Unsteady flow with separation behind a shock wave diffracting over curved walls

The unsteady separation of the compressible flow field behind a diffracting shock wave was investigated along convex curved walls, using shock tube experimentation at large length and time scales, complemented by numerical computation. Tests were conducted at incident shock Mach numbers of $M_{\hbox {s}} =$ 1.5 and 1.6 over a 100 mm radius wall over a dimensionless time range up to $\tau \le $ 6.45. The development of the near wall flow at $M_{\hbox {s}} =$ 1.5 has been described in detail and is very similar to that observed for slightly lower $\tau $ ’s at $M_{\hbox {s}} =$ 1.6. Computations were performed at wall radii of 100 and 200 mm and for incident shock Mach numbers from 1.5 up to and including Mach 2.0. Comparing dimensionless times for different size walls shows that for a given value of $\tau $ the flow field is very similar for the various wall radii published to date and tested in this study. Previously published results that were examined alongside the results from this study had typical values of $1.6 < \tau < 3.2$ . At the later times presented here, flow features were observed that previously had only been observed at higher Mach numbers. The larger length scales allowed for a degree of Reynolds number independence in the results published here. The effect of turbulence on the numerical and experimental results could not be adequately examined due to limitations of the flow imaging system used and a number of questions remain unanswered.     

Comparative characteristics of strong shock and detonation waves in bubble media by an electrical wire explosion

Strong shock and detonation waves in inert and chemically active bubble media, which are generated by a wire explosion initiated by a capacitor with a stored energy $W_0 =12.3$ –1,600 J, is experimentally studied. The measurements are performed near the wire and far from the wire in a vertical shock tube 4.5 m long with a volume fraction of the gas in the medium $\beta _0 =1$ –4 %. It is shown that in inert bubble medium, a short intensely decaying shock wave (SW) with intense pressure oscillations is formed in the vicinity of wire explosion point; near the explosion point at $\beta _0 \le 2$  % the SW propagates with the velocity of sound in a liquid. In chemically active bubble medium, an unsteady detonation wave generated by a wire explosion is formed. The pressure amplitude and the velocity of this wave are greater and the length is smaller than those of SW in an inert bubble medium in the same range of explosion energy. It is found that in the interval of low energy explosion from ${\sim }12$ to 64 J, the formation of the bubble detonation wave occurs faster than that at high energies ( $3\times 10^{2}$ – $10^{3}$  J).     

Effect of an impinging shock wave on a partially opened door

The behavior of an aluminum door hanging at the exit of an open shock tube at different angles, from 5 $^\circ $ to 85 $^\circ $ , and thereby providing partially open space for the exiting flow, was investigated experimentally. Experiments were conducted with an incident shock wave Mach number of $M_\mathrm{is}=1.1$ impinging on the partially opened door. Both pressure measurements in the vicinity of the door, on its center and inside the shock tube, and schlieren visualization were undertaken for studying the door movement and its maximum opening angle relative to its initial position. It was found that for an initial opening angle smaller than 25 $^\circ $ the door opened completely while for larger angles its motion is marginal. In addition, for an initial door opening angle of about 10 $^\circ $ the lowest pressures were recorded inside the shock tube behind the evolving waves after exiting of the incident shock wave. The present experimental results may be useful to numerical studies of fluid–structure interactions, e.g., in designing safety valves in jet engines. Such a device is needed for preventing rupture in the case when a sudden overpressure pulse is generated inside the aircraft engine compartment.     

Non-uniqueness of solutions in asymptotically self-similar shock reflections

The present study addresses the self-similar problem of unsteady shock reflection on an inclined wedge. The start-up conditions are studied by modifying the wedge corner and allowing for a finite radius of curvature. It is found that the type of shock reflection observed far from the corner, namely regular or Mach reflection, depends intimately on the start-up condition, as the flow “remembers” how it was started. Substantial differences were found. For example, the type of shock reflection for an incident shock Mach number $M=6.6$ and an isentropic exponent $\gamma =1.2$ changes from regular to Mach reflection between $44^\circ $ and $45^\circ $ when a straight wedge tip is used, while the transition for an initially curved wedge occurs between $57^\circ $ and $58^\circ $ .     

Experimental study on a heavy-gas cylinder accelerated by cylindrical converging shock waves

The Richtmyer–Meshkov instability behavior of a heavy-gas $(\text{ SF }_6)$ cylinder accelerated by a cylindrical converging shock wave is studied experimentally. A curved wall profile is well-designed based on the shock dynamics theory [Phys. Fluids, 22: 041701 (2010)] with an incident planar shock Mach number of 1.2 and a converging angle of $15^\circ $ in a $95\,\text{ mm }\times 95$ mm square cross-section shock tube. The $\text{ SF }_6$ cylinder mixed with the glycol droplets flows vertically through the test section and is illuminated horizontally by a laser sheet. The images obtained only one per run by an ICCD (intensified charge coupled device) combined with a pulsed Nd:YAG laser are first presented and the complete evolution process of the $\text{ SF }_6$ cylinder is then captured in a single test shot by a high-speed video camera combined with a high-power continuous laser. In this way, both the developments of the first counter-rotating vortex pair and the second counter-rotating vortex pair with an opposite rotating direction from the first one are observed. The experimental results indicate that the phenomena induced by the converging shock wave and the reflected shock formed from the center of convergence are distinct from those found in the planar shock case.     

Influence of \text{ CF }_{3}\text{ H } and \text{ CCl }_{4} additives on acetylene detonation

The influence of $\text{ CF }_{3}\text{ H }$ and $\text{ CCl }_{4}$ admixtures (known as detonation suppressors for combustible mixtures) on the development of acetylene detonation was experimentally investigated in a shock tube. The time-resolved images of detonation wave development and propagation were registered using a high-speed streak camera. Shock wave velocity and pressure profiles were measured by five calibrated piezoelectric gauges and the formation of condensed particles was detected by laser light extinction. The induction time of detonation development was determined as the moment of a pressure rise at the end plate of the shock tube. It was shown that $\text{ CF }_{3}\text{ H }$ additive had no influence on the induction time. For $\text{ CCl }_{4}$ , a significant promoting effect was observed. A simplified kinetic model was suggested and characteristic rates of diacetylene $\text{ C }_{4}\text{ H }_{2}$ formation were estimated as the limiting stage of acetylene polymerisation. An analysis of the obtained data indicated that the promoting species is atomic chlorine formed by $\text{ CCl }_{4}$ pyrolysis, which interacts with acetylene and produces $\text{ C }_{2}\text{ H }$ radical, initiating a chain mechanism of acetylene decomposition. The results of kinetic modelling agree well with the experimental data.     

Semianalytical Solution for \text{ CO}_{2} Plume Shape and Pressure Evolution During \text{ CO}_{2} Injection in Deep Saline Formations

The injection of supercritical carbon dioxide ( $\text{ CO}_{2})$ in deep saline aquifers leads to the formation of a $\text{ CO}_{2}$ rich phase plume that tends to float over the resident brine. As pressure builds up, $\text{ CO}_{2}$ density will increase because of its high compressibility. Current analytical solutions do not account for $\text{ CO}_{2}$ compressibility and consider a volumetric injection rate that is uniformly distributed along the whole thickness of the aquifer, which is unrealistic. Furthermore, the slope of the $\text{ CO}_{2}$ pressure with respect to the logarithm of distance obtained from these solutions differs from that of numerical solutions. We develop a semianalytical solution for the $\text{ CO}_{2}$ plume geometry and fluid pressure evolution, accounting for $\text{ CO}_{2}$ compressibility and buoyancy effects in the injection well, so $\text{ CO}_{2}$ is not uniformly injected along the aquifer thickness. We formulate the problem in terms of a $\text{ CO}_{2}$ potential that facilitates solution in horizontal layers, with which we discretize the aquifer. Capillary pressure is considered at the interface between the $\text{ CO}_{2}$ rich phase and the aqueous phase. When a prescribed $\text{ CO}_{2}$ mass flow rate is injected, $\text{ CO}_{2}$ advances initially through the top portion of the aquifer. As $\text{ CO}_{2}$ is being injected, the $\text{ CO}_{2}$ plume advances not only laterally, but also vertically downwards. However, the $\text{ CO}_{2}$ plume does not necessarily occupy the whole thickness of the aquifer. We found that even in the cases in which the $\text{ CO}_{2}$ plume reaches the bottom of the aquifer, most of the injected $\text{ CO}_{2}$ enters the aquifer through the layers at the top. Both $\text{ CO}_{2}$ plume position and fluid pressure compare well with numerical simulations. This solution permits quick evaluations of the $\text{ CO}_{2}$ plume position and fluid pressure distribution when injecting supercritical $\text{ CO}_{2}$ in a deep saline aquifer.     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

Predominant Gas Transport in Microporous Hydrotalcite–Silica Membrane

The prepared microporous hydrotalcite (HT)–silica membrane was found to exhibit the molecular sieving characteristic of pristine silica material and high $\mathrm{CO}_{2}$ adsorption capacity of HT. The combined properties made enhanced $\mathrm{CO}_{2}$ permeability and separability from $\mathrm{CH}_{4}$ possible. The gas transport in the membrane was predominantly surface adsorption. The porous membrane overcame the Knudsen limitation and yielded the highest separation selectivity of 120 at 40 % $\mathrm{CO}_{2}$ feed concentration, $30\,^{\circ }\mathrm{C}$ operating temperature, and 100 kPa pressure difference.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

Mixed Convection Boundary-Layer Flow Over a Vertical Surface Embedded in a Porous Material Subject to a Convective Boundary Condition

The mixed convection boundary-layer flow on one face of a semi-infinite vertical surface embedded in a fluid-saturated porous medium is considered when the other face is taken to be in contact with a hot or cooled fluid maintaining that surface at a constant temperature $T_\mathrm{{f}}$ . The governing system of partial differential equations is transformed into a system of ordinary differential equations through an appropriate similarity transformation. These equations are solved numerically in terms of a dimensionless mixed convection parameter $\epsilon $ and a surface heat transfer parameter $\gamma $ . The results indicate that dual solutions exist for opposing flow, $\epsilon <0$ , with the dependence of the critical values $\epsilon _\mathrm{{c}}$ on $\gamma $ being determined, whereas for the assisting flow $\epsilon >0$ , the solution is unique. Limiting asymptotic forms for both $\gamma $ small and large and $\epsilon $ large are also discussed.     

A New Insight into the Convective Boundary Condition

The steady mixed convection boundary layer flows over a vertical surface adjacent to a Darcy porous medium and subject respectively to (i) a prescribed constant wall temperature, (ii) a prescribed variable heat flux, $q_\mathrm{w} =q_0 x^{-1/2}$ q w = q 0 x ? 1 / 2 , and (iii) a convective boundary condition are compared to each other in this article. It is shown that, in the characteristic plane spanned by the dimensionless flow velocity at the wall ${f}^{\prime }(0)\equiv \lambda $ f ′ ( 0 ) ≡ λ and the dimensionless wall shear stress $f^{\prime \prime }(0)\equiv S$ f ′ ′ ( 0 ) ≡ S , every solution $(\lambda , S)$ ( λ , S ) of one of these three flow problems at the same time is also a solution of the other two ones. There also turns out that with respect to the governing mixed convection and surface heat transfer parameters $\varepsilon $ ε and $\gamma $ γ , every solution $(\lambda , S)$ ( λ , S ) of the flow problem (iii) is infinitely degenerate. Specifically, to the very same flow solution $(\lambda , S)$ ( λ , S ) there corresponds a whole continuous set of values of $\varepsilon $ ε and $\gamma $ γ which satisfy the equation $S=-\gamma (1+\varepsilon -\lambda )$ S = ? γ ( 1 + ε ? λ ) . For the temperature solutions, however, the infinite degeneracy of the velocity solutions becomes lifted. These and further outstanding features of the convective problem (iii) are discussed in the article in some detail.     

Multi-component Reynolds-averaged Navier–Stokes simulations of Richtmyer–Meshkov instability and mixing induced by reshock at different times

Turbulent mixing generated by shock-driven acceleration of a perturbed interface is simulated using a new multi-component Reynolds-averaged Navier–Stokes (RANS) model closed with a two-equation $K$ – $\epsilon $ model. The model is implemented in a hydrodynamics code using a third-order weighted essentially non-oscillatory finite-difference method for the advection terms and a second-order central difference method for the gradients in the source and diffusion terms. In the present reshocked Richtmyer–Meshkov instability and mixing study, an incident shock with Mach number $M\!a_{\mathrm{s}}=1.20$ is generated in air and progresses into a sulfur hexafluoride test section. The time evolution of the predicted mixing layer widths corresponding to six shock tube test section lengths are compared with experimental measurements and three-dimensional multi-mode numerical simulations. The mixing layer widths are also compared with the analytical self-similar power-law solution of the simplified model equations prior to reshock. A set of model coefficients and initial conditions specific to these six experiments is established, for which the widths before and after reshock agree very well with experimental and numerical simulation data. A second set of general coefficients that accommodates a broader range of incident shock Mach numbers, Atwood numbers, and test section lengths is also established by incorporating additional experimental data for $M\!a_{\mathrm{s}}=1.24$ , $1.50$ , and $1.98$ with $At=0.67$ and $M\!a_{\mathrm{s}}=1.45$ with $At=-0.67$ and previous RANS modeling. Terms in the budgets of the turbulent kinetic energy and dissipation rate equations are examined to evaluate the relative importance of turbulence production, dissipation and diffusion mechanisms during mixing. Convergence results for the mixing layer widths, mean fields, and turbulent fields under grid refinement are presented for each of the $M\!a_{\mathrm{s}}=1.20$ cases.     

Numerical and experimental investigations of pseudo-shock systems in a planar nozzle: impact of bypass mass flow due to narrow gaps

During previous investigations on pseudo-shock systems, we have observed reproducible differences between measurement and simulations for the pressure distribution as well as for size and shape of the pseudo-shock system. A systematic analysis of the deviations leads to the conclusion that small gaps of $\Delta z=O(10^{-4})$  m between quartz glass side walls and metal contour of the test section are responsible for this mismatch. This paper describes a targeted experimental and numerical study of the bypass mass flow within these gaps and its interaction with the main flow. In detail, we analyze how the pressure distribution within the channel as well as the size, shape and oscillation of the pseudo-shock system are affected by the gap size. Numerical simulations are performed to display the flow inside the gaps and to reproduce and explain the experimental results. Numerical and experimental schlieren images of the pseudo-shock system are in good agreement and show that especially the structure of the primary shock is significantly altered by the presence of small gaps. Extensive unsteady flow simulations of the geometry with gaps reveal that the shear layer between subsonic gap flow and supersonic core flow is subject to a Kelvin–Helmholtz instability resulting in small pressure fluctuations. This leads to a shock oscillation with a frequency of $f= O(10^5) \hbox {s}^{-1}$ . The corresponding time scale $\tau $  (s) is 16 times higher than the characteristic time scale $\tau _\delta =\delta /U_\infty $ of the boundary layer given by the ratio of the boundary layer thickness $\delta $ directly ahead of the shock and the undisturbed free stream velocity $U_\infty $ . To assess the reliability of our numerical investigations, the paper includes a grid study as well as an extensive comparison of several RANS turbulence models and their impact on the predicted shape of pseudo-shock systems.     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

Generalized Resolvent Estimates of the Stokes Equations with First Order Boundary Condition in a General Domain

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain ${\Omega}$ of the N-dimensional Eulidean space ${\mathbb{R}^N, N \geq 2}$ . This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter ${\lambda}$ varying in a sector ${\Sigma_{\sigma, \lambda_0} = \{\lambda \in \mathbb{C} \mid |\arg \lambda| < \pi-\sigma, \enskip |\lambda| \geq \lambda_0\}}$ , where ${0 < \sigma < \pi/2}$ and ${\lambda_0 \geq 1}$ . The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution ${p \in \hat{W}^1_{q, \Gamma}(\Omega)}$ to the variational problem: ${(\nabla p, \nabla \varphi) = (f, \nabla \varphi)}$ for any ${\varphi \in \hat W^1_{q', \Gamma}(\Omega)}$ . Here, ${1 < q < \infty, q' = q/(q-1), \hat W^1_{q, \Gamma}(\Omega)}$ is the closure of ${W^1_{q, \Gamma}(\Omega) = \{ p \in W^1_q(\Omega) \mid p|_\Gamma = 0\}}$ by the semi-norm ${\|\nabla \cdot \|_{L_q(\Omega)}}$ , and ${\Gamma}$ is the boundary of ${\Omega}$ . In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in ${(\lambda_0, \infty)}$ . Our assumption is satisfied for any ${q \in (1, \infty)}$ by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q =  2.     

A Comment on the Flow and Heat Transfer Past a Permeable Stretching/Shrinking Surface in a Porous Medium: Brinkman Model

An analytical solution is presented for the boundary-layer flow and heat transfer over a permeable stretching/shrinking surface embedded in a porous medium using the Brinkman model. The problem is seen to be characterized by the Prandtl number $Pr$ , a mass flux parameter $s$ , with $s>0$ for suction, $s=0$ for an impermeable surface, and $s<0$ for blowing, a viscosity ratio parameter $M$ , the porous medium parameter $\Lambda $ and a wall velocity parameter $\lambda $ . The analytical solution identifies critical values which agree with those previously determined numerically (Bachok et al. Proceedings of the fifth International Conference on Applications of Porous Media, 2013) and shows that these critical values, and the consequent dual solutions, can arise only when there is suction through the wall, $s>0$ .     

Super-equilibrium increase of chemical reaction rate in the detonation front and other effects in the detonation wave initiated by a shock wave

In the present work the problem of detonation wave formation in a shock tube was considered in one-dimensional formulation. The Monte Carlo non-stationary method of statistical simulation (MCNMSS), also known as DSMC, was used for simulation. The method automatically takes into account all details of mass and heat transfer. At an initial moment, the low-pressure channel (LPC) of the shock tube was filled with gas A while the high-pressure chamber (HPC) was filled with gas C. The cross-sections of the HPC and LPC, as well as the temperatures of gases A and C were equal to each other. At the beginning of the simulation the ratio of pressures in the HPC and LPC was equal to 100. It was assumed that chemical reactions $\mathrm{{A}}+\mathrm{{M}} \rightarrow \mathrm{{B}}+\mathrm{{M}}$ ( $\mathrm{{M}}=\mathrm{{A}},\, \mathrm{{B}}$ and $\mathrm{{C}}$ ) took place. The ratio of molecular masses of gases $\mathrm{{A}},\, \mathrm{{B}}$ and $\mathrm{{C}}$ was taken as 20:20:1. Different reaction thresholds were considered. For the case of a low reaction threshold, the velocity of the resulting detonation wave was found to be higher than the Chapman–Jouguet velocity. A region with constant values of flow parameters inside product was observed. An increase of the reaction threshold led to disappearance of this region and gave rise to something similar to an expansion wave, with peaks of flow parameters at the leading part of the detonation wave. The values of these peaks were found to be constant in time. The velocity of the detonation wave became appreciably lower than the Chapman–Jouguet velocity. Further increase of the reaction threshold led to disappearance of detonation. The reactions $\mathrm{{A}}+\mathrm{{B}} \rightarrow \mathrm{{B}}+\mathrm{{B}}$ and $\mathrm{{A}}+\mathrm{{C}}\rightarrow \mathrm{{B}}+\mathrm{{C}}$ turned out to be very important for initiation of detonation.     

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.