbf{x}} + {\mathbf{V}}^{\varvec{*}} \left( {\mathbf{x}} \right)} \right)\), where x is the coordinates of voxels in \({\mathbf{B}}^{\varvec{*}} \), with a 3D weighted average interpolation, which provides high accuracy, low memory requirement, and low computational time. To check the validity of the framework, we generate pairs of 3D brightness fields by employing Hill’s spherical vortex velocity field. \({\mathbf{B}}\) and the generated \({\mathbf{B}}^{\varvec{*}} \) are then processed by our in-house 3D particle image velocimetry software to obtain the interrelated velocity field. The results indicates that the computed and imposed velocity fields are in agreement.     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727–790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit \({\mathcal {O}}_{p}\) with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \). We prove that \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) is a three-dimensional \(C^{1}\)-submanifold of the phase space and admits a smooth global graph representation. Within \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \), there exist heteroclinic connections from \({\mathcal {O}}_{p}\) to three different periodic orbits. These connecting sets are two-dimensional \(C^{1}\)-submanifolds of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) and homeomorphic to the two-dimensional open annulus. They form \(C^{1}\)-smooth separatrices in the sense that they divide the points of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) into three subsets according to their \(\omega \)-limit sets.     

Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size \({\varepsilon}\) separated by distances \({d_{\varepsilon}}\) and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of \({\frac{d_{\varepsilon}}\varepsilon}\) when \({\varepsilon}\) goes to zero. If \({\frac{d_{\varepsilon}}\varepsilon \to \infty}\), then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, \({\frac{d_{\varepsilon}}\varepsilon \to 0}\), then the fluid cannot penetrate the porous region, namely, the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of \({\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}}\) where \({\gamma \in (0,\infty]}\) is related to the geometry of the lateral boundaries of the obstacles. If \({\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty}\), then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions; in particular, it is equal to \({\varepsilon^{3}}\) for balls.     

Further Dense Properties of the Space of Circle Diffeomorphisms with a Liouville Rotation Number

In continuation of Matsumoto’s paper (Nonlinearity 25:1495–1511, 2012) we show that various subspaces are \(C^{\infty }\)-dense in the space of orientation-preserving \(C^{\infty }\)-diffeomorphisms of the circle with rotation number \(\alpha \), where \(\alpha \in {\mathbb {S}}^1\) is any prescribed Liouville number. In particular, for every odometer \({\mathcal {O}}\) of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to \({\mathcal {O}}\).     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\):

$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$

where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.

     

Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients

We study the asymptotic behaviour of the resolvents \({(\mathcal{A}^\varepsilon+I)^{-1}}\) of elliptic second-order differential operators \({{\mathcal{A}}^\varepsilon}\) in \({\mathbb{R}^d}\) with periodic rapidly oscillating coefficients, as the period \({\varepsilon}\) goes to zero. The class of operators covered by our analysis includes both the “classical” case of uniformly elliptic families (where the ellipticity constant does not depend on \({\varepsilon}\)) and the “double-porosity” case of coefficients that take contrasting values of order one and of order \({\varepsilon^2}\) in different parts of the period cell. We provide a construction for the leading order term of the “operator asymptotics” of \({(\mathcal{A}^\varepsilon+I)^{-1}}\) in the sense of operator-norm convergence and prove order \({O(\varepsilon)}\) remainder estimates.     

Uniaxial experimental study of the acoustic emission and deformation behavior of composite rock based on 3D digital image correlation (DIC)

In this paper, uniaxial compression tests were carried out on a series of composite rock specimens with different dip angles, which were made from two types of rock-like material with different strength. The acoustic emission technique was used to monitor the acoustic signal characteristics of composite rock specimens during the entire loading process. At the same time, an optical non-contact 3 D digital image correlation technique was used to study the evolution of axial strain field and the maximal strain field before and after the peak strength at different stress levels during the loading process. The effect of bedding plane inclination on the deformation and strength during uniaxial loading was analyzed. The methods of solving the elastic constants of hard and weak rock were described. The damage evolution process, deformation and failure mechanism, and failure mode during uniaxial loading were fully determined. The experimental results show that the θ = 0?–45?specimens had obvious plastic deformation during loading, and the brittleness of the θ = 60?–90?specimens gradually increased during the loading process. When the anisotropic angle θincreased from 0?to 90?, the peak strength, peak strain,and apparent elastic modulus all decreased initially and then increased. The failure mode of the composite rock specimen during uniaxial loading can be divided into three categories:tensile fracture across the discontinuities(θ = 0?–30?), slid-ing failure along the discontinuities(θ = 45?–75?), and tensile-split along the discontinuities(θ = 90?). The axial strain of the weak and hard rock layers in the composite rock specimen during the loading process was significantly different from that of the θ = 0?–45?specimens and was almost the same as that of the θ = 60?–90?specimens. As for the strain localization highlighted in the maximum principal strain field, the θ = 0?–30?specimens appeared in the rock matrix approximately parallel to the loading direction,while in the θ = 45?–90?specimens it appeared at the hard and weak rock layer interface.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.     

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Representation of a Smooth Isometric Deformation of a Planar Material Region into a Curved Surface

We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset \({\mathcal{D}}\) of two-dimensional Euclidean point space \(\mathbb{E}^{2}\) into a surface \({\mathcal{S}}\) in three-dimensional Euclidean point space \(\mathbb{E}^{3}\). To be isometric, such a deformation must preserve the length of every possible arc of material points on \({\mathcal{D}}\). Characterizing the curves of zero principal curvature of \({\mathcal{S}}\) is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in \(\mathbb{E}^{2}\), based upon an à priori chosen pre-image form of the curves of zero principal curvature in \({\mathcal{D}}\), and use that coordinate system to construct the most general isometric deformation of \({\mathcal{D}}\) to a smooth surface \({\mathcal{S}}\). A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of \({\mathcal{S}}\) are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).     

Non-classical continuum theory for fluids incorporating internal and Cosserat rotation rates

This paper presents a non-classical continuum theory for fluent continua in which the conservation and balance laws are derived by incorporating both internal rotation rates arising from the velocity gradient tensor and the rotation rates of the Cosserats. Specifically, in this non-classical continuum theory we have (1) the usual velocities (\(\bar{ \pmb {\varvec{v }}}\)), (2) the three internal rotation rates (\({}_i^t\bar{ \pmb {\varvec{\Theta }}}\)) about the axes of a fixed triad whose axes are parallel to the x-frame arising from the velocity gradient tensor \((\bar{ \pmb {\varvec{L }}})\) that are completely defined by the antisymmetric part of the velocity gradient tensor, and (3) three additional rotation rates (\({}_e^t\bar{ \pmb {\varvec{\Theta }}}\)) about the axes of the same triad located at each material point as additional three unknown degrees of freedom, referred to as Cosserat rotation rates. This gives rise to \(\bar{ \pmb {\varvec{v }}}\) and \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) as six degrees of freedom at a material point. The internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\), often neglected in classical fluid mechanics, exist in all deforming fluent continua as these are due to velocity gradient tensor. When the internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) are resisted by deforming fluent continua, conjugate moment tensor arises that together with \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) may result in energy storage and/or dissipation, which must be considered in the conservation and balance laws. The Cosserat rotation rations \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) also result in conjugate moment tensor that together with \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) may also result in energy storage and/or dissipation. The main focus of this paper is a consistent derivation of conservation and balance laws for fluent continua that incorporate the aforementioned physics and associated constitutive theories for thermofluids using the conditions resulting from the entropy inequality. The material coefficients derived in the constitutive theories are clearly defined and discussed.