Carreau-Yasuda Nanofluid Research Articles

Repercussions of thermally stratified magnetic dipole for mixed convectively heated Darcy-Forchheimer Carreau-Yasuda nanofluid flow via viscous dissipation analysis

Nowadays nanotechnology is progressing speedily with the passage of time, consequently the research on innovative fluid termed as nanofluid is at its peak. Invention of nanofluid has made a revolution in engineering and chemical processes due to its enhanced heat transmission characteristics along with environmental friendliness as well as financially affordability of nanofluid are causes of its attraction among the research community. In current study the basic aim is to scrutinize a rheological ferrofluid model with magnetic effects. The procedure for conversion of coupled P.D. Es into coupled set of O.D.Es which is performed by use of similarity variables. The numerical scheme used here is bvp4c scheme for obtaining the required numerical outcomes. A comprehensive review of graphical illustration for fluid's temperature, velocity as well as concentration is delivered here. We inspected that with upsurge of parameter for Brownian motion resultantly concentration of fluid declines while increase in thermal profile is noted.

Entropy generation outcomes in peristalsis of Carreau–Yasuda nanofluid flow with activation energy

AbstractThis paper focuses on the study of activation energy and entropy generation in the peristaltic flow of a Carreau–Yasuda nanofluid. Peristaltic transport, coupled with entropy generation and activation energy in a curved geometry, has significant applications in biomedical and industrial processes involving non‐Newtonian fluids. Blood flow in the arteries, the movement of chyme in the gastrointestinal tract, and peristaltic motion replicate the natural muscular contractions that drive fluid flow in the physiological systems. The integration of entropy generation allows for the evaluation of energy efficiency and the analysis of losses due to viscous dissipation. Activation energy plays a crucial role in processes such as drug delivery, where temperature‐sensitive reactions or biochemical changes affect fluid behavior. In industrial applications, like peristaltic pumps in chemical reactors or polymer processing, the consideration of entropy and activation energy aids in optimizing thermal management and reaction rates. The mathematical model is developed under these assumptions, and the governing equations are numerically solved using the NDSolve function in Mathematica. Graphical results show that higher activation energy and concentration reduce the reaction rate, while the Bejan number and entropy generation increase with a larger Brinkman number. Velocity and temperature increase under slip conditions, whereas concentration decreases, and a stronger radial magnetic field reduces both velocity and the size of the trapped bolus.

Investigating thermal conduction dynamics in ternary Carreau–Yasuda nanofluids under convective boundary conditions and exponential heat source/sink effects

The analysis of the Carreau–Yasuda nanofluid (CYNF) across a stretching surface has practical applications in several fields, such as heat exchange, engineering, and material science. Scholars may discover novel perspectives for increasing heat transfer performance, establishing advanced materials, and enhancing the efficiency of multiple engineering systems by studying the behavior of CYNF in this particular instance. The energy transfer through trihybrid CYNF flow with the effect of magnetic dipole across a stretching sheet is examined in the present study. The ternary hybrid nanofluid (THNF) has been prepared by the addition of ternary nanoparticles (NPs) in the water (50%) and ethylene glycol (50%). Titanium dioxide (TiO2), silicon dioxide (SiO2), and aluminum oxide (Al2O3) are used in base fluid. The fluid velocity and heat transfer are examined under the impact of Darcy–Forchheimer, chemical reaction, convective condition, activation energy, and exponential heat source. The fluid flow has been stated in form of velocity, energy, and concentration equations. The set of modeled equations is simplified to non-dimensional form of ordinary differential equations (ODEs) by using similarity substitutions. Numerically, the system of lowest order ODEs is calculated through the parametric continuation method. It has been noticed that the temperature field augments against the variation of viscous dissipation and heat source term. Furthermore, the magnetic dipole has significant impact to enhance the thermal curve of THNF whereas falloffs the velocity profile. It can be noticed that the energy propagation rate enhances with the rising numbers of ternary NPs, from 1.22% to 5.98% (nanofluid), 2.30% to 9.61% (hybrid nanofluid), and 3.23% to 11.12% (trihybrid nanofluid).

Viscous dissipation and Joule heating in case of variable electrical conductivity Carreau–Yasuda nanofluid flow in a complex wavy asymmetric channel through porous media

This paper focuses on flow structures and thermal fields of the Carreau–Yasuda (CY) nanofluid model through a two-dimensional, wavy, complicated vertical asymmetrical conduit filled with porous elements. Formulations of the viscous dissipation in the case of CY nanofluids are derived and nonlinear radiation flux as well as joule heating are examined. Buongiorno’s nanofluid approach, which involves Brownian motion and thermophoresis aspects is considered. The electrical conductivity of the suspension is considered as a variable where it depends upon the ambient temperature and concentration distributions and the Joule heating impacts are not neglected. The approach of solving the problem is contingent upon converting the system to dimensionless form then the lubrication approach with low magnetic Reynold numbers is applied. Numerical solutions are found for the resultant system, and wide ranges are considered for Weissenberg number We, non-Newtonian parameter n and Darcy number [Formula: see text], namely, [Formula: see text], [Formula: see text] and [Formula: see text], respectively. The major results indicated that gradients of the pressure are higher in case of shear thickening [Formula: see text] comparing to in the instance of shear thinning [Formula: see text]. Also, the velocity is enhanced, close to the channel’s lowest portion, as the Weissenberg number is growing. The variable electrical conductivity gives a higher mass transfer rate compared to the constant property.

Theoretical investigation of MHD peristalsis of non‐Newtonian nanofluid flow under the impacts of temperature‐dependent thermal conductivity: Application to biomedical engineering

AbstractNanofluids have the tendency to improve thermal characteristics in a broader context, such as in medical processes, industrial cooling applications, the transportation industry, hybrid engines, gas temperature reduction, microelectromechanical systems, refrigerators, nuclear reactors, vehicle temperature control, pharmaceutical processes, thermal management of vehicles, microelectronics, and chillers, etc. Therefore, present study investigates the effects of a two‐dimensional magnetohydrodynamics (MHD) peristalsis of the Carreau–Yasuda nanofluid through conservation principles of energy, concentration, mass, and momentum, which is motivated by recent advances in biological engineering. The entire system is coupled through viscous dissipation, Joule heating, a heat sink/source, Brownian diffusion, a radially magnetic field, thermophoresis motion, and slip conditions in a curved geometry. Lubrication theory is employed in order to simplify governing equations. Resulting systems are nonlinear coupled differential equations whose exact solutions are difficult to obtain, therefore, numerical solutions are obtained through NDSolve in Mathematica. The key features of peristaltic movement and rates of heat transfer are disseminated in detail via graphical representations for variation of various physical parameters. According to the findings, for greater values of the Hartman number, temperature increases, and the velocity decreases. The validity of these results has been verified using the existing published articles of Hayat et al. (2018).

Analysis of entropy generation and activation energy on a convective MHD Carreau–Yasuda nanofluid flow over a sheet

In recent days, entropy generation has attracted the attention of several researchers due to its applications in manufacturing electronic devices, heat exchangers, conservation of energy, and generation of power. Further, activation energy is an essential requirement in automobile and chemical industries, nuclear reactors, and so on. This paper investigation aims to examine the entropy generation and the mass and energy transfer in the magneto-hydro dynamic movement of convective Carreau-Yasuda nano liquid past a porous stretching sheet with the consequences of activation energy, energy source, and binary chemical reaction. Also, the analysis of the impact of viscous and Joule dissipation in the presence of suction/injection into the Buongiorno model is an essential objective of this study. By introducing a similarity variable, the partial differential equations are altered into a system of ordinary differential equations. Numerical solutions to the modified equations are determined by utilizing a BVP5C MATLAB package. The findings are presented visually for several relevant flow characteristics and explained carefully. The thermophoresis parameter and activation energy parameter optimize the concentration of nanoparticles. These outcomes further indicate that as the larger Brinkmann and permeability parameters the entropy generation is broadened. It is discovered that an augmentation in thermal and solutal Grashof numbers is associated with greater velocity distribution.

Peristalsis of Carreau–Yasuda nanofluid possessing variable thermal conductivity regulated by electroosmosis via an expanding asymmetric channel

Peristalsis of non-Newtonian nanofluid via a non-uniform conduit has several applications in physiology and industry. This study proposes a mathematical model as well as exploration of the impacts of entropy generation for peristalsis of magneto nanofluid with temperature-dependent thermal conductivity via an expanding asymmetric channel. Buongiorno’s model for study of nanofluid flow has been adopted. Rheological aspects are accounted using the Carreau–Yasuda model due to its experimentally validated significance. Impacts of applied electric and magnetic fields have been taken into account. Thermophoresis, ohmic heating, viscous heating and Brownian motion effects are incorporated in the model. Numerical method is used via NDSolve in Mathematica after implementing the lubrication approach and Debye–Huckel linearization. The effects of embedded parameters on the flow, heat transfer, entropy generation and Bejan number are studied using graphical depictions. Outcomes indicate that a rise in electroosmotic parameter enhances heat transfer rate, whereas it reduces entropy generation and mass transfer rate at the boundary. Therefore, this study can provide basics to study nanofluidic systems which has applications in drug delivery. Increasing trends for temperature, heat transfer rate and entropy generation, while declining trends in the concentration of the particles are noted for higher joule heating.

Heat transfer analysis of Carreau–Yasuda nanofluid flow with variable thermal conductivity and quadratic convection

Abstract Brownian motions and Thermophoresis are primary sources of nanoparticle diffusion in nanofluids, having substantial implications for the thermo-physical characteristics of nanofluids. With such a high need, the 2D, laminar MHD (Magnetohydrodynamic) quadratic convective stream of Carreau–Yasuda nano liquid across the stretchy sheet has been reported. The flow is caused by surface stretching. The principal purpose of this extensive study is to enhance thermal transmission. The effects of variable thermal conductivity and heat source are considered as well. The governing boundary layer equations are transmuted using similarity parameters into a series of non-linear ODEs (ordinary differential equations). The bvp4c algorithm is adopted to fix the translated system numerically. The effects of prominent similarity variables over the temperature, velocity and concentration field are graphically visualized and verified via tables. It explored that fluid’s speed diminishes for the more significant inputs of the magnetic coefficient, Brownian motion coefficient and Prandtl number. The thermal efficiency is improved for larger values of thermophoretic constant, varying thermal conductance and heat-generating parameters. The concentration field has proved to be a decreasing function of nanofluid constants.

Mathematical modelling of entropy generation in radiative non-Darcy flow of a Carreau-Yasuda nanofluid in a rotating channel with activation energy, Hall current and the Cattaneo-Christov heat flux

The current research investigates entropy generation in magnetohydrodynamic rotating multiple slip flow of Carreau-Yasuda nanofluid between parallel plates within a non-Darcy porous medium, considering factors such as viscous dissipation, Joule heating, Hall current, and nonuniform heat generation. The Carreau-Yasuda model, applicable to non-Newtonian fluids like blood and shampoo, is employed to comprehend rheological behavior. This study holds significance in industrial, technological, and medical domains, impacting areas like drilling muds, metallurgy, and cardiac MRI. Nonlinear thermal radiation enhances model realism. The research delves into chemical reactions influenced by activation energy. The mathematical model is formed using nonlinear partial differential equations with multiple slip boundary conditions, and addressed using a compatibility-based similarity transformation. Numerical solutions employ the successive linearization coupled with Chebyshev spectral collocation method. MATLAB-generated graphs and tables illustrate the influence of flow parameters on the physical quantities. The investigation reveals inverse and favoring trends of the Weissenberg number to the energy field in shear-thinning and shear-thickening scenarios, respectively. Velocity slip is found to diminishes entropy production in the channel.

The impact of Carreau-Yasuda nanofluid flow on ciliated walls of the channel

The peristaltic transport of Carreau-Yasuda fluid model over a sinusoidal medium of non-Newtonian nano flow is examined. In this analysis, the channel walls are considered as ciliated with the higher and lower walls. The formulation of the 2D fluid model depends on the nonlinear (PDEs) that are transformed into simple ordinary (ODEs) by applying the dimensionless parameters and long-wavelength approximation. The numerical solution of the 2D system is taken out by applying the MATHEMATICA solver NDSolve function. The impact of various parameters, such as, flow rate parameterF, wave-length number δ, Brownian motion number Nb, thermophoresis number Nt, and account of eccentricity αon the velocity profile, temperature distribution, and concentration distribution and stream function are addressed with graphs.

Double diffusion Forchheimer flow of Carreau-Yasuda nanofluid with bioconvection and entropy generation: Thermal optimized analysis via non-Fourier model

The entropy generation phenomenon is important in thermodynamic systems, chemical engineering, industrial systems, reducing the energy consumptions and optimizing the resource utilization. Owing to such motivated applications, aim of current research is to present the applications entropy generation phenomenon in double diffusion flow Carreau-Yasuda nanofluid due to bidirectional stretched surface. The rheology of non-Newtonian fluid is observed by using the Carreau-Yasuda (CY) nonlinear model. The analysis is further supported with suspension of microorganisms with Carreau-Yasuda nanofluid. The formulated model is extended by entertaining the viscous dissipation, thermal radiation and activation energy consequences. The inertial effects are supported with Darcy Forchheimer approach. Additionally, non-Fourier and Fick theories are implanted to observe the thermal phenomenon. The equations resulted from the given model are analytically entertained via homotopy analysis scheme. The physical insight of thermal model is graphically presented. Moreover, the streamlines are plotted for flow model. An enhancing outcomes for entropy generation phenomenon are noted due to diffusion parameter and magnetic parameter. Moreover, the optimized phenomenon also increases due to Brinkman number.

Analysis of blood flow of unsteady Carreau-Yasuda nanofluid with viscous dissipation and chemical reaction under variable magnetic field

Blood flow analysis through arterial walls depicts unsteady non-Newtonian fluid flow behavior. Arterial walls are impacted by various chemical reactions and magnetohydrodynamic effects during treatment of malign and tumors, cancers, drug targeting and endoscopy. In this regard, current manuscript focuses on modeling and analysis of unsteady non-Newtonian Carreau-Yasuda fluid with chemical reaction, Brownian motion and thermophoresis under variable magnetic field. The main objective is to simulate the effect of different fluid parameters, especially variable magnetic field, chemical reaction and viscous dissipation on the blood flow to help medical practitioners in predicting the changes in blood to make diagnosis and treatment more efficient. Suitable similarity transformations are used for the conversion of partial differential equations into a coupled system of ordinary differential equations. Homotopy analysis method is used to solve the system and convergent results are drawn. Effect of different dimensionless parameters on the velocity, temperature and concentration profiles of blood flow are analyzed in shear thinning and thickening cases graphically. Analysis reveals that chemical reaction increases blood concentration which enhance the drug transportation. It is also observed that magnetic field elevates the blood flow in shear thinning and thickening scenarios. Furthermore, Brownian motion and thermophoresis increases temperature profile.

Interaction of variable diffusion coefficients with electrokinetically regulated peristalsis of Carreau-Yasuda nanofluid

In this analysis, the interaction of variable diffusion coefficients with electrokinetically modulated peristaltic motion of a radiative Carreau-Yasuda (CY) nanofluid is communicated. Here, ethylene glycol is used as the base fluid, and graphene nanoplatelets are considered nanoparticles. Unlike many existing studies, the thermophoretic diffusion term appearing in the equation of temperature and concentration is considered a variable term instead of manipulating their average values to obtain a more precise estimation of the conduct of the nanofluid. The Carreau-Yasuda (CY) model is considered to represent the non-Newtonian features of nanofluids. A mathematical model is developed under the effects of Joule heating, variable thermal conductivity, mixed convection, thermal radiation, electric field, Brownian motion, viscous dissipation, magnetic field, and thermophoresis assumptions. The entropy equation involving a variable diffusion coefficient and thermal conductivity is constructed using the second law of thermodynamics. The acquired dimensionless equations are numerically resolved with the help of built-in function. The graphical results indicate that due to Joule heating characteristics, the temperature profile increases. Manipulation of radiation and thermal conductivity parameters can improve the thermal efficiency and productivity of the system. An electroosmotic parameter causes an augmentation in the Bejan number. The nanofluid flow decreases at a higher electroosmotic velocity. The nanofluid chosen makes it possible to envisage industrial applications, for example in the field of heat extraction, without compromising mechanical performance.

Energy transmission through carreau yasuda fluid influenced by ethylene glycol with activation energy and ternary hybrid nanocomposites by using a mathematical model

The current study aims to assess the augmentation of energy transmission in the presence of magnetic dipole through trihybrid Carreau Yasuda nanofluid flow across a vertical sheet. The rheological properties and thermal conductivity of the based fluids are improved by framing an accurate combination of nanoparticles (NPs). The trihybrid nanofluid (Thnf) has been synthesized by the addition of ternary nanocomposites (MWCNTs, Zn, Cu) to the ethylene glycol. The energy and velocity conveyance has been observed in the context of the Darcy Forchhemier effect, chemical reaction, heat source/sink, and activation energy. The trihybrid nanofluid flow across a vertical sheet has been accurately calculated for velocity, concentration, and thermal energy in the form of a system of nonlinear PDEs. The set of PDEs is reduced to dimensionless ODEs by using suitable similarity replacements. The obtained set of non-dimensional differential equations is numerically computed through the Matlab package bvp4c. It has been perceived that the energy curve enhances by the influence of heat generation factor and viscous dissipation. It is also noted that the magnetic dipole has a momentous contribution to raising the transmission of thermal energy of trihybrid nanofluid and declines the velocity curve. The inclusion of multi-wall carbon nanotubes (MWCNTs), zinc (Zn), and copper (Cu) nano particulates to the base fluid “ethylene glycol”, augments the energy and velocity outlines.

Peristaltic flow of chemically reactive Carreau-Yasuda nanofluid with modified Darcy's expression

Nanofluids have recently been widely used in energy technologies and physiological processes. These fluids have already demonstrated significant interest in the thermal amplification of numerous industrial sectors. Therefore, the present work investigates the mixed convective peristaltic flow of Carreau-Yasuda nanoliquid in presence of slip conditions. Modified Darcy's expression for porous space is utilized. A symmetric channel with compliant wall characteristics is considered. The characteristics of Brownian motion and thermophoresis are included in the nanofluid model being considered. Outcomes of first-order chemical reactions and radiation are analyzed. The resulting problem is solved numerically by using the NDSolve technique via Mathematica. Features of pertinent variables against velocity, temperature and concentration are graphically discussed. Finally, heat transfer rate is examined through tabular values. Velocity of the fluid shows opposite behavior for porous medium and velocity slip parameter. Temperature is declines for radiation parameter. Heat transfer rate enhances for larger Brownian motion and thermophoresis variables.

Non-Fourier pseudoplastic nanofluidic transport under the impact of momentum slip and thermal radiation

In this modern era of research, fluid mechanics and particularly nanofluids flowing on the stretching surfaces are considered to be occurring more often due to their faster heat transfer mechanism. Mostly, the theoretical research related to nanofluids including this one contributed in studying the performance and reaction of nanomaterials against different physical constraints so that they have wide applications in several developed engineering industries like, atomic receptacles, transportation, microchip technology as well as biomedicine industry and food industry. Several fluids permeating in industrial and biological processes are pseudo-plastics. Their non-Newtonian behavior is usually related to structural reorganizations of the fluid molecules due to their flow. Keeping this in view, this study is an effort to investigate two-dimensional steady incompressible Cattaneo–Christov heat flux model of Carreau–Yasuda nanofluid in the presence of thermal radiation and velocity slip. The governed model is presented through partial differential equations which are transformed into ordinary differential equations by using similarity transformation. The final system of equations are solved numerically and the results are expressed through graphs and tables for fluid’s velocity temperature, concentration and physical quantities like heat and mass flux. It is observed that skin friction as well as local heat and mass flux decline with velocity slip parameter and heat transfer rises but mass flux at surface declines through increment in radiation parameter.

Analysis of entropy generation for Magnetohydrodynamics peristaltic motion of Carreau-Yasuda nanofluid through a curved channel with variable thermal conductivity and Joule heating

Present communication investigates the effects of entropy generation on the MHD peristaltic motion of Carreau–Yasuda nanofluid through a curved geometry with temperature-dependent thermal conductivity. Considered non-Newtonian model depicts the thermophysical properties of graphene nano-powder and Ethylene glycol. The assumed flow situation is modeled considering viscous dissipation, heat absorption/generation, Joule heating, thermophoresis effects, Brownian motion, and slip conditions using lubrication approximation. Numerical computations of resulting non-linear system of equations subject to the related boundary conditions are carried out using NDSolve. Physical behaviors of velocity profile, Bejan number, temperature profile, concentration profile, stresses at the wall, entropy production, mass, and heat transfer rates are computed and presented through graphs and tables. The results indicate that temperature rises and concentration of nanofluids reduces for larger values of thermal slip parameter. In the curved channel, the velocity of fluids decreases by enhancing the Hartman number. Entropy generation is reduced by increasing thermal conductivity parameter. Temperature distribution increases when the Eckert number is assigned higher values. Furthermore, the Hartman number and curvature parameter are observed to enhance the rates of mass and heat transfer at the wall. Additionally, the straight channel results can be obtained for the higher values of the curvature parameter.

Mixed convection and non-linear thermal radiative analysis for Carreau-Yasuda nanofluid in an endoscope

The understanding of flow behaviour of non-Newtonian fluids in the presence of nanoparticles is of great interest from an industrial point of view. Bulk addition of nano-sized particles in fluid enhance thermo-physical properties of fluid which is of crucial importance to industries such as medicine, agriculture, electronics etc. Current study discusses thermophoretic diffusion and Brownian motion in peristaltic flow of Carreau-Yasuda nanofluid in tube-like (endoscopic) structure. Unlike many existing studies, the nonlinear radiation effect in the presence of heat and mass transfer i.e., mixed convection is considered. Dissipation, Joule heating and heat source, sink effects are modelled. The governing mathematical problems are numerically solved. Non-dimensionalization is achieved via use of lubrication approach and streamline functions. Quantities like velocity, temperature, concentration, heat transfer rate and trapping are addressed. Observing effects of Weissenberg number (We = 0.1 − 0.3) on fluid flow conclude that velocity as well as the size of trapped bolus increases as We is increased. Heat transfer rate enhances as temperature ratio parameter (θ1 = 0.3 − 0.7) is increased.

Aiding or opposing electro-osmotic flow of Carreau–Yasuda nanofluid induced by peristaltic waves using Buongiorno model

Peristaltic flow in a symmetric channel containing Carreau–Yasuda (CY) nanofluid is studied through a Buongiorno model. The flow field is being assisted or opposed by electro-kinetic body force. Lubrication approximation along with Debye–Huckel linearization assumption is executed to simplify the governing model. Computational results are executed through a shooting method for values of parameters. The solutions contain a parameter measuring the relative strength of electro-osmotic force. In this study, values of are considered in the range . However, power-law index and Weissenberg number , appearing in the CY-model, are varied within and , respectively. Furthermore, Brinkman number is kept in the interval with Prandtl number assumed fixed at . A marked increase in the heat transfer coefficient is detected provided that is sufficiently large. No noticeable change in temperature profile is detected when Brownian motion and thermophoresis parameters are varied. For physically realistic results, these parameters are assumed to be of the order , as suggested in a recent study. A marked variation in the concentration field occurs whenever the ratio of Brownian to thermophoretic diffusion changes. Streamlines are also computed for a variety of parameter values, which reveal that the size of trapped boluses shrinks as we increase the strength of electro-osmotic force.

Overlapping grid spectral collocation approach for electrical MHD bioconvection Darcy–Forchheimer flow of a Carreau–Yasuda nanoliquid over a periodically accelerating surface

AbstractThe focus of this study is on computational grid‐manipulation to enhance the accuracy, convergence, and computational efficiency of the multidomain bivariate spectral local linearization method (MD‐BSLLM). The improved method is used in the scrutiny of Darcy–Forchheimer bioconvection flow of Carreau–Yasuda nanofluid induced by an oscillatory moving surface with cross‐diffusion, activation energy with binary chemical reactive species, combined electrical and magnetohydrodynamic field effects. The proposed method is utilized in solving the nondimensional form of the flow equations. Sensitivity and error analyses are provided to aid an understanding of the efficiency, stability, convergence rate, and accuracy of the iterative scheme. The impact of different parameter values on fluid properties and transport phenomena is discussed. Numerical simulation has indicated that the overlapping grid MD‐BSLLM is computational efficacious, and produces stable and sufficiently accurate results using a few collocation nodes in each respective subinterval and the entire computational domain. Other findings include the fact that fluid properties are enhanced with injection while flow characteristics are improved with suction. Using the Darcy–Forchheimer model in the flow analysis improves the temperature of the nanofluid. The imposition of electric field augments nanofluid velocity, the amplitude of skin friction coefficient, rates of mass, and motile microorganisms transfer while reducing the rate of heat transfer. The considered flow analysis can contribute towards engineering solicitations in paper production, polymer solution, metal extrusion, and glass blowing.