Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l ³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is any portion of?ω withlength γ ₀>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ ₀, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ ₀) is contained in a generatrix ofS. We show that, if the applied body force density isO(? ²) with respect to?, the fieldu(?)=(u _i (?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH ¹(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts _?1 ¹ u dx ₃, which belongs to the spaceV _F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η _i ) ∈V _F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2?, clamped along their entire lateral face, all having the same middle surfaceS=φ(?ω) ?R ³, whereω ?R ² is a bounded and connected open set with a Lipschitz-continuous boundaryγ, andφ ∈l ³ ( $\overline \omega$ ;R ³). We make an essential geometrical assumption on the middle surfaceS, which is satisfied ifγ andφ are smooth enough andS is “uniformly elliptic”, in the sense that the two principal radii of curvature are either both>0 at all points ofS, or both<0 at all points ofS. We show that, if the applied body force density isO(1) with respect to?, the fieldtu(?)=(u _i(?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, one “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges inH ¹(Ω)×H ¹(Ω)×L ²(Ω) as?→0 to a limitu, which is independent of the transverse variable. Furthermore, the averageξ=1/2ε _?1 ¹ u dx ₃, which belongs to the space $$V_M (\omega ) = H_0^1 (\omega ) \times H_0^1 (\omega ) \times L^2 (\omega ),$$ satisfies the (scaled) two-dimensional equations of a “membrane shell” viz., $$\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta )\gamma _{\alpha \beta } (\eta ) \sqrt \alpha dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\}\eta _i \sqrt a dy$$ for allη=(η _i) εV _M(ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\gamma _{\alpha \beta } (\eta ) = \frac{1}{2}\left( {\partial _{\alpha \eta \beta } + \partial _{\beta \eta \alpha } } \right) - \Gamma _{\alpha \beta }^\sigma \eta _\sigma - b_{\alpha \beta \eta 3} $$ are the components of the linearized change of metric tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_{\alpha \beta }$ are the components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “membrane shell” are therefore justified.     

Remnant functions

Remnant functions are defined, with \(\kappa = \sigma + \tau + \tfrac{1}{2}\) , by $$R_{\sigma \tau } (z) = [{{\Gamma (\sigma - [\kappa ])} \mathord{\left/ {\vphantom {{\Gamma (\sigma - [\kappa ])} {\Gamma (\sigma )}}} \right. \kern-\nulldelimiterspace} {\Gamma (\sigma )}}]\sum\limits_{r = 1}^\infty {r^{2\tau } \left[\kern-0.15em\left[ {(r^2 + z)^{\sigma - 1} } \right]\kern-0.15em\right]_\kappa }$$ where \(\left[\kern-0.15em\left[ \right]\kern-0.15em\right]_\kappa\) denotes subtraction of sufficiently many terms of the Taylor series in powers of z to yield a convergent sum; for integral σ a factor \([1 + ({z \mathord{\left/ {\vphantom {z {r^2 }}} \right. \kern-0em} {r^2 }})]\) may also enter. These functions arise in various contexts, in particular, in the calculation of uniform remainder terms for the approximation by integrals of sums with singular summands. Differential recurrence relations, Taylor expansions, and various integral representations are obtained. The full asymptotic expansions for ¦z¦→∞ with ¦arg z¦ <π are derived, and it is shown that for integral τ these converge exponentially fast.     

On the logarithmic-law constants and the turbulent boundary layer at low Reynolds numbers

It is shown that a family of formally derived similarity solutions describe to leading order the outer region of a turbulent boundary layer for all Reynolds numbers for which the layer satisfies the logarithmic law-of-the-wall. The family includes Coles' [1] hypothesis. For consistency with this hypothesis and the logarithmic law-of-the-wall, it is further shown that the constants in the latter form the product κC=2+O(ε), suggesting the logarithmic law of the wall be written $${U \mathord{\left/ {\vphantom {U {U_\tau = \kappa ^{ - 1} }}} \right. \kern-\nulldelimiterspace} {U_\tau = \kappa ^{ - 1} }}\ln \left( {e^2 U_\tau {y \mathord{\left/ {\vphantom {y \nu }} \right. \kern-\nulldelimiterspace} \nu }} \right) + O\left( \in \right).$$ A range of data are reprocessed to determine the skin friction coefficientC _f using κC = 2 and these collapse well when plotted against momentum thickness Reynolds number, Re _θ . It is also shown that the form parameter, Π, in Coles hypothesis is not unique but is determined by history effects peculiar to the boundary layer. Expressions are derived forC _f (Re _θ ) and the shape factorH (Re _θ ); both agree closely with the data and are valid over all Reynolds numbers for which the logarithmic law of the wall is satisfied.     

Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A ₁(x, D) and A ₂(x, D) be differential operators of the first order acting on l-vector functions ${u= (u_1, \ldots, u_l)}$ in a bounded domain ${\Omega \subset \mathbb{R}^{n}}$ with the smooth boundary ${\partial\Omega}$ . We assume that the H ¹-norm ${\|u\|_{H^{1}(\Omega)}}$ is equivalent to ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ and ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ , where B _i  = B _i (x, ν) is the trace operator onto ${\partial\Omega}$ associated with A _i (x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ${\partial\Omega}$ ). Furthermore, we impose on A ₁ and A ₂ a cancellation property such as ${A_1A_2^{\prime}=0}$ and ${A_2A_1^{\prime}=0}$ , where ${A^{\prime}_i}$ is the formal adjoint differential operator of A _i (i = 1, 2). Suppose that ${\{u_m\}_{m=1}^{\infty}}$ and ${\{v_m\}_{m=1}^{\infty}}$ converge to u and v weakly in ${L^2(\Omega)}$ , respectively. Assume also that ${\{A_{1}u_m\}_{m=1}^{\infty}}$ and ${\{A_{2}v_{m}\}_{m=1}^{\infty}}$ are bounded in ${L^{2}(\Omega)}$ . If either ${\{B_{1}u_m\}_{m=1}^{\infty}}$ or ${\{B_{2}v_m\}_{m=1}^{\infty}}$ is bounded in ${H^{\frac{1}{2}}(\partial\Omega)}$ , then it holds that ${\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}$ . We also discuss a corresponding result on compact Riemannian manifolds with boundary.     

A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents

In this paper, we consider the following PDE involving two Sobolev–Hardy critical exponents,

Flow behaviour of highly concentrated emulsions of supersaturated aqueous solution in oil

A set of highly concentrated water-in-oil emulsions with supersaturated dispersed phase were investigated in this work to verify and/or develop the models that have been presented both in the literature and in this work. The material used to form emulsions consisted of supersaturated oxidiser solution, hydrocarbon oil and PIBSA-based surfactants. The interfacial characteristics for different surfactant types were first examined. Then, the rheology of samples was studied, and different scaling methods and fitting of experimental data were studied. On the basis of flow curve measurements and observed $\tau _\emph{v} \sim \dot {\gamma }^{1/2}$ scaling, a modified version of Windhab model was suggested which showed excellent fitting of experimental results. The linear dependences of ?? _y0/?? versus 1/d ₃₂ for studied emulsions showed non-zero intercept which implies a non-linear dependence (resulting from interdroplet interaction) to fulfil the zero-intercept requirement. It was established that the zero intercept condition was fulfilled in the $\tau _{y0} \sim \sigma /d_{32}^2 $ scaling, although the experimental results for different surfactants were not superimposed.     

A Concentration Phenomenon for Semilinear Elliptic Equations

For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q _n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q _n  > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u _n is a nontrivial solution corresponding to Q _n , then the sequence (u _n ) concentrates at x ₀ with respect to the H ¹ and certain L ^q -norms. We also show that if the sets {Q _n  > 0} shrink to two points and u _n are ground state solutions, then they concentrate at one of these points.     

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

Consider a homogeneous, isotropic, hyperelastic body occupying the region ${A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}}$ in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation $${\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)$$ that maps each sphere ${S_R\subset\,A}$ , of radius R > 0, centred at the origin into another such sphere ${S_r={\bf u}^{\rm rad}(S_R)\subset\,A^*}$ that encloses the same volume as u(S _R ). Since the volumes enclosed by the surfaces u(S _R ) and u ^rad (S _R ) are equal, the classical isoperimetric inequality implies that ${{{\rm Area}( {\bf u}^{\rm rad} (S_R))\leqq {\rm Area}({\bf u} (S_R))}}$ . The equality of the enclosed volumes together with this reduction in surface area is then shown to give rise to a reduction in total energy for many of the constitutive relations used in nonlinear elasticity. These results are also extended to classes of Sobolev deformations and applied to prove that the radially symmetric solutions to these boundary-value problems are local or global energy minimisers in various classes of (possibly nonsymmetric) deformations of a thick spherical shell.     

Improved theoretical model for wavy filmwise condensation

This paper presents a numerical solution for wavy laminar film-wise condensation on vertical walls. Integral method is achieved based on the recently developed simple wave equations. Solutions are obtained for ranges of dimensionless groups as follows: $$1.5 \leqslant \left( {Pr = \frac{{^{\mu C} p}}{k}} \right) \leqslant 6.0$$ $$10 \leqslant \left( {G = \frac{{^h fg}}{{^{C_p \Delta T} }}} \right) \leqslant 400$$ $$100 \leqslant \left( {S = \left( {\frac{{\sigma ^2 \rho }}{{g_\rho \mu ^4 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} } \right) \leqslant 400$$ $$1000 \leqslant \left( {L = \frac{{{\rm H}_t }}{{^\delta cr}}} \right) \leqslant 10000$$ . Such ranges cover the expected situations in industrial applications. It is found that the Reynolds number (Re=h_LΔTH_t/h_fg) is a linear function of L on the log-log plane. It is also relatively insensitive to small variations of Pr at high values of this number. At situations where G less than 200 the Re appears to be dependent on S. Agreement with experimental observation is improved over that obtained from previous analytical theories.     

Recoil in macromolecular solutions according to rigid dumbbell kinetic theory

A macromolecular solution is represented by the simple model of rigid dumbbells suspended in a Newtonian fluid with Brownian motion included. Hydrodynamic interaction is not taken into account. It is found that for this model there will be recoil after the cessation of steady shearing flow. The ultimate shear recovery S _∞ is developed as a power series in κ^?, the shear rate prior to the cessation of the steady shear flow: $$S_\infty = (\theta _0 /2\eta _0 ) \kappa ^\user1{ - } + O(\kappa ^\user1{ - } )^3$$ where η₀ and θ₀ values of the viscosity and primary normal stress functions respectively at zero-shear rate. The coefficient of the term in (κ^?)³ is calculated. In addition, the behavior of the normal stresses during the recoil process is found; during recoil τ₂₂?τ₃₃ has the opposite sign from τ₁₁?τ₂₂.     

Modified dissipativity for a non-linear evolution equation arising in turbulence

We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ?>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\) . If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\) , smoothness of initial data persists indefinitely. If 0≦α<α _cr, there exist positive constants ν₁(α) and ν₂(α), depending on the data, such that global regularity persists for ν>ν₁(α), whereas, for 0≦ν<ν₂(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α _cr, similar results hold for the Navier-Stokes equation.     

Pressure drop in alternating curved tubes

Pressure drop measurements in the laminar and turbulent regions for water flowing through an alternating curved circular tube (x=h sin 2πz/λ) are presented. Using the minimum radius of curvature of this curved tube in place of that of the toroidally curved one in calculating the Dean number (ND=Re(D/2R _c )², it is found that the resulting Dean number can help in characterizing this flow. Also, the ratio between the height and length of the tube waves which represents the degree of waveness affects significantly the pressure drop and the transition Dean number. The following correlations have been found:

1. For laminar flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.03,\operatorname{Re}< 2000.$$
2. For turbulent flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.005,2000< \operatorname{Re}< 15000.$$
3. The transition Dean number: $$ND_{crit} = 5.012 \times 10^3 \left( {\frac{D}{{2R}}} \right)^{2.1} ,0.0111< {D \mathord{\left/ {\vphantom {D {2R_c }}} \right. \kern-\nulldelimiterspace} {2R_c }}< 0.71.$$

     

A gradient based iterative algorithm for solving structural dynamics model updating problems

The procedure of updating an existing but inaccurate model is an essential step toward establishing an effective model. Updating damping and stiffness matrices simultaneously with measured modal data can be mathematically formulated as following two problems. Problem 1: Let M _a ∈SR ^n×n be the analytical mass matrix, and Λ=diag{λ ₁,…,λ _p }∈C ^p×p , X=[x ₁,…,x _p ]∈C ^n×p be the measured eigenvalue and eigenvector matrices, where rank(X)=p, p<n and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in\nobreak{\mathbf{C}} $ , $x_{2j} = \bar{x}_{2j-1} \in{\mathbf{C}}^{n} $ for j=1,…,l, and λ _k ∈R, x _k ∈R ⁿ for k=2l+1,…,p. Find real-valued symmetric matrices D and K such that M _a XΛ ²+DXΛ+KX=0. Problem 2: Let D _a ,K _a ∈SR ^n×n be the analytical damping and stiffness matrices. Find $(\hat{D}, \hat{K}) \in\mathbf{S}_{\mathbf{E}}$ such that $\| \hat{D}-D_{a} \|^{2}+\| \hat{K}-K_{a} \|^{2}= \min_{(D,K) \in \mathbf{S}_{\mathbf{E}}}(\| D-D_{a} \|^{2} +\|K-K_{a} \|^{2})$ , where S _E is the solution set of Problem 1 and ∥?∥ is the Frobenius norm. In this paper, a gradient based iterative (GI) algorithm is constructed to solve Problems 1 and 2. A sufficient condition for the convergence of the iterative method is derived and the range of the convergence factor is given to guarantee that the iterative solutions consistently converge to the unique minimum Frobenius norm symmetric solution of Problem 2 when a suitable initial symmetric matrix pair is chosen. The algorithm proposed requires less storage capacity than the existing numerical ones and is numerically reliable as only matrix manipulation is required. Two numerical examples show that the introduced iterative algorithm is quite efficient.     

A note on the soil-water conductivity of a fractal soil

We show that for a fractal soil the soil-water conductivity, K, is given by $$\frac{K}{{K_\varepsilon }} = (\Theta /\varepsilon )^{2D/3 + 2/(3 - D)}$$ where $K_\varepsilon$ is the saturated conductivity, θ the water content, ? its saturated value and D is the fractal dimension obtained from reinterpreting Millington and Quirk's equation for practical values of the porosity ?, as $$D = 2 + 3\frac{{\varepsilon ^{4/3} + (1 - \varepsilon )^{2/3} - 1}}{{2\varepsilon ^{4/3} \ln ,{\text{ }}\varepsilon ^{ - 1} + (1 - \varepsilon )^{2/3} \ln (1 - \varepsilon )^{ - 1} }}$$ .     

An experimental study on turbulence modification in the near-wall boundary layer of a dilute gas-particle channel flow

Turbulence modifications of a dilute gas-particle flow are experimentally investigated in the lower boundary layer of a horizontal channel by means of a simultaneous two-phase PIV measurement technique. The measurements are conducted in the near-wall region with y ⁺?<?250 at Re _τ (based on the wall friction velocity u _τ and half channel height h)?=?430. High spatial resolution and small interrogation window are used to minimize the PIV measurement uncertainty due to the velocity gradient near the wall. Polythene beads with the diameter of 60?μm (d _p ⁺ ?=?1.71, normalized by the fluid kinematic viscosity ν and u _τ) are used as dispersed phase, and three low mass loading ratios (Φ _m ) ranging from 10^?4 to 10^?3 are tested. It is found that the addition of the particles noticeably modifies the mean velocity and turbulent intensities of the gas-phase, as well as the turbulence coherent structures, even at Φ _m ?=?0.025?%. Particle inertia changes the viscous sublayer of the gas turbulence with a smaller thickness and a larger streamwise velocity gradient, which increases the peak value of the streamwise fluctuation velocity ( $ u_{\text{rms}}^{ + } $ ) of the gas-phase with its location shifting to the wall. Particle sedimentation increases the roughness of the bottom wall, which significantly increases the wall-normal fluctuation velocity ( $ v_{\text{rms}}^{ + } $ ) and Reynolds shear stress ( $ - \langle u^{ \prime } v^{\prime } \rangle^{ + } $ ) of the gas-phase in the inner region of the boundary layer (y ⁺?<?10). Under effect of particle–wall collision, the Q2 events (ejections) of the gas-phase are slightly increased by particles, while the Q4 events (sweeps) are obviously decreased. The spatial scale of the coherent structures near the wall shrinks remarkably with the presence of the particles, which may be attributed to the intensified crossing-trajectory effects due to particle saltation near the bottom wall. Meanwhile, the $ v_{\text{rms}}^{ + } $ and $ - \langle u^{ \prime } v^{\prime } \rangle^{ + } $ of the gas-phase are significantly reduced in the outer region of the boundary layer (y ⁺?>?20).     

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \| a_0 \|_{L^\infty} + \|u_0^h\|_{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\|u_0^d\|_{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c ₀, C _r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L ^p (L ^q ) regularity theorem for the heat kernel plays an essential role in this context.