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Bemisia tabaci is one of the most important agricultural pests worldwide, and the combined application of multiple natural enemies such as predators and parasitoids can potentially control B. tabaci. The study examined whether the predator Orius similis and the parasitoid Encarsia formosa can synergistically control B. tabaci (crop: kidney bean). The greenhouse cage method was used to release O. similis and E. formosa alone or in combination in different ratios. The combined release of O. similis and E. formosa synergistically decreased the B. tabaci population when compared with O. similis or E. formosa alone. Additionally, O. similis + E. formosa decreased the number of E. formosa black pupae and adults in each crop stage. However, the niche overlap index of E. formosa with B. tabaci nymphs in the O. similis + E. formosa group was higher than in the E. formosa group. Grey correlation analysis revealed that the correlation degree between natural enemies and B. tabaci was the highest when the O. similis and E. formosa release ratio was 1:3. These findings indicate that the combined release of O. similis and E. formosa synergistically controlled B. tabaci with the release ratio 1:3 being optimal for field application.
In this paper, a novel series–parallel stable platform is proposed, and its kinematic and dynamic models are established. The relationship between the length, speed, and acceleration of rolling and pitching electric push rods is analyzed. The workspace of the series–parallel stable platform is determined, and the singularity and interference are analyzed. The state-machine-based control system of the stable platform is designed. An experimental environment of the principle of the real-time control system based on dSPACE was built. A position–speed double closed-loop experiment, simulating mounting carrier of the random signal tracking, and system comprehensive performance experiment were conducted to verify the accuracy of the kinematics and dynamics model of the series–parallel stable platform and the rationality and stability of the control system.
Sugar beet production demands sustainable intensification approaches to enhance both yield and quality. This study examined the effects of foliar nano-ammonium nitrate (NH4NO3) applications on five sugar beet cultivars – JAMPOL, BTS 9830, DEL 1135, ASEEL and Raspoly – over two growing seasons in Egypt’s Nile Delta, using a split-plot randomized complete block design. Three nano-NH4NO3 concentrations (0, 50 and 100 ppm) were sprayed twice each season to assess impacts on growth, yield and quality parameters. The results indicated limited influence of treatments on primary yield metrics; taproot yields remained unaffected across all cultivars and seasons. Nonetheless, the 50 ppm treatment preserved optimal sugar quality, with sucrose content reaching 19.4 %, compared to 18 % in the controls. Carotenoid levels increased by 12 % under the 100 ppm treatment, reflecting enhanced nutritional quality. Among cultivars, ASEEL yielded the best results with taproot outputs of 34 t/ha and sugar yields of 6.1 t/ha under optimal conditions, demonstrating significant cultivar-dependent variation. Multivariate analysis revealed distinct response patterns among cultivars and treatments, with three-way interactions (Season × Cultivar × Nanoparticle) affecting several traits. Clustering identified four trait groups and three treatment clusters, highlighting sugar beet’s complex response to nanoparticles. Economic analysis shows limited benefits, with no significant increase in taproot yield, despite a rise in secondary metabolites. While nano-NH4NO3 can modify biochemical parameters, the lack of yield improvements casts doubt on its economic feasibility. Cultivar choice primarily influences sugar beet performance, with environmental conditions also affecting treatment efficacy.
Mitochondrial dysfunction has been implicated in the pathogenesis of major depressive disorder (MDD); however, the causal contributions of specific mitochondrial genes across regulatory layers remain unclear.
Methods
We integrated genome-wide association study summary statistics from the Psychiatric Genomics Consortium and FinnGen with quantitative-trait-locus (QTL) datasets for DNA methylation, gene expression (eQTL), and protein abundance. Mitochondrial genes were annotated using the MitoCarta3.0 database. Summary-based Mendelian randomization and Bayesian colocalization were applied to assess causal relationships, with colocalization determined by the posterior probability of a shared causal variant (PPH4), and the false discovery rate used for multiple-testing correction. Brain-specific effects were evaluated using Genotype-Tissue Expression eQTL data. Prioritized genes were ranked based on cross-omics consistency and replication evidence.
Results
Five mitochondrial genes were prioritized. TDRKH showed consistent associations across methylation, transcription, and protein levels, with hypermethylation at cg24503712 linked to reduced expression and a lower risk of MDD (Tier 1). METAP1D (Tier 2) demonstrated protective effects at both the transcript and protein levels. LONP1, FIS1, and SCP2 (Tier 3) exhibited consistent but complex regulatory patterns. Several signals were replicated in brain tissues, including TDRKH in the caudate and METAP1D in the cortex.
Conclusions
This study provides multi-omics evidence for the causal involvement of mitochondrial genes in MDD. TDRKH and METAP1D emerged as key candidates, offering promising targets for future mechanistic research and therapeutic development.
In the fields of meal-assisting robotics and human–robot interaction (HRI), real-time and accurate mouth pose estimation is critical for ensuring interaction safety and improving user experience. The complexity arises from the diverse opening degrees of mouths, variations in orientation, and external factors such as lighting conditions and occlusions, which pose significant challenges for real-time and accurate posture estimation of mouths. In response to the above-mentioned issues, this paper proposes a novel method for point cloud fitting and posture estimation of mouth opening degrees (FP-MODs). The proposed method leverages both RGB and depth images captured from a single viewpoint, integrating geometric modeling with advanced point cloud processing techniques to achieve robust and accurate mouth posture estimation. The innovation of this work lies in the hypothesis that different states of mouth openings can be effectively described by distinct geometric shapes: closed mouths are modeled by spatial quadratic surfaces, half-open mouths by spatial ellipses, and fully open mouths by spatial circles. Then, based on these hypotheses, we developed algorithms for fitting geometric models to point clouds obtained from mouth regions, respectively. Specifically, for the closed mouth state, we employ an algorithm based on least squares optimization to fit a spatial quadratic surface to the point cloud data. For the half-open or fully open mouth states, we combine inverse projection methods with least squares fitting to model the contour as a spatial ellipse and circle, respectively. Finally, to evaluate the effectiveness of the proposed FP-MODs method, extensive actual experiments were conducted under varying conditions, including different orientations and various types of mouths. The results demonstrate that the proposed FP-MODs method achieves high accuracy and robustness. This study can provide a theoretical foundation and technical support for improving HRI and food delivery safety in the field of robotics.
Investigations into the effects of polymers on small-scale statistics and flow patterns were conducted in a turbulent von Kármán swirling (VKS) flow. We employed the tomographic particle image velocimetry technique to obtain full information on three-dimensional velocity data, allowing us to effectively resolve dissipation scales. Under varying Reynolds numbers ($R_\lambda =168{-}235$) and polymer concentrations ($\phi =0{-}25\ {\textrm{ppm}}$), we measured the velocity gradient tensor (VGT) and related quantities. Our findings reveal that the ensemble average and probability density function (PDF) of VGT invariants, which represent turbulent dissipation and enstrophy along with their generation terms, are suppressed as polymer concentration increases. Notably, the joint PDFs of the invariants of VGT, which characterise local flow patterns, exhibited significant changes. Specifically, the third-order invariants, especially the local vortex stretching, are greatly suppressed, and strong events of dissipation and enstrophy coexist in space. The local flow pattern tends to be two-dimensional, where the eigenvalues of the rate-of-strain tensor satisfy a ratio $1:0:-1$, and the vorticity aligns with the intermediate eigenvector of the rate-of-strain tensor, while it is perpendicular to the other two. We find that these statistics observations can be well described by the vortex sheet model. Moreover, we find that these vortex sheet structures align with the symmetry axis of the VKS system, and orient randomly in the horizontal plane. Further investigation, including flow visualisation and conditional statistics on vorticity, confirms the presence of vortex sheet structures in turbulent flows with polymer additions. Our results establish a link between single-point statistics and small-scale flow topology, shedding light on the previously overlooked small-scale structures in polymeric turbulence.
In the field of parafoil airdrop path planning, the inherent complexity and time-sensitive nature of mission requirements necessitate rapid path generation through low-order mathematical models that approximate the system’s true dynamics. This study presents a novel sparse identification framework for constructing a parafoil path planning approximate model. Leveraging high-fidelity 9-degree-of-freedom (9 DOF) dynamic simulation data as training inputs, our method identifies simple nonlinear relationships between 3D positional coordinates (for spatial targeting) and yaw angle (for directional control), which are critical path planning parameters. Compared to conventional 4 DOF models, experimental validation using field airdrop data reveals that the proposed sparse model achieves enhanced predictive accuracy while maintaining computational efficiency. Quantitative analysis demonstrates reductions in root mean square error (RMSE) by approximately 12.96% (horizontal position), 54.44% (height) and 37.96% (yaw angle). The efficacy is further confirmed through successful fixed-point homing across diverse initial deployment scenarios, underscoring its potential for parafoil path planning.
Schizophrenia progresses through high-risk, first-episode, and chronic stages, each associated with altered spontaneous brain activity. Resting state functional MRI studies highlight these changes, but inconsistencies persist, and the genetic basis remains unclear.
Methods
A neuroimaging meta-analysis was conducted to assess spontaneous brain activity alterations in each schizophrenia stage. The largest available genome-wide association study (GWAS) summary statistics for schizophrenia (N = 53,386 cases, 77,258 controls) were used, followed by Hi-C-coupled multimarker analysis of genomic annotation (H-MAGMA) to identify schizophrenia-associated genes. Transcriptome-neuroimaging association and gene prioritization analyses were performed to identify genes consistently linked to brain activity alterations. Biological relevance was explored by functional enrichment.
Results
Fifty-two studies met the inclusion criteria, covering the high-risk (Nhigh-risk = 409, Ncontrol = 475), first-episode (Ncase = 1842, Ncontrol = 1735), and chronic (Ncase = 1242, Ncontrol = 1300) stages. High-risk stage showed reduced brain activity in the right median cingulate and paracingulate gyri. First-episode stage revealed increased activity in the right putamen and decreased activity in the left gyrus rectus and right postcentral gyrus. Chronic stage showed heightened activity in the right inferior frontal gyrus and reduced activity in the superior occipital gyrus and right postcentral gyrus. Across all stages, 199 genes were consistently linked to brain activity changes, involved in biological processes such as nervous system development, synaptic transmission, and synaptic plasticity.
Conclusions
Brain activity alterations across schizophrenia stages and genes consistently associated with these changes highlight their potential as universal biomarkers and therapeutic targets for schizophrenia.
Arabinoxylans (AX), the primary hemicellulose found in cereals and grasses, play a crucial role in regulating immunity, metabolism and various physiological processes, underscoring their value as essential components in dietary nutrition. Considering the extensive research on AX in piglet nutrition, this paper systematically reviews their impacts on gut health and microbiota in piglets, as well as the underlying mechanisms of action. AX have been shown to mediate gut barrier fortification through tight junction protein upregulation and orchestrate mucosal immunity homeostasis, consequently ameliorating early-weaning-associated diarrheal pathogenesis in piglets. Additionally, AX function as microbial ecological modulators through selective enrichment of beneficial commensal microbiota (e.g. Bifidobacterium spp. and Lactobacillus spp.), while simultaneously stimulating microbial biosynthesis of SCFA and ferulic acid exhibiting potent antioxidant and anti-inflammatory activity, thereby maintaining the intestinal health of piglets. This review offers valuable insights into their potential as a dietary intervention to support gut health and immune function in early-weaned piglets. However, most studies focus on single-source AX such as wheat or maize, with limited exploration of novel sources or comparative effects of source combinations. Future research should systematically investigate the molecular mechanisms of AX action, provide data-driven guidance for selecting AX sources in feed formulations and establish optimal inclusion levels in practical feeding regimens. Such efforts will further solidify the precision nutrition potential of AX in promoting sustainable and healthy growth in piglets.
Turbulence amplification is crucial in shock-wave/turbulent boundary layer interaction (SWTBLI). To examine the impact of interaction intensity on turbulence amplification and inter-component energy transfer, direct numerical simulations of impinging oblique shock reflections at strong ($37^\circ$) and weak ($33.2^\circ$) incident angles are conducted. The results indicate that strong interaction generates a larger permanent separation zone, featuring the unique ‘oblique platform’ in Reynolds stress peaks and ‘secondary turbulence amplification’ downstream. Reynolds stress budget and spanwise spectral analyses reveal that $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ amplify primarily by production terms. $u''$, $v''$ and $w''$ represent the streamwise, wall-normal and spanwise velocity fluctuations. At the investigated Reynolds number, deceleration effect dominates the initial amplification of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$, influencing multi-scale wall-bounded turbulence structures, while shear effect remains active along the shear layer and may primarily affects streaky structures. The initial amplification of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is driven by the adverse pressure gradient, which reshapes the velocity profile and affects the wall-normal velocity. The primary energy for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ amplification originates from $\widetilde{ u^{\prime \prime}u^{\prime \prime}}$ via the pressure-strain term. The delayed amplification of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ is influenced by its production term and energy redistribution, with $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ exhibiting higher spectral consistency with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and receiving more energy. In strong interaction, the ‘oblique platform’ serves as a stable dissipation region, formed by increased separation–incident shock distance, characterised by progressively concentrated stress spectra and the transition to large-scale streaks. The downstream ‘secondary amplification’ process resembles the initial amplification near the separation shock foot, driven by intermittent compression waves that strengthen shear instabilities and the deceleration effect. These findings detail the streamwise stress evolution, providing a more comprehensive turbulence amplification mechanism in SWTBLI.
Remote injury assessment during natural disasters poses major challenges for healthcare providers due to the inaccessibility of disaster sites. This study aimed to explore the feasibility of using artificial intelligence (AI) techniques for rapid assessment of traumatic injuries based on gait analysis.
Methods
We conducted an AI-based investigation using a dataset of 4500 gait images across 3 species: humans, dogs, and rabbits. Each image was categorized as either normal or limping. A deep learning model, YOLOv5—a state-of-the-art object detection algorithm—was trained to identify and classify limping gait patterns from normal ones. Model performance was evaluated through repeated experiments and statistical validation.
Results
The YOLOv5 model demonstrated high accuracy in distinguishing between normal and limp gaits across species. Quantitative performance metrics confirmed the model’s reliability, and qualitative case studies highlighted its potential application in remote, fast traumatic assessment scenarios.
Conclusions
The use of AI, particularly deep convolutional neural networks like YOLOv5, shows promise in enabling fast, remote traumatic injury assessment during disaster response. This approach could assist healthcare professionals in identifying injury risks when physical access to patients is restricted, thereby improving triage efficiency and early intervention.
While flow confinement effects on a shear layer of an one-sided or submerged vegetation array’s interface have been widely studied, turbulent interactions between shear layers in channels with vegetation on both sides remain unclear. This study presents laboratory experiments investigating flow adjustments and turbulent interaction within a symmetrical vegetation–channel–vegetation system, considering varying array widths and densities. In the outer shear layer, the shear stress is primarily balanced by the pressure gradient. As the array extends laterally, the outer penetration of the shear layer reduces from a fully developed thickness to the half-width of the open region, resulting in flow confinement. Flow confinement enhances the pressure gradient, which increases the interior velocity and shear stress at the interface. Despite the time-averaged shear stress being zero at the centreline when the shear layer is confined, the shear instabilities from both sides interact, producing significant turbulent events at the centreline with equal contributions from each side. Furthermore, the two parallel vortex streets self-organised and created a wave response with a $\pi$-radian phase shift , where alternating vortex cores amplify the pressure gradient, intensifying coherent structures and facilitating momentum exchange across the channel centreline. Although the turbulent intensity is enhanced, the decreased residence time for turbulent flow events may limit transport distance. Overall, the shear layer that develops on one interface acts as an additional resistance to shear turbulence on the other interface, leading to a more rapid decline of shear stress in the open region, despite a higher peak at the interface.
Loneliness is a common public health concern, particularly among mid- to later-life adults. However, its impact on early mortality (deaths occurring before reaching the oldest old age of 85 years) remains underexplored. This study examined the predictive role of loneliness on early mortality across different age groups using data from the Health and Retirement Study (HRS).
Methods
A retrospective cohort study was conducted using data from the 2010–2020 waves of the HRS, restricted to participants aged 50–84 years at baseline. Loneliness was measured using the 11-item UCLA Loneliness Scale, categorized into four levels: low/no loneliness (scores 11–13), mild loneliness (14–16), moderate loneliness (17–20) and severe loneliness (21–33). Cox proportional hazards models and time-varying Cox regression models with age as the time scale were created to evaluate the relationship between loneliness and early mortality, adjusting for sociodemographic, lifestyle, and physical and mental health factors.
Results
Among 6,392 participants, the overall mortality rate before the age of 85 years was 19.1 per 1,000 person-years. A dose–response relationship was observed, with moderate and severe loneliness associated with 23% (adjusted hazard ratio [aHR]: 1.23, 95% confidence interval [CI] = 1.02–1.48) and 36% (aHR: 1.36, 95% CI = 1.13–1.65) higher mortality risk, respectively. Significant associations existed for the 65–74-year-old (aHR = 1.37, 95% CI = 1.03–1.83) and 75–84-year-old (aHR = 1.77, 95% CI = 1.23–2.56) age groups in the fully-adjusted models, but not for the 50–64-year-old age group. Time-varying Cox models showed a stronger association for severe loneliness (aHR = 1.65, 95% CI = 1.37–1.99).
Conclusions
Loneliness is a significant predictor of mortality among older adults. Preventive and interventional programs targeting loneliness may promote healthy ageing.
Elasto-inertial turbulence (EIT) has been demonstrated to be able to sustain in two-dimensional (2-D) channel flow; however the systematic investigations on 2-D EIT remain scarce. To address this gap, this study conducts direct numerical simulations of 2-D EIT at a modest Reynolds number ($Re=2000$) to examine its statistical characteristics and dynamic mechanisms. Meanwhile, this paper explores the similarities and differences between 2-D EIT with the maximum drag reduction (MDR) state in three-dimensional (3-D) flow. We demonstrate that statistical characteristics of 2-D EIT follow distinct trends compared to those in viscoelastic drag-reducing turbulence as nonlinear elasticity increases. These differences can be attributed to two different underlying dynamical processes: the gradual suppression of inertial turbulence in 3-D flow, and the progressive enhancement of EIT in 2-D flow. Also, we present the role of pressure, energy budget and spectral characteristics of 2-D EIT, which show significant similarities to those in the MDR state, thus providing compelling evidence for the 2-D nature of EIT. More strikingly, we identify an anomalous Reynolds stress in 2-D EIT that contributes negatively to flow resistance, which differs from the extremely small but positive Reynolds stress observed in the MDR state. Although with small values of Reynolds stress, the correlation analysis indicates clearly moderate positive correlation between the streamwise and normalwise velocity fluctuations rather than their being uncorrelated. Moreover, quadrant analysis of velocity fluctuations reveals the predominance of motions in the first and third quadrants, which are closely associated with the typical polymer extension sheet-like structures.
This paper presents a detailed technical overview of the femtosecond precision timing and synchronization systems implemented at the Shanghai high repetition rate XFEL and extreme light facility (SHINE). These systems are designed to deliver stabilized optical references to multiple receiver clients, ensuring high-precision synchronization between the optical master oscillator (OMO) and optical/RF subsystems. The core components include an OMO, fiber length stabilizers and laser-to-laser synchronization modules that achieve femtosecond-level accuracy. Our discussion extends to the various subsystems that comprise the synchronization infrastructure, including the OMO, fiber length stabilizer and advanced phase detection techniques. Finally, we highlight ongoing research and development efforts aimed at enhancing the functionality and efficiency of these systems, thereby contributing to the advancement of X-ray free-electron laser technology and its applications in scientific research.
Social determinants of health (SDHs) exert a significant influence on various health outcomes and disparities. This study aimed to explore the associations between combined SDHs and mortality, as well as adverse health outcomes among adults with depression.
Methods
The research included 48,897 participants with depression from the UK Biobank and 7,771 from the US National Health and Nutrition Examination Survey (NHANES). By calculating combined SDH scores based on 14 SDHs in the UK Biobank and 9 in the US NHANES, participants were categorized into favourable, medium and unfavourable SDH groups through tertiles. Cox regression models were used to evaluate the impact of combined SDHs on mortality (all-cause, cardiovascular disease [CVD] and cancer) in both cohorts, as well as incidences of CVD, cancer and dementia in the UK Biobank.
Results
In the fully adjusted models, compared to the favourable SDH group, the hazard ratios for all-cause mortality were 1.81 (95% CI: 1.60–2.04) in the unfavourable SDH group in the UK Biobank cohort; 1.61 (95% CI: 1.31–1.98) in the medium SDH group and 2.19 (95% CI: 1.78–2.68) in the unfavourable SDH group in the US NHANES cohort. Moreover, higher levels of unfavourable SDHs were associated with increased mortality risk from CVD and cancer. Regarding disease incidence, they were significantly linked to higher incidences of CVD and dementia but not cancer in the UK Biobank.
Conclusions
Combined unfavourable SDHs were associated with elevated risks of mortality and adverse health outcomes among adults with depression, which suggested that assessing the combined impact of SDHs could serve as a key strategy in preventing and managing depression, ultimately helping to reduce the burden of disease.
Previous L1 syntactic processing studies have identified the crucial left frontotemporal network, whereas research on L2 syntactic processing has shown that learner factors, such as L2 proficiency and linguistic distance, can modulate the related networks. Here, we developed a function-word-based jabberwocky sentence reading paradigm to investigate the neural correlates underlying Chinese L2 syntactic processing. Twenty Chinese L2 Korean native speakers were recruited in this fMRI study. Chinese proficiency test scores and Chinese-Korean syntactic similarity scores were measured to quantify the learner factors, respectively. The imaging results revealed an effective left frontoparietal network involving superior parietal lobule (SPL), posterior inferior frontal gyrus (pIFG) and precentral gyrus (PreCG). Moreover, the signal intensity of SPL as well as the connectivity strength between SPL and PreCG significantly correlated with the learner factors. These findings shed light on the neurobiological relationships between L1 and L2 syntactic processing and on the modulation of L2 learner factors.
For each $n\geq 1$, let $FT_n$ be the free tree monoid of rank n and $E_n$ the full extensive transformation monoid over the finite chain $\{1, 2, \ldots , n\}$. It is shown that the monoids $FT_n$ and $E_{n+1}$ satisfy the same identities. Therefore, $FT_n$ is finitely based if and only if $n\leq 3$.
The incorporation of trace metals into land snail shells may record the ambient environmental conditions, yet this potential remains largely unexplored. In this study, we analyzed modern snail shells (Cathaica sp.) collected from 16 sites across the Chinese Loess Plateau to investigate their trace metal compositions. Our results show that both the Sr/Ca and Ba/Ca ratios exhibit minimal intra-shell variability and small inter-shell variability at individual sites. A significant positive correlation is observed between the shell Sr/Ca and Ba/Ca ratios across the plateau, with higher values being recorded in the northwestern sites where less monsoonal rainfall is received. We propose that shell Sr/Ca and Ba/Ca ratios, which record the composition of soil solution, may be controlled by the Rayleigh distillation in response to prior calcite precipitation. Higher rainfall amounts may lead to a lower degree of Rayleigh distillation and thus lower shell Sr/Ca and Ba/Ca ratios. This is supported by the distinct negative correlation between summer precipitation and shell Sr/Ca and Ba/Ca ratios, enabling us to reconstruct summer precipitation amounts using the Sr/Ca and Ba/Ca ratios of Cathaica sp. shells. The potential application of these novel proxies may also be promising for other terrestrial mollusks living in the loess deposits globally.
Cavitation bubble pulsation and liquid jet loads are the main causes of hydraulic machinery erosion. Methods to weaken the load influences have always been hot topics of related research. In this work, a method of attaching a viscous layer to a rigid wall is investigated in order to reduce cavitation pulsations and liquid jet loads, using both numerical simulations and experiments. A multiphase flow model incorporating viscous effects has been developed using the Eulerian finite element method (EFEM), and experimental methods of a laser-induced bubble near the viscous layer attached on a rigid wall have been carefully designed. The effects of the initial bubble–wall distance, the thickness of the viscous layer, and the viscosity on bubble pulsation, migration and wall pressure load are investigated. The results show that the bubble migration distance, the normalised thickness of the oil layer and the wall load generally decrease with the initial bubble–wall distance or the oil-layer parameters. Quantitative analysis reveals that when the initial bubble–wall distance remains unchanged, there exists a demarcation line for the comparison of the bubble period and the reference period (the bubble period without viscous layer under the same initial bubble–wall distance), and a logarithmic relationship is observed that $\delta \propto \log_{10} \mu ^*$, where $\delta =h/R_{max}$ is the thickness of the viscous layer h normalised by the maximum bubble radius $R_{max}$, $\mu ^* = \mu /({R_{max }}\sqrt {{\rho }{{\mathop {P}\nolimits } _{{atm}}}})$ is the dynamic viscosity $\mu$ normalised by water density $ \rho $ and atmospheric pressure $P_{atm}$. The results of this paper can provide technical support for related studies of hydraulic cavitation erosion.