Casson Parameter Research Articles

Heat Transfer in Magnetohydrodynamic Convective Flow of Hybrid Nanofluid Over a Revolving Cone with Heat Generation/Absorption

The contemporary study’s goal is to investigate the role of the Casson hybrid nanofluid on boundary layer flow and heat transfer over a vertical rotating cone using various base fluids. The dynamic effects of the magnetic field and heat generation/absorption are taken into account in the modeling of hybrid nanofluids. Flow-related PDEs are remodeled to ODEs through use of similarity transmutations. Furthermore, the numerical results are explained using the fourth order Runge-Kutta scheme in conjunction with the shooting technique. The solution depends on a Lorentz force, Casson parameter, heat generation/absorption and spin parameter. The dependency of the skin friction coefficient and local Nusselt number on these four parameters is numerically explored. To the best of the author’s knowledge, the presence of three types of hybrid nanoparticles (Al2O3– TiO2, TiO2–Cu and Al2O3–Cu) with Newtonian/non-Newtonian base fluids has not yet been examined. The findings of the study reveal that increasing the magnetic parameter values reduces flow velocities (tangential and swirl) and increases the tangential velocity profile of Newtonian based hybrid nanofluid. Additionally, the thermal profile and non-Newtonian based hybrid nanofluid indicate a rising trend in heat generation/absorption parameter. Furthermore, the thermal transfer rate of water-based hybrid nanofluid is increased, while the skin friction coefficient reverses.

Stagnation point on MHD boundary layer flow of heat and mass transfer over a non-linear stretching sheet with effect of Casson nanofluid

ABSTRACT The present study analyzes the effects of the magnetohydrodynamic stagnation point and Casson fluid flow considered over the stretching surface. Due to the growing need for non-Newtonian nanofluid flows in industry and engineering areas, the present work focuses on the Casson nanofluid flow over a non-linear stretching surface with effects. The Casson nanofluid is more helpful for cooling and friction-reducing agents compared to Newtonian-based nanofluid flow. The mathematical explanation of the problem is elaborated with the help of partial differential equations. The coupled nonlinear form of ordinary differential equations (ODEs) is solved numerically by the Keller-Box method using a suitable MATLAB program. On the effect of the Casson parameter on the velocity parameter, it is observed that for large values of the Casson parameter, the velocity profile decreases. The reason behind this behavior is that increasing the values of the Casson parameter increases the fluid viscosity, i.e. reduces the yield stress. Therefore, the momentum boundary layer thickness reduces and the thermal boundary layer thickness increases.

Casson fluid flow with heat generation and radiation over a non-linear stretching sheet

This study investigates Casson fluid flow with heat generation and radiation over a non-linear stretching sheet. The resulting partial differential equations are converted to a system of ordinary differential equations using similarity transformation and solved numerically using shooting technique with fourth order Runge-Kutta method. The effect of radiation, heat source, and Casson parameters on the momentum and thermal boundary layers are examined through graphs using MATLAB. From the graphs it is seen that increase in the values of heat source and Prandtl number parameter increases the temperature profiles. The momentum boundary layer thickness decreases while the rate of heat transfer increases with increasing non-linear parameter. The temperature and thermal boundary layer thickness decreases with increasing radiation parameter values.

MHD Casson fluid flow with chemical reaction and suction in a porous medium over an exponentially stretching sheet

MHD Casson fluid flow with chemical reaction and suction in a porous medium over an exponentially stretching sheet was studied. The mathematical problem was solved numerically using shooting technique with fourth order Runge-Kutta method after converting the resulting partial differential equations to a system of ordinary differential equations and applying similarity transformation on the latter. The effects of some fluid parameters on the momentum and concentration profiles are examined through graphs using MATLAB. The results from the graphs show that increase in the values of Casson parameter decreases the velocity profiles and causes increase in the concentration profiles. The momentum boundary layers decrease while the solute boundary layers increase with increasing values of magnetic parameter and the permeability parameter. The concentration profile reduces with increasing values of reaction parameters.

Computational study of magnetized and dual stratified effects on Non-Darcy Casson nanofluid flow: An activation energy analysis

The current research looks into magnetohydrodynamics Casson nanofluid flow and suction/injection implications for a nonlinearly stretched interface. To improve heat transport, joules of heat, radiation impacts and thermal stratification are used. A Darcy-Forchheimer porous media is used to conduct fluid flow. Chemical reactions involving energies of activation and solutal stratum are additionally considered. The Keller-box technique is used to solve the resulting non-linear set of ordinary differential equations. The impacts of a number of variables are examined employing diagrams of quantity, velocity, and heat. Increased assessments of porousness, Darcy-Forchheimer, and Casson fluid characteristics result in a decrease in velocity trends. For improved measurements of Lewis number and solutal fractionated factors, the intensity pattern reduces. The skin friction factor increases with increasing Casson parameter estimate. In the instance of the porous and nonlinear stretching parameters, the Nusselt number rises. For increasing levels of the Lewis number and Brownian motion factors, the Sherwood number drops. Furthermore, Nusselt number ratios are estimated with Prandtl numbers with regard to current information available, which shows remarkable consistency. For varied incorporated parameters, streamlines, heatlines, and masslines additionally displayed.

Unsteady Magnetohydrodynamic Radiative Casson Nanofluid within Chemically Reactive Flow over a Stretchable Surface with Variable Thickness through a Porous Medium

This study presents a mathematical investigation into the phenomena of radiative heat with an unsteady MHD electrically conducting boundary layer of chemically reactive Casson nanofluid flow due to a pored stretchable sheet immersed in a porous medium in the presence of heat generation, thermophoretic force, and Brownian motion. The surface is assumed to be not flat, and has variable thickness. The magnetic field is time-dependent, and the chemical reaction coefficient is inversely varied with the distance. The nanofluid’s velocity, heat, and concentration at the surface are nonlinearly varied. A similarity transformation is introduced, and the controlling equations are converted into nondimensional forms involving many significant physical factors. The transformed forms are analyzed numerically using a computational method based on the finite difference scheme and Newton’s linearization procedure. The impact of the involved physical parameters is performed in graphical and tabular forms. Some special cases of the current work are compared with published studies, and an excellent agreement is obtained. The main results of the present work indicate that the higher values of the Casson parameter cause an increase in both the shear stress and heat flux, but a decrease in the mass flux. Also, it is noted that the chemical reaction, the nanoparticles’ volume, and the permeability factor enhance the effect the of Casson parameter on both the shear stress and heat flux, while the variable thickness and thermal radiation field reduce it; on the other hand, the variable thickness and nanoparticles’ volume enforce the influence of the Casson parameter on mass flux, but thermal radiation, the permeability factor, and chemical reaction decrease it. The present study has important applications in mechanical engineering and natural sciences. In addition, it has significant applications in devices used for blood transfusion, dialysis and cancer therapy.

Electroosmosis‐modulated Darcy–Forchheimer flow of Casson nanofluid over stretching sheets in the presence of Newtonian heating

A review of the literature revealed that nanofluids are more effective in transferring heat than conventional fluids. Since there are significant gaps in the illumination of existing methods for enhancing heat transmission in nanomaterials, a thorough investigation of the previously outlined models is essential. This study’s main objective is to examine the Casson nanofluid’s Darcy-Forchheimer flow across a stretching sheet. Investigations are being conducted on the viscous and Joule dissipations that the electroosmosis forces (EOF) have on the casson nanofluid boundary layer. The method transforms partial differential equations originating in nanofluidic systems into nonlinear differential equation systems with the proper degree of similarity. With a precision of order 4 to 5, the nonlinear nanofluid problem is solved using the (FDM) finite difference approach (Lobatto IIIA), which is accomplished using a number of collocation locations. The ability of Lobatto IIIA to handle coupled differential equations that are very nonlinear is one of its strengths. The boundary value dilemma (bvp4c) solver, which is a component of the MATLAB software programme, is used to reduce the higher order differential equations into a first order technique and computationally analyze the simplified mathematical model. When compared to previously published studies, the data acquired showed a high degree of accuracy and symmetry. The study’s primary results included that when the Casson fluid expands, the velocity field decreases, but the electric parameter, Forchheimer number, local Reynolds number, and permeability parameter show the opposite trend. Furthermore, High temperature is connected with the non-Newtonian heating parameter and the electric parameter. This work provides insights into practical applications such nanofluidic, energy conservation, friction reduction, and power generation. However, the work makes a significant point that the flow of a Casson fluid including nanoparticles can be regulated by appropriately modifying the Casson parameter, thermophoresis parameter, and Brownian motion parameter.

Numerical analysis of convective transport mechanisms in two-layer ternary (TiO 2 − SiO 2 − Al 2 O 3) Casson hybrid nanofluid flow in a vertical channel with heat generation effects

In this article, we investigate the flow dynamics and convective transport of nanofluid and ternary hybrid nanofluid in a two-layer vertical channel. Employing a Casson model with blood as the base fluid in both layers, the first layer incorporates nanofluid while the second layer comprises ternary hybrid nanofluid Laminar, incompressible flow with extra viscous dissipation effects is included in the formulation of this problem. The velocity and temperature are assumed to be continuous at the interface. The governing equations are converted into non-dimensional equations by using variables in the dimensionless form. The effective method known as homotopy analysis method (HAM) scheme is used to deal with the mathematical formulations. The behavior of velocity and temperature profile is sketched for various physical parameters and Nusselt number variations are tabulated. The results reveal that increasing the value of Eckert number raises the temperature. Temperature rises as and – – nanoparticles volume percentage rises. Nusselt number increases as Eckert number and Prandtl number increases while decreases as Casson parameters increase at the right wall. Rising values of and exhibit dissimilar behavior in the velocity profile. This study could be useful for medical purposes, like making better ways to deliver medicine through our blood vessels to treat cardiovascular diseases.

Heat generation (absorption) in 3D bioconvection flow of Casson nanofluid via a convective heated stretchable surface

In the past few years, the bioconvection phenomenon has become increasingly important in various fields such as genetic systems, manufacturing, biotechnology, and ecological sciences. Bioconvection refers to the large-scale movement of fluid driven by a density gradient resulting from the collective spinning of microorganisms, like bacteria and algae. These self-propelled motile microorganisms increase the fluid's density as they swim in a coordinated direction, giving rise to the bioconvection phenomenon. The current study's goals are to investigate the characteristics of heat generation and absorption in a three-dimensional mixed bioconvection flow of Casson nanofluid inside a stretching cylinder. To analyze the system, a set of transformed nonlinear equations is numerically solved using the Runge-Kutta algorithm. The graphical representation helps explain the impact of different variables on physical properties. Notably, higher values of the Casson parameter are associated with increased viscosity and flow resistance, while a higher bioconvection Rayleigh number leads to elevated skin friction and motile density. Also, Skin friction is decreased by increasing the stretching and mixed convection characteristics. Heat generating processes achieve higher motile densities, Nusselt, and Sherwood amounts than heat absorbing processes.

Effect of rotation on thermal instability under Hele-Shaw cell saturated by Casson nanofluid

This paper examines the effect of rotation on thermal instability under Hele-Shaw cell saturated by Casson nanofluid using both linear and nonlinear ways. The nanofluid model incorporates Brownian and thermophoresis diffusion. While conducting an analysis of nonlinear stability numerically using the truncated Fourier series method, analysis of linear stability is performed analytically using the normal mode methodology. The outcomes are all displayed graphically. The results demonstrate that the rotation has dual effect on Hele-Shaw parameter as well as Casson parameter, for higher value of rotation it has destabilizing effect and it stabilizing the system for lower values of rotation. Lewis number and concentration Rayleigh number promote the onset of convective motion within the system. On the other hand, rotation stabilize the system. Understanding the behavior of heat and mass transportation, the concentration of nanoparticles and fluid phase, utilize the Nusselt number when Nusselt numbers are assessed as a function of time, it is found that the variation of the rotation, Hele-Shaw and Casson parameter has a major influence on the heat and mass transfer. Both steady and unsteady weakly nonlinear analyses are performed to understand the heat transport in the system. It is concluded that the Casson nanofluid parameter has both stabilizing and destabilizing impact depending upon the rate of rotation and therefore this work can be possibly utilized in both places, where heat removal and heat conservation are required.

A 3D flow analysis of Casson Nano-fluid flow over a stretching surface with non-Fourier heat and non-Fick’s mass flux

The current study delves into the analysis of 3-D flow of a Casson nanofluid over a convectively heated stretching sheet. This investigation takes into account the effects of non-Fick’s mass and non-Fourier heat flux. An efficient semi-analytical technique called homotopy analysis method (HAM) with BVPh2.0 package is utilized to obtain the series solution and calculate the residual errors of the governing system of equations. Influences of the flow governing parameters were examined through graphs and tables. The influences of the different arbitrary constants in velocity, temperature, and concentration profiles were graphically shown and interpreted while the friction coefficients, Nusselt number, and Sherwood numbers were numerically evaluated. It is found that the Prandtl number Pr and the temperature profile θ(η) are inversely related, while the Biot number, the magnetic and Casson fluid parameters, M and β are all favorably correlated. The Casson parameter β and the magnetic parameter M have positive effects on the concentration profile φ(η) whereas Prandtl number Pr has a negative effect. When the Casson fluid parameter is raised, the skin friction coefficient decreases. On the contrary, at increasing values of the Prandtl number Pr, the local Nusselt number drops. In addition, the local Sherwood number decreases as the Brownian motion parameter rises.

Radiative heat transport of Cattaneo‐Christov double diffusive Casson nanofluid flow between two rotating disks with Hall current and activation energy

AbstractThe present research uncovers the mathematical exploration of the magnetohydrodynamic radiative Casson nanofluid flow produced due to stretching and coaxially rotating surfaces of two disks inside a non‐Darcy porous medium under the dominance of the diacritic Hall current and heat generation. The energy field is displayed under the dominance of radiative and dissipative heat transport by involving the consequences of distinctive nonlinear thermal radiation, viscous dissipation, and Joule heating. The present study overcomes the barrier of heat and mass transport by annexing Cattaneo‐Christov double diffusion effects. The current exploration shows the characteristics of the concentration field due to arising chemical reaction incited by activation energy. A substantive mathematical problem is modeled by assigning nonlinear partial differential equations together with convective boundary conditions. A compatible similarity transformation comprised in the current study is exerted to produce a set of nonlinear ordinary differential equations with competent boundary conditions. The resulting mathematical model is numerically solved via dispensing the SLM. The present article deals with an in‐depth exploration of diagnostic flow parameters' attributes against the flow field and efficient physical quantities with the help of distinctive graphs and tables. As per the current investigation, the energy field becomes potent due to enhancing the stretching of each of the disks. Besides, augmenting the thermophoretic diffusion and chemical reaction boosts diluting the concentration of nanoparticles. The flow along the radial direction gets controlled near the upper disk under the increasing influence of the Hall current, intensity of the magnetic field, and stretching rate of the lower disk. On the other hand, the enlarging Casson parameter, rotation parameter, and stretching parameter for the lower disk lead to controlling the flow along the radial direction adjacent to the lower disk. Apart from that, the intense effects of the stretching rate of the upper disk and rotation rein axial flow throughout the flow region.

Optimal Transient Energy Growth of Two-Dimensional Perturbation in a Magnetohydrodynamic Plane Poiseuille Flow of Casson Fluid

Abstract The study investigates the influence of the Casson fluid parameter and the spanwise uniform magnetic field on the onset of instability against infinitesimal disturbances in an electrically conducting fluid flow between two parallel nonconducting rigid plates. The fourth-order linearized disturbance equation governing stability is solved using the spectral collocation method with Chebyshev-based polynomials. The aim is to analyze in detail the effect of the parameters involved in the problem using both modal and nonmodal linear stability analysis. The modal analysis provides accurate values of the critical Reynolds number, critical wave number, and critical wave speed, denoted as critical triplets (Rc, αc, cc). Additionally, it examines the eigen-spectrum, growth rate curves, and neutral stability curves. On the other hand, the nonmodal analysis investigates the transient energy growth G(t) of two-dimensional (2D) optimal perturbations, the pseudospectrum of the non-normal Orr–Sommerfeld (O–S) operator (ℒ), and the regions of stability, instability, and potential instability of the fluid flow system. The extensive examination of both long-term behavior through modal analysis and short-term behavior through nonmodal analysis reveals that the Hartmann number (Ha) acts as a stabilizing agent, delaying the onset of instability. Conversely, the Casson parameter (η) acts as a destabilizing agent, advancing the onset of instability. The results obtained here are verified to be in good agreement with the existing literature in the absence of the Casson fluid flow parameter.

Investigation of dissipative heat transfer and peristaltic pumping on MHD Casson fluid flow in an inclined channel filled with porous medium

A theoretical investigation is performed for free convective peristaltic pumping of a Casson fluid through an inclined porous wavy channel. The flow is subjected to uniform magnetic field in the transverse direction. In addition, the energy equation contains porous and viscous dissipation effects. The governing flow problem is modeled for Casson fluid with the help of conservation laws of mass, momentum, and energy under the long wavelength assumption. Using a regular perturbation method, we obtained the analytical expressions for the axial velocity, temperature, pressure rise per one wavelength, and heat transfer rate. The consequences of various effects on the flow quantities are demonstrated in the form of graphical representations and discussed in detail. The findings reveal that the rise in the thermal Grashof number and permeability parameter leads to an increment in the velocity and thermal fields. The heat transfer rate strengthened when the Casson parameter and magnetic parameter were increased. The pressure rise exhibits an enhancing trend for the Brinkmann number and permeability parameter. Further, we observed a decreasing behavior on streamlines for increasing magnetic field strength. Moreover, the obtained findings are applicable to a variety of fields in the bioengineering and medical sciences, such as targeted drug delivery, heart-lung machines, MRI, cancer therapy, power conversion devices, and micromanufacturing processes.

Unsteady MHD Casson Fluid Through a Porous Medium over an Inclined Plate in the Present of Chemical Reaction

The unsteady of MHD Casson fluid flow through an infinite inclined plate in the presence of chemical reaction is investigated. In this research, Casson fluid model is choose to be used in this study to characterize the non-Newtonian fluid behaviour. We are employed the Laplace transform technique with an appropriate boundary condition to convert the governing partial differential equations into ordinary differential equations. All the transformed equations are then solved numerically by using Mathematica. Finally, we were obtained the exact solution of momentum, energy and concentration of the cases. The results of flow features for different values of the governing parameters, unsteadiness parameter, Casson parameter are analysed in graphs and has been discussed in the result section. The results presented that fluid velocity rises with the increment of magnetic parameter due to the present of Lorentz force. Lorentz force helps in reducing heat for electronic system and radiators. Furthermore, the increasing of inclination angle and chemical reaction make the velocity increases. The result for Casson fluid explained that this fluid helps to lower the resistance of the yield stress so that the velocity become higher.

Numerical treatment via the spectral collocation method for Casson–Williamson nanofluid flow due to a stretching sheet with slip conditions

The main goal of the research is to investigate the effects of temperature and concentration slip on the MHD Casson–Williamson nanofluid flow through a porous sheet, with a focus on viscous dissipation. This research has a novel approach, as it seeks to incorporate various factors that have not been previously studied in this context. Moreover, an inclined magnetic field from the outside is applied to the stretched surface. Besides this, it is considered that the viscosity of nanofluids depends on temperature while all other physical characteristics are taken to be constant. It is hypothesized that the expanding surface that stretched exponentially is what caused the flow motions of the nanofluid. There are several engineering applications for this type of study, which is based on MHD non-Newtonian nanofluid flow across a stretched surface with heat generation and viscous dissipation. They include solar energy storage, energy distribution, chemical reactors, and the manufacturing of polymers. To examine the heat transmission and mass transfer rates, tools like thermophoresis and Brownian motion were implemented. The flow problem is formally represented as a set of nonlinear PDEs that are then, through the use of dimensionless variables, converted into a set of ODEs. To numerically obtain the solution to the problem, the shifted Chebyshev polynomials of the third-kind approximation along with the spectral collocation technique are utilized. This process transforms the existing model into an algebraic equation system that was created as a restricted optimization problem, which is then optimized to obtain the solution and the unknown coefficients. For a variety of pertinent parameter values, the graphic and tabular representations of the concentration, temperature, velocities, rate of heat mass transfer, and shear stresses of nano-fluids at the surface of the sheet are shown. Finally, there is a remarkable amount of agreement between the earlier investigation and the comparison research that was conducted. The findings suggest that higher values of the magnetic parameter, Casson parameter, viscosity parameter, and suction parameter lead to a decrease in both velocity and boundary layer thickness. Furthermore, an increase in the viscosity parameter, Casson parameter, and magnetic parameter results in elevated values for both temperature and concentration.

Multi-objective optimization of electrokinetic energy conversion efficiency and entropy generation for streaming potential driven electromagnetohydrodynamic flow of couple stress Casson fluid in microchannels with slip-dependent zeta potentials

The present study develops a new mathematical approach to the pressure-mediated streaming potential flow of couple-stress Casson fluids with the influence of slip-dependent zeta potential between two parallel plates in a microchannel. The lateral electric and transverse magnetic fields are externally applied to control the fluid flow. A multi-objective optimization technique was implemented by utilizing the non-dominated sorting genetic algorithm (NSGA-II) by varying different constraints to maximize the electrokinetic energy conversion efficiency and minimize the irreversibility of the system. The present results show that the couple stress and Casson parameters influence the development of streaming potential, electrokinetic energy conversion (EKEC) efficiency, temperature, Nusselt number, and Bejan number for asymmetric velocity slip and temperature jump circumstances. The viscous dissipation, Joule heating, and thermal radiation parameters significantly affect the heat transfer. According to the present results, the EKEC efficiency increases with a slip-dependent zeta potential of couple stress Casson fluid, couple stress fluid, Casson fluid, and Newtonian fluid by 45.92 %, 47.10 %, 42.81 %, and 37.68 %, respectively, compared to that for the slip-independent zeta potential. At the centre of the microchannel for the jump boundary conditions, temperature increases by 2.95 % compared to no jump boundary conditions.

Shape factor analysis in stagnation point flow of Casson nanofluid over a stretching/shrinking sheet using Cattaneo–Christov model

In this article, the stagnation point flow of an incompressible Au–blood fluid over a stretching sheet in porous medium is examined with suction/injection and Cattaneo–Christov effects. The transformed ODEs are solved numerically by the shooting-based bvp4c method with MATLAB software. The flow phenomenon is influenced by thermal relaxation with slip boundary conditions. The shape of gold (Au) nanoparticle is considered as: cylindrical, platelet, and blade, and the Hamilton–Crosser model of thermal conductivity is taken. The Casson parameter increases the nanofluid velocity and decreases the fluid temperature. The increment in the thermal relaxation parameter increases the fluid temperature profiles. The particle shape can also enhance the temperature. The nanoparticles of blade shape have huge influences on temperature rise due to greater thermal properties. A comparative analysis is made in accordance with previous works and is in good agreement.

Internal heat source effects on thermosolutal convection of Casson nanofluids embedded by Darcy-Brinkman model

The current study investigates Darcy-Brinkman convective instability for a non-Newtonian binary nanofluid layer in the presence of internal heat source. Non-Newtonian fluid behavior is characterized by Casson model. Normal mode technique is employed to convert the relevant PDEs into ODEs and Boussinesq approximation is used. The phenomenon is explored for both stationary as well as oscillatory modes of convection. The novelty and significance of the work lie in the fact that oscillatory motions have come into existence due to the presence of internal heat source term in the thermal energy equation which were non-existent otherwise for top-heavy configuration of nanoparticles. The impact of various nanofluid and solute parameters except solute Rayleigh number is found to destabilize the nanofluid layer while solute Rayleigh number, Darcy number and porosity parameters are found to postpone the onset of convection. Non-Newtonian Casson parameter and internal heat source strength are found to hasten the convection for the present arrangement of nanoparticles.

Magnetized casson SA-hybrid nanofluid flow over a permeable moving surface with thermal radiation and Joule heating effect

Here in, heat transfer characteristics of a magnetized Casson sodium alginate-alumina/copper hybrid nanofluid flowing through the permeable moving plane are investigated considering the effects of thermal radiation and Joule heating with Tiwari and Das model. The primary objective of the present study is to investigate the influence of the velocity ratio parameter on the solid volume fraction and the Casson parameter. In addition, the velocity and temperature profiles used in the current study illustrate the effects of the velocity ratio parameter R, solid volume fraction, suction effect, magnetic field, Joule heating, and thermal radiation. To apply the three-stage Lobatto IIIa with boundary value problems (bvp4c) approach to implement the appropriate similarity variables to transform the partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs). It has been determined that there are two branch solutions for the velocity ratio parameter against using the Casson parameter and copper volume fraction values. Non-unique branch at R < Rci however, no branch happens at R > Rci. As the Eckert number and radiation grew, the rate of heat transmission surged. The increase of Casson hybrid nanofluid parameter values caused to drop the skin friction in both upper and lower branch.