A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries

In this paper we investigate a free boundary problem for the classical Lotka–Volterra type predator–prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as \(t\rightarrow \infty \) at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.     

Persistence of Exponential Trichotomy for Linear Operators: A Lyapunov–Perron Approach

In this article we revisit the perturbation of exponential trichotomy of linear difference equation in Banach space by using a Perron–Lyapunov fixed point formulation for the perturbed evolution operator. This approach allows us to directly re-construct the perturbed semiflow without using shift spectrum arguments. These arguments are presented to the case of linear autonomous discrete time dynamical system. This result is then coupled to Howland semigroup procedure to obtain the persistence of exponential trichotomy for non-autonomous difference equations.     

Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.     

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      6. A spectral solver for the Navier–Stokes equations in the velocity–vorticity formulation      M.F.M. Speetjens  H.J.H. Clercx 《International Journal of Computational Fluid Dynamics》2013,27(3):191-209 A numerical algorithm intended for the study of flows in a cylindrical container under laminar flow conditions is proposed. High resolution of the flow field, governed by the Navier–Stokes equations in velocity–vorticity formulation relative to a cylindrical frame of reference, is achieved through spatial discretisation by means of the spectral method. This method is based on a Fourier expansion in the azimuthal direction and an expansion in Chebyshev polynomials in the (nonperiodic) radial and axial directions. Several regularity constraints are used to take care of the coordinate singularity. These constraints are implemented, together with the boundary conditions at the top, bottom and mantle of the cylinder, via the tau method. The a priori unknown boundary values of the vorticity are evaluated by means of the influence-matrix technique. The compatibility between the mathematical and numerical formulation of the Navier–Stokes equations is established through a tau-correction procedure. The resolved flow field exhibits high-precision satisfaction of the incompressibility constraints for velocity and vorticity and the definition of the vorticity. The performance of the solver is illustrated by resolution of several configurations representative of generic three-dimensional laminar flows.      7. Free Boundary Problems for a Lotka–Volterra Competition System      Mingxin Wang  Jingfu Zhao 《Journal of Dynamics and Differential Equations》2014,26(3):655-672 In this paper we investigate two free boundary problems for a Lotka–Volterra type competition model in one space dimension. The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the right-half-space as time \(t\) goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of the solutions and criteria for spreading and vanishing are also obtained. This paper is an improvement and extension of Guo and Wu (J Dyn Differ Equ 24:873–895, 2012).      8. A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions      Yonggang Zhao  Mingxin Wang 《Journal of Dynamics and Differential Equations》2018,30(2):743-777 We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.      9. Experimental-Theoretical Approach to Determining the Film–Substrate Adhesion      S. N. Yakupov 《Mechanics of Solids》2017,52(5):587-593 There are well-known methods and algorithms for determining the film adhesion to a plane substrate, which have their own advantages and drawbacks and the applicability ranges. The substrates can have a good initial shape (spherical, cylindrical, etc,) covered, for example, by a thin film layer. But there are practically no studies of the film adhesion to a substrate with a nonplane surface. A two-dimensional approach to determining the film adhesion to a plane or a nonplane substate is developed, which permits increasing the accuracy of determining the adhesion and decreasing the scattering of the obtained results. An example is considered.      10. A New Optimal Growth Constant for the Hele–Shaw Instability      Pasa  Gelu 《Transport in Porous Media》2002,49(1):27-40 We study the immiscible displacement of the oil from a homogeneous porous medium by using a less viscous fluid (water). We use the Hele–Shaw model, then a sharp interface exists between the fluids. The fingering phenomenon appears, first studied by Saffman and Taylor (1958). Gorell and Homsy (1983) consider an intermediate region (I. R.) between water and oil, containing a polymer mixture. The unknown viscosity in I. R. is a parameter which can improve the stability of the I. R.–oil interface. A numerical optimal viscosity profile in I. R. is given. Carasso and Paa (1998) obtain an explicit formula for an optimal viscosity profile in I. R. An upper estimation of the growth constant is given. In this paper, a very slow viscosity profile in I. R. is defined and an optimal formula for the growth constant is obtained, less than the previous estimation of Carasso and Paa. Moreover, this formula is similar with the Saffman–Taylor result, only the water viscosity is replaced by the limit value of viscosity in I. R. on the interface with the oil. We explain the apparent contradiction between the previous results of Gorell and Homsy (1983) and Paa and Polisevski (1992).      11. Sharp Trace Hardy–Sobolev-Maz’ya Inequalities and the Fractional Laplacian      Stathis Filippas  Luisa Moschini  Achilles Tertikas 《Archive for Rational Mechanics and Analysis》2013,208(1):109-161 In this work we establish trace Hardy and trace Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space, our results cover the full range of the exponent ${s \in}$ (0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L 2 setting to an open problem raised by Frank and Seiringer (“Sharp fractional Hardy inequalities in half-spaces,” in Around the research of Vladimir Maz’ya. International Mathematical Series, 2010).      12. A space-time lower–upper symmetric Gauss–Seidel scheme for the time-spectral method      Lei Zhan  Juntao Xiong  Feng Liu 《International Journal of Computational Fluid Dynamics》2016,30(5):337-355 The time-spectral method (TSM) offers the advantage of increased order of accuracy compared to methods using finite-difference in time for periodic unsteady flow problems. Explicit Runge–Kutta pseudo-time marching and implicit schemes have been developed to solve iteratively the space-time coupled nonlinear equations resulting from TSM. Convergence of the explicit schemes is slow because of the stringent time-step limit. Many implicit methods have been developed for TSM. Their computational efficiency is, however, still limited in practice because of delayed implicit temporal coupling, multiple iterative loops, costly matrix operations, or lack of strong diagonal dominance of the implicit operator matrix. To overcome these shortcomings, an efficient space-time lower–upper symmetric Gauss–Seidel (ST-LU-SGS) implicit scheme with multigrid acceleration is presented. In this scheme, the implicit temporal coupling term is split as one additional dimension of space in the LU-SGS sweeps. To improve numerical stability for periodic flows with high frequency, a modification to the ST-LU-SGS scheme is proposed. Numerical results show that fast convergence is achieved using large or even infinite Courant–Friedrichs–Lewy (CFL) numbers for unsteady flow problems with moderately high frequency and with the use of moderately high numbers of time intervals. The ST-LU-SGS implicit scheme is also found to work well in calculating periodic flow problems where the frequency is not known a priori and needed to be determined by using a combined Fourier analysis and gradient-based search algorithm.      13. A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range      A. Cheskidov  R. Shvydkoy 《Journal of Mathematical Fluid Mechanics》2014,16(2):263-273 Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.      14. On the Inequalities of Babuška–Aziz,Friedrichs and Horgan–Payne      Martin Costabel  Monique Dauge 《Archive for Rational Mechanics and Analysis》2015,217(3):873-898       15. A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions      Dag Chun Standnes  Steinar Evje  Pål Østebø Andersen 《Transport in Porous Media》2017,119(3):707-738 A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.      16. A two-step back–forth algorithm for the solution of the unsteady adjoint equations      Dimitrios I. Papadimitriou 《International Journal of Computational Fluid Dynamics》2016,30(3):272-284 A new algorithm for the solution of the unsteady adjoint equations is proposed in this article, aiming at overcoming the excessive computational cost and memory requirements of the conventional adjoint approach for the optimisation of unsteady problems in computational mechanics. The total cost is equal to four times the cost of the unsteady state solution, which is twice the cost of the conventional backward-in-time adjoint calculation but the memory requirements are very small, equivalent to those of a steady-state problem, while stability is acceptable. The proposed algorithm is validated in the case of the 1D unsteady Burgers equation with non-smooth source terms.      17. On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach      Eleni Agiasofitou  Markus Lazar 《Journal of Elasticity》2010,99(2):163-178 A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.      18. A priori study for the modelling of velocity–interface correlations in the stratified air–water flows      《International Journal of Heat and Fluid Flow》2015 This work concerns the modelling of stratified two-phase turbulent flows with interfaces. We consider an equation for an intermittency function $\alpha (\mathbf{x}\text{,}t)$ which denotes the probability of finding an interface at a given time t and a given point $\mathbf{x}$. In Wacławczyk and Oberlack (2011) a model for the unclosed terms in this equation was proposed. Here, we investigate the performance of this model by a priori tests, and finally, based on the a priori data discuss its possible modification and improvements.      19. A pressure-based velocity decomposition procedure for the incompressible Navier–Stokes equations      E. M. Wahba 《Meccanica》2016,51(8):1675-1684       20. A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations      Stefano Bosia  Vittorino Pata  James C. Robinson 《Journal of Mathematical Fluid Mechanics》2014,16(4):721-725 We give simple proofs that a weak solution u of the Navier–Stokes equations with H 1 initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L s (0, T;L r,∞(Ω)) or if its L s,∞(0, T;L r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.

A spectral solver for the Navier–Stokes equations in the velocity–vorticity formulation

A numerical algorithm intended for the study of flows in a cylindrical container under laminar flow conditions is proposed. High resolution of the flow field, governed by the Navier–Stokes equations in velocity–vorticity formulation relative to a cylindrical frame of reference, is achieved through spatial discretisation by means of the spectral method. This method is based on a Fourier expansion in the azimuthal direction and an expansion in Chebyshev polynomials in the (nonperiodic) radial and axial directions. Several regularity constraints are used to take care of the coordinate singularity. These constraints are implemented, together with the boundary conditions at the top, bottom and mantle of the cylinder, via the tau method. The a priori unknown boundary values of the vorticity are evaluated by means of the influence-matrix technique. The compatibility between the mathematical and numerical formulation of the Navier–Stokes equations is established through a tau-correction procedure. The resolved flow field exhibits high-precision satisfaction of the incompressibility constraints for velocity and vorticity and the definition of the vorticity. The performance of the solver is illustrated by resolution of several configurations representative of generic three-dimensional laminar flows.     

Free Boundary Problems for a Lotka–Volterra Competition System

In this paper we investigate two free boundary problems for a Lotka–Volterra type competition model in one space dimension. The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the right-half-space as time \(t\) goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of the solutions and criteria for spreading and vanishing are also obtained. This paper is an improvement and extension of Guo and Wu (J Dyn Differ Equ 24:873–895, 2012).     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

Experimental-Theoretical Approach to Determining the Film–Substrate Adhesion

There are well-known methods and algorithms for determining the film adhesion to a plane substrate, which have their own advantages and drawbacks and the applicability ranges. The substrates can have a good initial shape (spherical, cylindrical, etc,) covered, for example, by a thin film layer. But there are practically no studies of the film adhesion to a substrate with a nonplane surface. A two-dimensional approach to determining the film adhesion to a plane or a nonplane substate is developed, which permits increasing the accuracy of determining the adhesion and decreasing the scattering of the obtained results. An example is considered.     

A New Optimal Growth Constant for the Hele–Shaw Instability

We study the immiscible displacement of the oil from a homogeneous porous medium by using a less viscous fluid (water). We use the Hele–Shaw model, then a sharp interface exists between the fluids. The fingering phenomenon appears, first studied by Saffman and Taylor (1958). Gorell and Homsy (1983) consider an intermediate region (I. R.) between water and oil, containing a polymer mixture. The unknown viscosity in I. R. is a parameter which can improve the stability of the I. R.–oil interface. A numerical optimal viscosity profile in I. R. is given. Carasso and Paa (1998) obtain an explicit formula for an optimal viscosity profile in I. R. An upper estimation of the growth constant is given. In this paper, a very slow viscosity profile in I. R. is defined and an optimal formula for the growth constant is obtained, less than the previous estimation of Carasso and Paa. Moreover, this formula is similar with the Saffman–Taylor result, only the water viscosity is replaced by the limit value of viscosity in I. R. on the interface with the oil. We explain the apparent contradiction between the previous results of Gorell and Homsy (1983) and Paa and Polisevski (1992).     

Sharp Trace Hardy–Sobolev-Maz’ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space, our results cover the full range of the exponent ${s \in}$ (0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L ² setting to an open problem raised by Frank and Seiringer (“Sharp fractional Hardy inequalities in half-spaces,” in Around the research of Vladimir Maz’ya. International Mathematical Series, 2010).     

A space-time lower–upper symmetric Gauss–Seidel scheme for the time-spectral method

The time-spectral method (TSM) offers the advantage of increased order of accuracy compared to methods using finite-difference in time for periodic unsteady flow problems. Explicit Runge–Kutta pseudo-time marching and implicit schemes have been developed to solve iteratively the space-time coupled nonlinear equations resulting from TSM. Convergence of the explicit schemes is slow because of the stringent time-step limit. Many implicit methods have been developed for TSM. Their computational efficiency is, however, still limited in practice because of delayed implicit temporal coupling, multiple iterative loops, costly matrix operations, or lack of strong diagonal dominance of the implicit operator matrix. To overcome these shortcomings, an efficient space-time lower–upper symmetric Gauss–Seidel (ST-LU-SGS) implicit scheme with multigrid acceleration is presented. In this scheme, the implicit temporal coupling term is split as one additional dimension of space in the LU-SGS sweeps. To improve numerical stability for periodic flows with high frequency, a modification to the ST-LU-SGS scheme is proposed. Numerical results show that fast convergence is achieved using large or even infinite Courant–Friedrichs–Lewy (CFL) numbers for unsteady flow problems with moderately high frequency and with the use of moderately high numbers of time intervals. The ST-LU-SGS implicit scheme is also found to work well in calculating periodic flow problems where the frequency is not known a priori and needed to be determined by using a combined Fourier analysis and gradient-based search algorithm.     

A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range

Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

A two-step back–forth algorithm for the solution of the unsteady adjoint equations

A new algorithm for the solution of the unsteady adjoint equations is proposed in this article, aiming at overcoming the excessive computational cost and memory requirements of the conventional adjoint approach for the optimisation of unsteady problems in computational mechanics. The total cost is equal to four times the cost of the unsteady state solution, which is twice the cost of the conventional backward-in-time adjoint calculation but the memory requirements are very small, equivalent to those of a steady-state problem, while stability is acceptable. The proposed algorithm is validated in the case of the 1D unsteady Burgers equation with non-smooth source terms.     

On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.     

A priori study for the modelling of velocity–interface correlations in the stratified air–water flows

This work concerns the modelling of stratified two-phase turbulent flows with interfaces. We consider an equation for an intermittency function $\alpha (\mathbf{x}\text{,}t)$ which denotes the probability of finding an interface at a given time t and a given point $\mathbf{x}$. In Wacławczyk and Oberlack (2011) a model for the unclosed terms in this equation was proposed. Here, we investigate the performance of this model by a priori tests, and finally, based on the a priori data discuss its possible modification and improvements.     

A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

We give simple proofs that a weak solution u of the Navier–Stokes equations with H ¹ initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L ^s (0, T;L ^r,∞(Ω)) or if its L ^s,∞(0, T;L ^r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.