Axisymmetric Flow Research Articles

Curved detonation equations with analysis and its applications

In this paper, curved detonation equations with gradients for the pre-wave and post-wave are constructed followed by analysis, verification and applications. The study focuses on shock induced chemical reaction such as detonation, with the energy effect for the main attention. Equations consider both planar and transverse curvature to accommodate both planar and axisymmetric flow problems. Influence coefficients are derived and used to analyse the effect of energy and curvature on the post-wave gradient. Good agreement with the simulation results demonstrates that the equations presented in this paper can calculate various post-wave gradients accurately. After verification, the equations can be applied to applications, including not only solution and analysis but also in the inverse design. First, the method can be applied with polar analysis to provide a new perspective and higher order parameters for the study of detonation. Second, the equations can be used for the capture of detonation waves, where both planar and axisymmetric examples show better performance. Furthermore, the equations can be used in the inverse design of detonation waves in combination with the method of characteristics, which is one of the unique benefits of the present equations.

Thermal instability of three-dimensional boundary layer stagnation point flow towards a rotating disc

In this paper, the thermal instability of a three-dimensional boundary layer axisymmetric stagnation point flow towards a heated horizontal rotating disk is considered. A large number of works have been done on stability analysis. However, they did not check the thermal stability of the non-parallel-flow in the face of small disturbances that occur in the vicinity of the heated rotating disk. The governing equations of the basic flow are reduced to three coupled nonlinear partial differ-ential equations, and solved numerically with the fourth-order Runge-Kutta method. Thermal stability is examined by making use of linear stability theory based on the decomposition of the normal mode of Görtler-Hammerlin. The resulting eigenvalue problem is solved numerically using a pseudo-spectral method based on the expansion of Laguerre’s polyno-mials. The obtained results are discussed in detail through multiple configurations. As the main result, for large Prandtl numbers (Pr), the rotation disk parameter (Ω) has a destabilizing effect while for small Pr (around the unity) it tends to stabilize the basic flow. It was found that as the disk radius r→0, the flow is linearly stable, and the disturbances grow rapidly away from the stagnation point. For low values of Pr, the flow becomes more stable, and strong thermal gradients are necessary to destabilize it. However, an increase in Pr leads to a significant expansion of the instability region.

Binary collision dynamics of immiscible Newtonian and non-Newtonian fluid droplets

This experimental and theoretical study is devoted to the investigation of head-on collisions of two immiscible Newtonian and non-Newtonian droplets. The density of the two droplets is similar, and the viscosity of 0.3% carboxymethyl cellulose droplet is slightly larger than 10 cSt silicone oil. The sizes and relative velocity of the colliding droplets close to the point of impact are measured by means of image processing. The deformed states after the impact and their evolution with time are studied by experimental visualization and the energy evolution with time are discussed by numerical results. The accuracy of the two-dimensional axisymmetric three-phase flow computational model is validated. We study the effects of collisions of non-Newtonian droplets with Newtonian droplets and the subsequent retraction kinetics. Droplet “cannibalization” is commonly observed: after collision and spreading, the droplet retracts rapidly, resulting in a Newtonian droplet wrapping around a non-Newtonian droplet. We show the whole process of droplet collision captured by a high-speed camera and obtain the cloud and velocity vector maps of the droplets by numerical simulation. The droplet wrapping phenomenon is produced by different three-phase interfacial tensions and viscosities. We delineate the different phases of the collision process and discuss the dominant forces in each phase. We calculate the energy evolution of the spreading phase and use it to derive a predictive model for the dimensionless maximum spreading diameter and spreading time.

Isolator shape transition impact on hypersonic internal waverider intake flow distortion

This paper examines the impact of scramjet isolator shape transition on hypersonic internal waverider (IWR) intake. The IWR intake is designed using the osculating axisymmetric flows and streamline-tracing methods. The new Internal Conical Flow “M” basic flowfield is utilized to provide the flow information for the design method. The intake is equipped with three isolators: one with a constant cross section and two with variable cross sections with circular and rectangular exits. The entrance shape and area of the three isolators are fixed to the intake throat shape and area. The exit area of the three isolators is maintained as the entrance one. Numerical computations of three-dimensional configurations reveal that the isolators with variable cross section shapes demonstrate a higher uniformity index than those with constant cross section shape. Thus, the isolator shape transition has decreased the flow distortion of the hypersonic IWR intake system. The three isolators exhibit varied wall pressure distribution depending on the isolator cross section shape, and the total pressure recovery ratios at the three isolators' exit planes are similar. The wall pressure distributions and key performance parameters at the intake throat section, including total pressure recovery, compression ratio, and Mach number, remained consistent across the first part of the intakes. Therefore, changing the cross section shape of the isolator while keeping the area constant could enhance the flow uniformity of compressed air without negatively impacting the intake system's performance. This allows a separate shape selection of the IWR intake throat and the scramjet combustor entrance to fulfill their special requirements.

Radiative influence on axisymmetric ternary hybrid nanofluid flow with convective boundary conditions over a nonlinearly permeable stretching/shrinking disk

PurposeIn recent times, ternary hybrid nanofluid has garnered attention from scientist and researchers due to its improved thermal efficiency. This study aims to delve into the examination of ternary hybrid nanofluid (Al2O3–Cu–TiO2/water), particularly concerning axisymmetric flow over a nonlinearly permeable stretching/shrinking disk. In addition, the investigation of convective boundary conditions and thermal radiation effects is also considered within the context of the described flow problem.Design/methodology/approachMathematical formulations representing this problem are reduced into a set of ordinary differential equations (ODEs) using similarity transformation. The MATLAB boundary value problem solver is then used to solve the obtained set of ODEs. The impact of considered physical parameters such as suction parameter, radiation parameter, nonlinear parameter, nanoparticle volumetric concentration and Biot number on the flow profiles as well as the physical quantities is illustrated in graphical form.FindingsThe findings revealed the thermal flux for the nonlinearly shrinking disk is approximately 1.33%, significantly higher when compared to the linearly shrinking disk. Moreover, the existence of dual solutions attributed to the nonlinear stretching/shrinking disk is unveiled, with the first solution being identified as the stable and reliable solution through temporal stability analysis.Practical implicationsUnderstanding ternary hybrid nanofluid behavior and flow has applications in engineering, energy systems and materials research. This study may help develop and optimize nanofluid systems like heat exchangers and cooling systems.Originality/valueThe study of flow dynamics across nonlinear stretching/shrinking disk gains less attention compared to linear stretching/shrinking geometries. Many natural and industrial processes involve nonlinear changes in boundary shapes or sizes. Understanding flow dynamics over nonlinear shrinking/stretching disks is therefore essential for applications in various fields such as materials processing, biomedical engineering and environmental sciences. Hence, this study highlights the axisymmetric flow over a nonlinear stretching/shrinking disk using ternary hybrid nanofluid composed of alumina (Al2O3), copper (Cu) and titania (TiO2). Besides, this study tackles a complex problem involving multiple factors such as suction, radiation and convective boundary conditions. Analyzing such complex systems can provide valuable insights into real-world phenomena where multiple factors interact.

Laminar swirling slender pipe flows

Swirling flows are extensively used to enhance mixing in chemical and physical processes. Although many fundamental studies involve decaying laminar swirls in circular ducts with constant area—for which several theoretical descriptions exist—converging and diverging pipe sections are often employed, demanding the extension of the analysis to swirling laminar flows in pipes with varying areas. This work analytically investigates axial swirl decay in laminar axisymmetric flows in converging and diverging pipes at moderate Reynolds numbers. The theoretical model extends the classical theory for laminar low swirling flows in straight pipes with impermeable walls to pipes with slowly varying areas. Inlet Reynolds number and converging/diverging angle effects are analyzed for incompressible axisymmetric laminar flows in slender pipes (lengths much greater than the radius). Results show that diverging pipes enhance the swirl decay rate, whereas converging systems have a weakening effect for flows with the same inlet Reynolds number. Conversely, increasing the inlet angle in converging flows decreases the decay rate, whereas larger diverging pipe inlet angles promote swirl decay. A simplified eigenvalue analysis shows that the relationship between the swirl decay lengths in diverging (ℓd,div), straight (ℓd,st), and converging (ℓd,conv) pipes is given by ℓd,div<ℓd,st<ℓd,conv for flows with similar inlet Reynolds numbers. Comparisons with numerical simulations show that the model predictions are almost exact for nondimensional pipe lengths up to 5×10−2 times the inlet Reynolds number. Approximating the axial velocity profile through a fully developed function representative of laminar flows with heat transfer through the walls shows that the narrower axial velocity profile in diverging flows increases viscous effects, enhancing the swirl decay rate. On the contrary, the flatter axial velocity profile in converging pipes further extends the decay length.

Influences of Newtonian heating and Darcy's law in micropolar fluid flow over a magnetized stretchable disk: A Bayesian analysis

A laminar two-dimensional magnetized micropolar fluid flow passed through a stretchable surface is examined. The axi-symmetric time-independent thermal flow experienced the impacts of Darcy's law, thermal conductivity, and Newtonian heating. The dimensionalized flow model is governed by similarity transformations. The resulting non-linear flow system is numerically solved through Runge-Kutta-Fehlberg (RKF-45) built in scheme. The graphics illustration of velocity and thermal fields for multiple physical parameters is presented. The numerical values at the disk surface for couple stresses, thermal rate, and shear stresses are also calculated. A Bayesian approach is used to assess the association degree amongst the under-study constraints and variables. The magnitude of the microrotational profiles is enhanced by the combined effect of micropolar parameters while an opposite effect is observed in microrotational field against magnetic and porosity parameters. The temperature field is modified through enhancing values of conjugate parameters. Moreover, such thermal field is larger for variable thermal conductivity as assumed to constant fluid property.

Numerical investigation of an axisymmetric, dissipative, and unstable micropolar fluid flow over a stretched Riga plate incorporating mixed convection and chemical reaction

AbstractThe demand to improve the thermal performance of industrial working fluids and materials for better quality in engineering and household devices has spurred numerous studies on non‐Newtonian viscous fluids under various conditions. Researchers have explored different structures and geometries to understand and enhance the thermal propagation of fluid particles. As such, this research aims to investigate the effects of two‐dimensional unsteady boundary layer flow phenomena of a dissipative micropolar fluid over a radially stretching Riga plate when first‐order slip, thermal and solutal boundary conditions, mixed convection, and thermal radiation are all considered. Ordinary differential equations are the governing equations resulting from converting partial differential equations. The boundary layer conditions are altered and numerically solved using the Runge–Kutta fourth‐order shooting approach. Graphs and tables are used to visually analyze the physical attributes of temperature, concentration, velocity distributions, skin friction, Nusselt number, and Sherwood in relation to various factors. The results reveal significant insights into the impact of mixed convection and chemical reactions on the flow characteristics and dimensions. These findings have important implications for various engineering applications, particularly in enhancing industrial systems' heat and mass transfer processes, providing practical knowledge for improving thermal performance in real‐world applications.

Thermal Diffusion and Diffusion Thermo Effects on Axi-Symmetric Boundary Layer Flow of Nanofluid Due to Non-Linear Stretching Sheet along the Radial Direction in presence of Magnetic field

Thermal Diffusion and Diffusion Thermo Effects on Axi-Symmetric Boundary Layer Flow of Nanofluid Due to Non-Linear Stretching Sheet along the Radial Direction in presence of Magnetic field

Numerical method research and characteristics analysis on frozen start-up process of high-temperature heat pipe

Future space exploration technology requires a long-life and reliable power source that is not reliant on solar energy. Space micro-reactors are able to meet this need, with Heat Pipe Cooled Reactors (HPR) emerging as a notable type of space micro-reactor that has attracted widespread attention in recent years. The HPR utilizes high-temperature alkali metal heat pipes for heat transfer, which presents certain complexities due to the solid state of the alkali metals working medium at room temperature. This results in a three-phase transition during the high-temperature heat pipes start-up process, which significantly impacts the heat transfer characteristics and dynamic behavior of the HPR start-up process. Consequently, thorough research is necessary in this area. Numerical simulation is a crucial tool that can effectively analyze, predict, and guide experiments. This article utilizes the Finite Volume Method (FVM) to develop a simulation code for high-temperature heat pipe frozen start-up. Various physical models are integrated to describe different components of the heat pipe: the container is represented by a two-dimensional axisymmetric heat conduction equation, the wick region utilizes a Fixed Grid Method (FGM) to depict the melting process of the medium, and the vapor channel is described through a two-dimensional axisymmetric compressible laminar flow. The wick region and vapor channel are coupled through the evaporation and condensation of the medium. For the vapor channel, numerical methods such as SIMPLE and PISO are used for solving. Adaptive time step and OpenMP acceleration are employed in the code to enhance computational efficiency. Finally, by comparing the calculated results with experimental data, the feasibility and accuracy of the code are assessed, highlighting special phenomena during the start-up process. The findings confirm that the developed code accurately predicts parameter changes during start-up, and can serve as a heat pipe analysis module for multi-physics coupling analysis of HPR.

The effect of slip parameter in an axisymmetric oscillatory Stokes flow

A general solution of Stokes equations for the problem of an axisymmetric oscillatory flow of an incompressible, viscous fluid past a sphere satisfying general boundary conditions is obtained. The behavior of the magnitude of drag is observed with the variation of the slip parameter. Some more interesting behaviors are detailed, and several existing results pertaining to steady flows and flows with rigid and shear free boundary conditions are recovered as special cases. The corresponding results are discussed for four different axisymmetric oscillatory Stokes flows, namely, uniform flow, flows generated due to a dipole, a source, and a Stokeslet. A few interesting streamline patterns like formation, elongation, and disappearance of viscous eddies in the vicinity of the sphere with a periodic reversal of the flow are observed at different frequencies for different values of the slip parameter.

Axisymmetric phase-field-based lattice Boltzmann model for incompressible two-phase flow with phase change

In this work, a phase-field-based lattice Boltzmann equation (LBE) model for axisymmetric two-phase flow with phase change is proposed. Two sets of discrete particle distribution functions are employed to match the conserved Allen–Cahn equation and the hydrodynamic equations with phase change effect, respectively. Since phase change occurs at the interface, the divergence-free condition of the velocity field is no longer satisfied due to mass transfer, and the conserved Allen–Cahn equation needs to be equipped with a source term dependent on the phase change model. To deal with these, a novel source term in the hydrodynamic LBE is delicately designed to recover the correct target governing equations. Meanwhile, the LBE for the Allen–Cahn equation is modified with a discrete force term to model mass transfer. In particular, an additional correction term is added into the hydrodynamic LBE to reduce the spurious velocity and improve numerical stability. Several axisymmetric benchmark multiphase problems with phase change, including bubble growing in superheated liquid, D2 law, film boiling, bubble rising in superheated liquid under gravity, and droplet impact on a hot surface, have been conducted to test the performance of the proposed model. Numerical results agree well with analytical solutions and available published data in the literature.

Memory Effects in the Magnetohydrodynamic Axial Symmetric Flows of Oldroyd-B Fluids in a Porous Annular Channel

In this article, we analytically investigate the isothermal magnetohydrodynamic axial symmetric flows of ordinary and fractional incompressible Oldroyd-B fluids through a porous medium in an annular channel. The fluid’s motion is generated by an outer cylinder, which moves along its symmetry axis with an arbitrary time-dependent velocity Vh(t). Closed-form expressions are established for the dimensionless velocity fields of both kinds of fluids, generating exact solutions for any motion of this type. To illustrate the concept, two particular cases are considered, and the velocity fields corresponding to the flow induced by the outer cylinder are presented in simple forms, with the results validated graphically. The motion of fractional and ordinary fluids becomes steady over time, and their corresponding velocities are presented as the sum of their steady and transient components. Moreover, the steady components of these velocities are identical. The influence of magnetic fields and porous media on the flow of fractional fluids is graphically depicted and discussed. It was found that a steady state is reached earlier in the presence of a magnetic field and later in the presence of a porous medium. Moreover, this state is obtained earlier in fractional fluids compared with ordinary fluids.

Numerical investigations of shock/boundary-layer interaction and control for Mach 2.5 flow in an axisymmetric inlet

The present work investigates the application of vortex generators for separation control in axisymmetric isolator flows. Reynolds-Averaged Navier-Stokes computations were performed to simulate a Mach 2.5 flow past a half-isolator geometry with a 20∘ axisymmteric compression ramp based on the experiments of Funderburk and Narayanaswamy at North Carolina State University. Single vortex generator and a vortex generator array placed upstream of the shock-induced separation region were investigated. Near wall streamlines, surface pressure contours and density contours on crosswise planes were compared with experiments for flow control using a single vortex generator. Results indicate that the present computations are able to capture the wake and shock structures, and also predict reduction in the streamwise extent of flow separation downstream of the device trailing edge. Finally, the effects of the shape/orientation (forward facing/backward facing) of a single vortex generator, and the design of an array of three vortex generators (with devices of similar and mixed orientations) on the flow separation were also investigated. Contours of near-surface streamwise velocity showed that device orientation had a strong effect on separation control, which is attributed to the differences in the primary streamwise vortices shed in the two cases. Further, the overall reduction in the footprint of separated flow was determined to be most with the use of an array of vortex generators with mixed orientations.

Mechanism of active flow control using a novel spike aerodome-channel concept in the hypersonic flow: A numerical study

Among the design requirements of hypersonic vehicles, reducing aerodynamic heating and drag force simultaneously is the main challenge. This paper proposes a novel spike aerodome-channel combination concept to realize the flow field reconstruction around the hypersonic blunt body. The novel configuration is investigated in the axisymmetric flow at a Mach number of 6 at zero angle of attack. The two-dimensional Reynold-averaged Navier–Stokes equations are numerically solved, and the shear-stress transport k–ω model is the turbulence model implemented in this study. Parameters such as spike length and lateral jet location are investigated to explore the drag and heat reduction performance and the flow control features. The obtained results show that the application of the novel spike aerodome-channel concept alters the flow field by eliminating or replacing the strong bow shock wave, and the design of hypersonic vehicles can benefit from the application of the proposed concept. The blunt body coupled with a frustum of cone-tipped spike-channel configuration provides a remarkable drag reduction effect of 20.71% with respect to the case without channel. Considering the effect of lateral jet location, the drag reduction performance of the case with LR = 0.75 is superior to that of the root jet case at the same spike length, and a considerable drag reduction of 28.93% is obtained with L/D = 2.4. In addition, longer spike length is beneficial for improving drag reduction performance, while excellent efficiency of heat protection is obtained in a certain spike length range. For the case of L/D = 1.6 with root jet, the peak Stanton number is significantly decreased by 33.51%.

Improved Design Method for Hypersonic Intakes Using Pressure-Corrected Osculating Axisymmetric Flows Method

The efficient design of air intake is crucial for high-speed airbreathing propulsion systems. This paper presents a novel design method that integrates the recently introduced internal conical flow M (ICFM) basic flowfield (Musa et al., “New Parent Flowfield for Streamline-Traced Intakes,” AIAA Journal, Vol. 61, No. 7, 2023, pp. 2906–2921) with the pressure-corrected osculating axisymmetric flows and streamline tracing techniques. The new design method takes into account the effects of lateral pressure gradients by incorporating pressure corrections into the original design method using crossflow velocity information from the ICFM flowfield. Additionally, new entrance and throat profiles are introduced for designing two hypersonic internal waverider intakes with shape transition. One intake is designed using the new method, and the other using the original method. Viscous simulations have been performed at design (i.e., Mach 6.0) and off-design (i.e., Mach 4.0 and 3.5) conditions. The results indicate that the pressure-corrected intake exhibits superior performance in terms of total pressure recovery and reduced flow distortion. This reveals that the new design method can achieve a smooth shape and efficient aerodynamic transitions between the intake entrance and throat. The new entrance and throat profiles realized better performance than the typical ones. The startability analysis reveals that the Van Wie curve controls the considered intakes. Thus, combining the pressure-corrected osculating axisymmetric flows with the ICFM flowfield enhances the performance of hypersonic intakes, making this approach a promising avenue for future hypersonic vehicles.

Heat and mass transport micropolar Maxwell and Williamson nanofluids flow past a perpendicular cylinder using combined convective flow

The current study aims to investigate the heat and mass transport characteristics of micropolar Maxwell and Williamson nanofluids flowing past a perpendicular cylinder under the influence of combined convective flow using the Buongiorno nanofluid model. The objective is to analyze the axisymmetric flow of these nanofluids around an orthogonal cylinder, highlighting the effects of various physical parameters on temperature profiles and velocity distributions. Maple 23 software was employed to solve the coupled nonlinear differential equations derived from appropriate similarity transformations. The numerical results are presented in tabular and graphical form to show the impacts of key parameters on the selected micropolar nanofluids. The significant outcomes show that the skin friction coefficient, as well as the Nusselt and Sherwood numbers, increase along the axial direction, indicating enhanced heat and mass transfer capabilities. Additionally, the study emphasizes the roles of micro polarity, relaxation time, and viscoelastic properties in modulating these transfer processes. These findings have significant implications for applications in biomechanics, polymer manufacturing, aerosol deposition, and thermal treatment processes, offering valuable insights for future research and industrial practices.

Axisymmetric stagnation-point flow of a fractional Oldroyd-B hybrid nanofluid over a vertical cylinder by improved Buongiorno's model

This paper investigated the unsteady axisymmetric stagnation-point flow, heat and mass transfer of Oldroyd-B hybrid nanofluid over a vertical cylinder by using Caputo time fractional derivatives for the first time, with the consideration of second-order velocity slip conditions. Improved Buongiorno's model was adopted to build the governing equations and finite difference combined with L1-algorithm was employed to solve the numerical solutions. Effects of key physical parameters on velocity, heat and mass transfer were presented graphically and discussed in detail. Outcomes show that the fractional derivatives are capable to describe the memory effect and thermoelasticity of viscoelastic nanofluids. Both Brownian motion and thermophoresis, especially Brownian motion, improve the heat transfer on the cylinder wall. In addition, the sensitivity of average Nusselt (or Sherwood) number to velocity fractional derivates α and β rises with increasing α and β, while the sensitivity to temperature fractional derivate γ declines with increasing γ. When γ increases from 0.2 to 0.3, the relative growth rate of average Nusselt number is about 9.26 %. This study provides a reference for exploring the stationary-point flow of fractional order nanofluids on vertical cylinders.

Newton-Krylov continuation of amplitude-modulated rotating waves in sheared annular electroconvection.

We present an approach for studying the primary, secondary, and tertiary flow transitions in sheared annular electroconvection. In particular, we describe a Newton-Krylov method based on time integration for the computation of rotating waves and amplitude-modulated rotating waves, and for the continuation of these flows as a parameter of the system is varied. The method exploits the rotational nature of the flows and requires only a time-stepping code of the model differential equations, i.e., it does not require an explicit code for the discretization of the linearized equations. The linear stability of the solutions is computed to identify the parameter values at which the transitions occur. We apply the method to a model of electroconvection that simulates the flow of a liquid crystal film in the smectic A phase suspended between two annular electrodes and subjected to an electric potential difference and a radial shear. Due to the layered structure of the smectic A phase, the fluid can be treated as two-dimensional (2D) and is modeled using the 2D incompressible Navier-Stokes equationscoupled with an equationfor charge continuity. The system is a close analog to laboratory-scale geophysical fluid experiments and thus represents an ideal system in which to apply the method before its application to these other systems that exhibit similar flow transitions. In the model for electroconvection, we identify the parameter values at which the primary transition from steady axisymmetric flow to rotating waves occurs, as well as at which the secondary transition from the rotating waves to amplitude-modulated rotating waves occurs. In addition, we locate the tertiary transition, which corresponds to a transition from the amplitude-modulated waves to a three-frequency flow. Of particular interest is that the method also finds a period-doubling bifurcation from the amplitude-modulated rotating waves and a subsequent transition from the flow resulting from this bifurcation.

Consistent and conservative lattice Boltzmann method for axisymmetric multiphase electrohydrodynamic flows

In this work, we first develop a consistent and conservative mathematical model to study multiphase electrohydrodynamic (EHD) flows, including the reduction-consistent and conservative Allen-Cahn equation for the multiphase field, the Laplace equation for the electric potential, and the consistent and conservative Navier–Stokes equations for the flow field. Owing to the reduction-consistent property, the present model can be used to handle a set of M (1≤M≤N) EHD fluids in N-phase system. Then a two-relaxation-time lattice Boltzmann (LB) method is proposed for axisymmetric multiphase EHD flows, which can correctly recover the axisymmetric governing equations through the direct Taylor expansion analysis. To test the capacity of the LB method, the deformations of a single two-phase droplet and a ternary-phase compound droplet under a uniform electric field are considered. It is found that different from a single droplet with two deformation modes, the compound droplet has four deformation modes, and additionally, the effects of the electric strength, the conductivity ratio and the permittivity ratio on the compound droplet are also investigated. Finally, the compound droplet in the quaternary-phase system is also explored, and there are eight deformation modes that have not been reported in the available literature.