We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli’s principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi’s method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.

This study quantitatively investigates the two-dimensional pseudosteady shock refraction at an inclined air–water interface, referred to as the water wedge, in the weak and strong incident shock strength groups. Numerical simulations are employed to validate the predicted refraction sequences from a previous study (Anbu Serene Raj et al. 2024 J. Fluid Mech. 998, A49). A distinctive irregular refraction pattern, referred to as the bound precursor refraction with a Mach reflection, is numerically validated in the weak shock group. Based on the numerical simulations, an enhanced formulation is proposed to determine the sonic line of the incident flow Mach number ($M_b$) in water, thereby providing an appropriate transition condition for an irregular refraction with a Mach reflection to a free precursor refraction with a Mach reflection transition. Furthermore, comparative studies on solid and water wedges of wedge angle $20^\circ$ reveal discernible differences in the shock reflection patterns. The interplay of the energy dissipation due to the transmitted shock wave and the Richtmyer–Meshkov instability at the air–water interface results in the variation of the triple-point trajectory and transition angles between single Mach reflection (SMR) to transitional Mach reflection (TMR) occurring in air.

A prediction framework for the mean quantities in a compressible turbulent boundary layer (TBL) with given Reynolds number, free-stream Mach number and wall-to-recovery ratio as inputs is proposed based on the established scaling laws regarding the velocity transformations, skin-friction coefficient and temperature–velocity (TV) relations. The established velocity transformations that perform well for collapsing the compressible mean profiles onto incompressible ones in the inner layer are used for the scaling of such inner-layer components of mean velocity, while the wake velocity scaling is determined such that self-consistency is achieved under the scaling law for the skin-friction coefficient. A total of 44 compressible TBLs from six direct numerical simulations databases are used to validate the proposed framework, with free-stream Mach numbers ranging from 0.5 to 14, friction Reynolds numbers ranging from 100 to 2400, and wall-to-recovery ratios ranging from 0.15 to 1.9. When incorporated with the scaling laws for velocity transformation from Griffin et al. (2021, Proc. Natl Acad. Sci., vol. 118, e2111144118), the skin-friction coefficient from Zhao & Fu (2025, J. Fluid Mech., vol. 1012, R3) and the TV relation from Duan & Martín (2011, J. Fluid Mech., vol. 684, pp. 25–59), the prediction errors in the mean velocity and temperature profiles remain within $4.0\,\%$ and $6.0\,\%$, respectively, across all tested cases. Correspondingly, the skin-friction and wall-heat-transfer coefficients are also accurately predicted, with root mean square prediction errors of approximately $3 \,\%$. When adopting different velocity transformation methods that are valid for inner-layer scaling, the root mean square prediction errors in the mean velocity and temperature profiles remain below $2.3\,\%$ and $3.6\,\%$, respectively, which further highlights the universality of the proposed framework.

The spatio-temporal evolution of very large-scale coherent structures, also known as superstructures, is investigated in both smooth- and rough-wall boundary layers by means of direct numerical simulations up to a frictional Reynolds number of ${\textit{Re}}_\tau = 3\,150$. One smooth-wall and four rough-wall cases are considered, all developing over a region as long as $\sim$60 times the incoming boundary-layer thickness in the streamwise direction. Bio-inspired, biofouling-type topographies are employed for the rough-wall cases, following the previous work of Womack et al. (2022 J. Fluid Mech. vol. 933, p. A38) and Kaminaris et al. (2023 J. Fluid Mech. vol. 961, p. A23). We utilise three-dimensional time series, as well as multiple two-point correlation functions along the boundary layer to capture the detailed length- and time-scale evolution of the superstructures. The results suggest that the presence of roughness significantly amplifies both the strength and the streamwise growth rate of superstructures. Interestingly, however, their ratios relative to the local boundary-layer thickness, $\mathscr{L}_{\!x}/\delta$ and $\mathscr{L}_z/\delta$, remain constant and independent of the streamwise coordinate, indicating that such scaled length scales might constitute a possible flow invariant. Volumetric correlations revealed that all cases induce structures inclined with respect to the mean-flow direction, with those over the rough-wall topographies exhibiting steeper inclination angles. Finally, via proper orthogonal decomposition, pairs of counter-rotating roll modes were detected and found to flank the high- and low-speed superstructures, supporting the conjecture in the literature regarding the mechanisms responsible for the lateral momentum redistribution. The latter also suggests that the way momentum organises itself in high Reynolds number wall-bounded flows might be independent of the roughness terrain underneath.

Supersonic jets impinging on a ground plane produce a highly unsteady jet shear layer, often resulting in extremely high noise level. The widely accepted mechanism for this jet resonance involves a feedback loop consisting of downstream-travelling coherent structures and upstream-propagating acoustic waves. Despite the importance of coherent structures, often referred to as disturbances, that travel downstream, a comprehensive discussion on the disturbance convection velocity has been limited due to the challenges posed by non-intrusive measurement requirements. To determine the convection velocity of disturbances in the jet shear layer, a high-speed schlieren flow visualisation is carried out, and phase-averaged wave diagrams are constructed from the image sets. The experiments are conducted using a Mach 1.5 jet under various nozzle pressure ratios and across a range of impingement distances. A parametric analysis is performed to examine the influence of nozzle pressure ratio on the convection velocity and phase lead/lag at specific impingement distances. The results reveal that impingement tonal frequency is nearly independent of the disturbance convection velocity, except in cases of staging behaviour. They also demonstrated that slower downstream convection velocity of the disturbance corresponds to larger coherent structures, resulting in increased noise levels. Based on the observation of acoustic standing waves, an acoustic speed-based frequency model has been proposed. With the help of the allowable frequency range calculated from the vortex-sheet model, this model can provide a good approximation for the majority of axisymmetric impingement tonal frequencies.

The present article investigates the stability of Rayleigh–Bénard convection in a composite system consisting of a horizontal fluid layer overlying a fluid-saturated Darcy porous layer subjected to a time-periodic temperature distribution. The bottom surface is heated periodically with time, whereas a Biot number-dependent thermal boundary condition represents the heat transfer at the upper surface. The Beavers–Joseph–Saffman–Jones condition describes the ‘slip’ at the interface of the domains, and the Lions interface condition governs the normal force balance, incorporating a dynamic pressure term. The Chebyshev tau method and Fourier analysis are utilised to obtain linear instability bounds, which are compared with strong global and asymptotic limits derived from the nonlinear analysis using the energy method. Four deliberately chosen configurations of superposed fluid- and porous-layer systems are investigated. Two configurations validate the analysis through the limiting cases of the classical Darcy–Bénard and Rayleigh–Bénard systems obtained by setting the fluid-to-porous depth ratio $(\hat {d})$ to zero and infinity, respectively. The other two configurations involve layers with equal depths $(\hat {d} =1)$ and a shallow fluid layer overlying a porous layer $(\hat {d} \sim 0.1)$. For these cases, modulation substantially influences the onset of convection. In the last case, the linear theory points out that modulation parameters can control the dominant convective mode (fluid/porous). Furthermore, unlike the previously reported studies, the nonlinear stability bounds are found to be significantly lower than the linear instability bounds, indicating the possibility of subcritical instabilities in the presence of modulation. The region of subcritical instabilities increases with modulation amplitude.

Solid atmospheric particles, such as ice crystals, pollen, dust, ash and microplastics, strongly influence Earth’s climate, ecosystems and air quality. Previous studies have typically relied on analytical models valid only for very small particles or experiments in liquids, where the particle-to-fluid density ratio $R$ is much lower than values encountered in the atmosphere. We combine a novel experimental set-up with particle-resolved direct numerical simulations to study the settling of sub-millimetre ellipsoids in still air. Particle shapes span elongation and flatness values $ 0.2 \leqslant {\textit{EL}}, {\textit{FL}} \leqslant 1.0$ at a density ratio $ R = 1000$ and particle Reynolds numbers $ 2.1 \lt {\textit{Re}}_{\!p} \lt 4.5$, a regime well below the onset of wake-induced instabilities. Nonetheless, we observe unexpectedly rich dynamics: all non-spherical particles exhibit damped oscillatory motion, and some triaxial ellipsoids follow fully three-dimensional, non-planar trajectories due to rotation about all three axes. Simulations at lower density ratios ($ R = 10, 100$) confirm that these behaviours are driven by strong lateral forces happening only at $R=1000$. Key settling characteristics exhibit nonlinear and non-trivial dependencies on shape. In the two-dimensional phase space of elongation and flatness, settling velocity is symmetric about the principal diagonal ($ {\textit{EL}} = {\textit{FL}}$), while oscillation frequency and damping rate show symmetry about the anti-diagonal. Flatness strongly influences pressure drag, while elongation governs lateral drift and swept volume, which can reach up to ten times the particle diameter and four times the volume-equivalent sphere, respectively.

One-degree-of-freedom flow-induced vibration (FIV) and energy harvesting through FIV of an elastically mounted circular cylinder with mechanically coupled rotation were investigated numerically for low Reynolds number 100, mass ratio 8 and a wide range of reduced velocities. The aims of this study are to investigate the effect of the flow direction angle $\beta$ on the vibration and energy harvesting through FIV. Two types of lock-in are found: vortex-induced vibration (VIV) and galloping. The response amplitude increases with the increase of $\beta$ in both regimes. Both VIV response and galloping regimes are found for $\beta$ = 45° to $\beta$ = 90°. For $\beta$ = −90° to $\beta$ = 0°, only VIV response regimes are found. The fluid force and fluid torque play different roles in exciting/damping the vibration. In the high-amplitude gallop regime, the fluid force excites the vibration, and the torque damps the vibration. Energy harvesting at flow direction angle 90° is investigated as this flow direction has the maximum galloping amplitude. The energy harvesting is achieved by a linear electric damping coefficient in the numerical model. The maximum harvestable power in the galloping regime is significantly greater than that in the VIV regime, and it increases with the increase of the reduced velocity. When the reduced velocity is 20, the harvested power is over 20 times that in the VIV regime, and can further increase if reduced velocity further increases. The maximum efficiency over all simulated parameters is 0.424, occurring when the reduced velocity is 20, and electric damping factor is 0.04.

This research examines in detail the complex nonlinear forces generated when steep waves interact with vertical cylindrical structures, such as those typically used as offshore wind turbine foundations. These interactions, particularly the nonlinear wave forces associated with the secondary load cycle, present unanswered questions about how they are triggered. Our experimental campaigns underscore the occurrence of the secondary load cycle. We also investigate how the vertical distributions of the scattering force, pressure field and wave field affect the nonlinear wave forces associated with the secondary load cycle phenomena. A phase-based harmonic separation method isolates harmonic components of the scattering force’s vertical distribution, pressure field and wave field. This approach facilitates the clear separation of individual harmonics by controlling the phase of incident waves, which offers new insights into the mechanisms of the secondary load cycle. Our findings highlight the importance of complex nonlinear wave–structure interactions in this context. In certain wave regimes, nonlinear forces are locally larger than the linear forces, highlighting the need to consider the secondary load cycle in structural design. In addition, a novel discovery emerges from our comparative analysis, whereby very high-frequency (over the fifth in harmonic and order) oscillations, strongly correlated to wave steepness, have the potential to play a role in structural fatigue. This new in-depth analysis provides a unique insight regarding the complex interplay between severe waves and typical cylindrical offshore structures, adding to our understanding of the secondary load cycle for applications related to offshore wind turbine foundations.

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.

Floating particles deform the liquid–gas interface, which may lead to capillary repulsion or attraction and aggregation of nearby particles (e.g. the Cheerios effect). Previous studies employed the superposition of capillary multipoles to model interfacial deformation for circular or ellipsoidal particles. However, the induced interfacial deformation depends on the shape of the particle and becomes more complex as the geometric complexity of the particle increases. This study presents a generalised solution for the liquid–gas interface near complex anisotropic particles using the domain perturbations approach. This method enables a closed-form solution for interfacial deformation near particles with an anisotropic shape, as well as the varying height of the pinned liquid–gas contact line. We verified the model via experiments performed with fixed particles held at the water level with shapes such as a circle, hexagon and square, which have either flat or sinusoidal pinned contact lines. Although in this study we concentrate on the equilibrium configuration of the liquid–gas interface in the vicinity of particles placed at fixed positions, our methodology paves the way to explore the interactions among multiple floating anisotropic particles and, thus, the role of particle geometry in self-assembly processes of floating particles.

A new statistical definition for the mean turbulent boundary layer (TBL) thickness is introduced, based on identification of the wall-normal location where the streamwise velocity skewness changes sign, from negative to positive, in the outermost region of the boundary layer. Importantly, this definition is independent of arbitrary thresholds, and broadly applicable, including to past single-point measurements. Furthermore, this definition is motivated by the phenomenology of streamwise velocity fluctuations near the turbulent/non-turbulent interface (TNTI), whose local characteristics are shown to be universal for TBLs under low free-stream turbulence conditions (i.e. with or without pressure gradients, surface roughness, etc.) through large-scale experiments, simulations and coherent structure-based modelling. The new approach yields a TBL thickness that is consistent with previous definitions, such as those based on Reynolds shear stress or ‘composite’ mean velocity profiles, and which can be used practically, e.g. to calculate integral thicknesses. Two methods are proposed for estimating the TBL thickness using this definition: one based on simple linear interpolation and the other on fitting a generalised Fourier model to the outer skewness profile. The robustness and limitations of these methods are demonstrated through analysis of several published experimental and numerical datasets, which cover a range of canonical and non-canonical TBLs. These datasets also vary in key characteristics such as wall-normal resolution and measurement noise, particularly in the critical TNTI region.

A long-standing conceptual debate regarding the identification and independence of first Mack and cross-flow instabilities is clarified over a Mach 5.9 sharp wing at zero angle of attack and varying sweep angles. Their receptivity of the boundary layers to three-dimensional slow acoustic and vorticity waves is investigated using linear stability theory, direct numerical simulation and momentum potential theory (MPT). Linear stability theory demonstrates that the targeted slow mode appears as the oblique first mode at small sweep angles ($0^\circ$ and $15^\circ$) and transitions to the cross-flow mode at larger sweep angles ($30^\circ$ and $45^\circ$). Direct numerical simulation indicates that both the oblique first mode and cross-flow mode share identical receptivity pathways: for slow acoustic waves, the pathway comprises ‘leading-edge damping–enhanced exponential growth–linear growth’ stages. For vorticity waves, it consists of ‘leading-edge damping–non-modal growth–linear growth’ stages. Momentum potential theory decomposes the fluctuation momentum density into vortical, acoustic and thermal components, revealing unified receptivity mechanisms: for slow acoustic waves, the leading-edge damping is caused by strong acoustic components generated through synchronization. The enhanced exponential growth stage is dominated by steadily growing vortical components, with acoustic and thermal components remaining at small amplitudes. For vorticity waves, leading-edge disturbances primarily consist of vortical components, indicating a distinct mechanism from slow acoustic waves. Non-modal stages originate from adjustments among MPT components. Vortical components dominate the linear growth stage for both instabilities. These uniform behaviours between first Mack and cross-flow modes highlight their consistency.

This study applies the scaling patch approach to investigate the influence of pressure gradients on the mean-momentum balance in turbulent boundary layers (TBLs). Under strong pressure gradients, the force balance in the outer region is dominated by advective and pressure forces, with gradients of Reynolds stresses playing a minimal role. To retain the relevance of Reynolds stress gradients within the scaling patch framework, we propose a redistribution of the component $U_e \textrm {d}U_e/\textrm {d}x$ from the advective term to the pressure-gradient term. Here, $U_e$ is the mean streamwise velocity at the boundary layer edge. This reformulation enhances the outer-scaling framework of Wei & Knopp (2023 J. Fluid Mech. 958, 1–21), ensuring consistency across a wide range of pressure gradients, including those involving flow separation. Remarkably, the new outer-scaled gradient of Reynolds shear stress in TBLs under a pressure gradient closely resembles that observed in zero-pressure-gradient TBLs. In the inner region, the impact of pressure gradient is well captured by the Stratford–Mellor parameter $\beta _{\textit{in}}$. For weak pressure gradients ($|\beta _{\textit{in}}| \ll 0.07$), traditional inner scaling remains valid. However, for stronger pressure gradients $|\beta _{\textit{in}}| \gtrsim 0.07$, the near-wall dynamics is governed by a balance between pressure gradient and viscous force, as described by Stratford (1959 J. Fluid Mech. 5, 1–16) and Mellor (1966 J. Fluid Mech. 24, 255–274). In this sub-layer, viscosity and the imposed wall pressure gradient dictate the relevant velocity and length scales. Moreover, when $|\beta _{\textit{in}}| \gtrsim 0.7$ and the wall pressure $P_{w\textit{all}}$ gradient $\textrm { d}P_{w\textit{all}}/\textrm {d}x \gt 0$, a distinct sub-layer emerges outside the pressure–viscous balance region, characterised by a dominant balance between the imposed pressure gradient and the gradient of the Reynolds shear stress. In this region, the Reynolds shear stress increases linearly with distance from the wall. These findings provide new insights into the structure of TBLs under pressure gradients and establish a refined framework for modelling their dynamics.

Surface quasi-geostrophic (SQG) theory describes the two-dimensional active transport of a scalar field, such as temperature, which – when properly rescaled – shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with fully developed three-dimensional turbulence. In particular, the Kraichnan – Leith – Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling for both scalar and velocity fields, and the presence of intermittency through multifractal scaling was pointed out by Sukhatme & Pierrehumbert (2002 Chaos 12, 439–450), in unforced settings. In this work, we refine the discussion of these statistical analogies, using numerical simulations with up to $16\,384^2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, coupling velocity increments with scalar differences, develop well-defined scaling ranges, highlighting the role of anomalous fluxes of all the scalar moments. However, the clean multiscaling properties of SQG transport are blurred when considering velocity and scalar fields separately. In particular, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires us to resort to generalised notions of scale invariance, such as extended self-similarity and a specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: they evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions that have been repeatedly observed in SQG numerics for the past 20 years.

Directional freezing of brine is widely found in numerous environmental and industrial settings. Despite extensive studies, the microscopic evolution of ice-brine structures remains unclear. By combining in situ micro-computed tomography visualisation and theoretical analyses, we reveal new details inside the porous ice structure and its evolution towards a cleaner ice layer. We identify three distinct stages characterised by different brine exclusion rates during solidification: a rapid initial stage possibly lasting seconds from nucleation to local equilibrium without long-range heat or mass transfer; a second stage where the system reaches global thermal equilibrium, involving brine expulsion by volume expansion and convection associated with gravity; and a final prolonged stage dominated by diffusion. Comparison between analytical solutions and the migration rates of microstructural features such as brine stripes, columns and pockets extracted from photographic images confirms these understandings. Morphologically, we capture the formation of random striped patterns together with brine columns during downward freezing and brine skirts during upward freezing, all of which gradually transform into vertically aligned polygonal patterns. The volume fraction of brine pockets in porous ice near the cold end reduces to less than 10 % after 22 h in most experiments. The residual brine pockets, however, are not rejected out of the porous ice as fast as predicted by diffusion and remain persistent. Our findings provide new insights into the brine freezing dynamics, with implications ranging from sea ice formation to freeze desalination and general solidification of binary melts.

Flows enabled by phoretic mechanisms are of significant interest in several biological and biomedical processes, such as bacterial motion and targeted drug delivery. Here, we develop a homogenisation-based macroscopic boundary condition that describes the effective flow across a diffusio-phoretic microstructured membrane, where the interaction between the membrane walls and the solute particles is modelled via a potential approach. We consider two cases where potential variations occur (i) at the pore scale and (ii) only in the close vicinity of the boundary, allowing for a simplified version of the macroscopic flow description, in the latter case. Chemical interactions at the microscale are rigorously upscaled to macroscopic phoretic solvent velocity and solute flux contributions, and added to the classical permeability and diffusivity properties of the membrane. These properties stem from the solution of Stokes advection–diffusion problems at the microscale, some of them forced by an interaction potential term. Eventually, we show an application of the macroscopic model to develop minimal phoretic pumps, showcasing its suitability for efficient design and optimisation procedures.

Space–time correlations of velocity and high-Schmidt-number ($Sc \approx 2000$) passive scalar fields are investigated in turbulent pipe flow using particle image velocimetry and planar laser-induced fluorescence, respectively. Both the velocity and scalar fields exhibit characteristic elliptical patterns in their respective space–time correlations. The elliptic approximation model, originally developed for the velocity field, is applied to estimate convection and sweeping velocities for both fields. In both fields, the convection velocity decreases, while the sweeping velocity increases, along the pipe radius. The convection velocity ratio between the scalar and velocity fields shows that high-Schmidt-number scalar fluctuations are advected faster than the velocity fluctuations. Similarly, the sweeping velocity of the scalar fluctuations is found to be larger than that of the velocity fluctuations. Furthermore, the high-Schmidt-number scalar is found to decorrelate more rapidly than the corresponding velocity, with the scalar Taylor microscale distinctly smaller than the velocity Taylor microscale.

The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph & Fosdick (1973, Arch. Rat. Mech. Anal. vol. 49, pp. 321–380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Giesekus model. We begin by neglecting inertia to focus on the interaction between gravity, viscoelasticity and surface tension. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface shape on this short time scale, and recover the steady-state shape on a longer time scale. In contrast to the work of Joseph and Fosdick, which used the method of successive approximations to determine the steady shape of the interface, we explicitly derive the interface shape for both steady and transient cases. Subsequently, we reintroduce small but finite inertial effects to investigate their interaction with viscoelasticity, and propose a criterion for determining the conditions under which rod climbing occurs. Through numerical computations, we obtain the transient interface shapes, highlighting the interplay between time-dependent viscoelastic and inertial effects.