Von Kármán rotating flow of Maxwell nanofluids featuring the Cattaneo-Christov theory with a Buongiorno model

This research paper analyzes the transport of thermal and solutal energy in the Maxwell nanofluid flow induced above the disk which is rotating with a constant angular velocity.The significant features of thermal and solutal relaxation times of fluids are studied with a Cattaneo-Christov double diffusion theory rather than the classical Fourier's and Fick's laws.A novel idea of a Buongiorno nanofluid model together with the Cattaneo-Christov theory is introduced for the first time for the Maxwell fluid flow over a rotating disk.Additionally,the thermal and solutal distributions are controlled with the impacts of heat source and chemical reaction.The classical von Karman similarities are used to acquire the non-linear system of ordinary differential equations(ODEs).The analytical series solution to the governing ODEs is obtained with the well-known homotopy analysis method(HAM).The validation of results is provided with the published results by the comparison tables.The graphically presented outcomes for the physical problem reveal that the higher values of the stretching strength parameter enhance the radial velocity and decline the circumferential velocity.The increasing trend is noted for the axial velocity profile in the downward direction with the higher values of the stretching strength parameter.The higher values of the relaxation time parameters in the Cattaneo-Christov theory decrease the thermal and solutal energy transport in the flow of Maxwell nanoliquids.The higher rate of the heat transport is observed in the case of a larger thermophoretic force.     

Rotational flow of Oldroyd-B nanofluid subject to Cattaneo-Christov double diffusion theory

A nanofluid is composed of a base fluid component and nanoparticles, in which the nanoparticles are dispersed in the base fluid. The addition of nanoparticles into a base fluid can remarkably improve the thermal conductivity of the nanofluid, and such an increment of thermal conductivity can play an important role in improving the heat transfer rate of the base fluid. Further, the dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The present predominately predictive modeling studies the flow of the viscoelastic Oldroyd-B fluid over a rotating disk in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov heat and mass flux expressions. The characteristic of the Lorentz force due to the magnetic field applied normal to the disk is studied. The Buongiorno model together with the Cattaneo-Christov theory is implemented in the Oldroyd-B nanofluid flow to investigate the heat and mass transport mechanism. This theory predicts the characteristics of the fluid thermal and solutal relaxation time on the boundary layer flow. The von K′arm′an similarity functions are utilized to convert the partial differential equations(PDEs) into ordinary differential equations(ODEs). A homotopic approach for obtaining the analytical solutions to the governing nonlinear problem is carried out. The graphical results are obtained for the velocity field, temperature, and concentration distributions. Comparisons are made for a limiting case between the numerical and analytical solutions, and the results are found in good agreement. The results reveal that the thermal and solutal relaxation time parameters diminish the temperature and concentration distributions, respectively. The axial flow decreases in the downward direction for higher values of the retardation time parameter. The impact of the thermophoresis parameter boosts the temperature distribution.     

Thermal analysis in swirl motion of Maxwell nanofluid over a rotating circular cylinder

In this paper, the mechanism of thermal energy transport in swirling flow of the Maxwell nanofluid induced by a stretchable rotating cylinder is studied. The rotation of the cylinder is kept constant in order to avoid the induced axially secondary flow. Further, the novel features of heat generation/absorption, thermal radiation, and Joule heating are studied to control the rate of heat transfer. The effects of Brownian and thermophoretic forces exerted by the Maxwell nanofluid to the transport of thermal energy are investigated by utilizing an effective model for the nanofluid proposed by Buongiorno. The whole physical problem of fluid flow and thermal energy transport is modelled in the form of partial differential equations(PDEs) and transformed into nonlinear ordinary differential equations(ODEs) with the help of the suitable flow ansatz.Numerically acquired results through the technique bvp4c are reported graphically with physical explanation. Graphical analysis reveals that there is higher transport of heat energy in the Maxwell nanoliquid for a constant wall temperature(CWT) as compared with the prescribed surface temperature(PST). Both thermophoretic and Brownian forces enhance the thermal energy transport in the flowing Maxwell nanofluid. Moreover, the temperature distribution increases with increasing values of the radiation parameter and the Eckert number. It is also noted that an increase in Reynolds number reduces the penetration depth, and as a result the flow and transport of energy occur only near the surface of the cylinder.     

Boundary layer flow of Maxwell fluid due to torsional motion of cylinder: modeling and simulation

This paper investigates the boundary layer flow of the Maxwell fluid around a stretchable horizontal rotating cylinder under the influence of a transverse magnetic field. The constitutive flow equations for the current physical problem are modeled and analyzed for the first time in the literature. The torsional motion of the cylinder is considered with the constant azimuthal velocity E. The partial differential equations (PDEs) governing the torsional motion of the Maxwell fluid together with energy transport are simplified with the boundary layer concept. The current analysis is valid only for a certain range of the positive Reynolds numbers. However, for very large Reynolds numbers, the flow becomes turbulent. Thus, the governing similarity equations are simplified through suitable transformations for the analysis of the large Reynolds numbers. The numerical simulations for the flow, heat, and mass transport phenomena are carried out in view of the bvp4c scheme in MATLAB. The outcomes reveal that the velocity decays exponentially faster and reduces for higher values of the Reynolds numbers and the flow penetrates shallower into the free stream fluid. It is also noted that the phenomenon of stress relaxation, described by the Deborah number, causes to decline the flow fields and enhance the thermal and solutal energy transport during the fluid motion. The penetration depth decreases for the transport of heat and mass in the fluid with the higher Reynolds numbers. An excellent validation of the numerical results is assured through tabular data with the existing literature.     

Transportation of heat through Cattaneo-Christov heat flux model in non-Newtonian fluid subject to internal resistance of particles

Thermal conduction which happens in all phases(liquid,solid,and gas) is the transportation of internal energy through minuscule collisions of particles and movement of electrons within a working body.The colliding particles comprise electrons,molecules,and atoms,and transfer disorganized microscopic potential and kinetic energy,mutually known as the internal energy.In engineering sciences,heat transfer comprises the processes of convection,thermal radiation,and sometimes mass transportation.Typically,more than one of these procedures may happen in a given circumstance.We use the Cattaneo-Christov(CC) heat flux model instead of the Fourier law of heat conduction to discuss the behavior of heat transportation.A mathematical model is presented for the Cattaneo-Christov double diffusion(CCDD) in the flow of a non-Newtonian nanofluid(the Jeffrey fluid) towards a stretched surface.The magnetohydrodynamic(MHD) fluid is considered.The behaviors of heat and mass transportation rates are discussed with the CCDD.These models are based on Fourier's and Fick's laws.The convective transportation in nanofluids is discussed,subject to thermophoresis and Brownian diffusions.The nonlinear governing flow expression is first altered into ordinary differential equations via appropriate transformations,and then numerical solutions are obtained through the built-in-shooting method.The impact of sundry flow parameters is discussed on the velocity,the skin friction coefficient,the temperature,and the concentration graphically.It is reported that the velocity of material particles decreases with higher values of the Deborah number and the ratio of the relaxation to retardation time parameter.The temperature distribution enhances when the Brownian motion and thermophoresis parameters increase.The concentration shows contrasting impact versus the Lewis number and the Brownian motion parameter.It is also noticed that the skin friction coefficient decreases when the ratio of the relaxation to retardation time parameter increases.     

Cattaneo-Christov heat and mass flux model for 3D hydrodynamic flow of chemically reactive Maxwell liquid

This research focuses on the Cattaneo-Christov theory of heat and mass flux for a three-dimensional Maxwell liquid towards a moving surface. An incompressible laminar flow with variable thermal conductivity is considered. The flow generation is due to the bidirectional stretching of sheet. The combined phenomenon of heat and mass transport is accounted. The Cattaneo-Christov model of heat and mass diffusion is used to develop the expressions of energy and mass species. The first-order chemical reaction term in the mass species equation is considered. The boundary layer assumptions lead to the governing mathematical model. The homotopic simulation is adopted to visualize the results of the dimensionless flow equations. The graphs of velocities, temperature, and concentration show the effects of different arising parameters. A numerical benchmark is presented to visualize the convergent values of the computed results. The results show that the concentration and temperature fields are decayed for the Cattaneo-Christov theory of heat and mass diffusion.     

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

     

Boundary layer ?ow of Maxwell ?uid due to torsional motion of cylinder: modeling and simulation

This paper investigates the boundary layer ?ow of the Maxwell ?uid around a stretchable horizontal rotating cylinder under the in?uence of a transverse magnetic?eld. The constitutive ?ow equations for the current physical problem are modeled and analyzed for the ?rst time in the literature. The torsional motion of the cylinder is considered with the constant azimuthal velocity E. The partial di?erential equations(PDEs)governing the torsional motion of the Maxwell ?uid together with energy transport are simpli?ed with the boundary layer concept. The current analysis is valid only for a certain range of the positive Reynolds numbers. However, for very large Reynolds numbers, the ?ow becomes turbulent. Thus, the governing similarity equations are simpli?ed through suitable transformations for the analysis of the large Reynolds numbers. The numerical simulations for the ?ow, heat, and mass transport phenomena are carried out in view of the bvp4 c scheme in MATLAB. The outcomes reveal that the velocity decays exponentially faster and reduces for higher values of the Reynolds numbers and the ?ow penetrates shallower into the free stream ?uid. It is also noted that the phenomenon of stress relaxation, described by the Deborah number, causes to decline the ?ow ?elds and enhance the thermal and solutal energy transport during the ?uid motion. The penetration depth decreases for the transport of heat and mass in the ?uid with the higher Reynolds numbers. An excellent validation of the numerical results is assured through tabular data with the existing literature.     

Unsteady three-dimensional MHD flow of the micropolar fluid over an oscillatory disk with Cattaneo-Christov double diffusion

Cattaneo-Christov heat and mass flux models are considered rather than Fourier and Fick laws due to the presence of thermal and concentration transport hyperbolic phenomena. The generalized form of the Navier-Stokes model is considered in hydromagnetic flow. Three-dimensional(3 D) unsteady fluid motion is generated by the periodic oscillations of a rotating disk. Similarity transformations are used to obtain the normalized fluid flow model. The successive over relaxation(SOR) method with finite difference schemes are accomplished for the numerical solution of the obtained partial differential non-linear system. The flow features of the velocity, microrotation, temperature,and concentration fields are discussed in pictorial forms for various physical flow parameters. The couple stresses and heat and mass transfer rates for different physical quantities are explained via tabular forms. For better insight of the physical fluid model, 3 D fluid phenomena and two-dimensional(2 D) contours are also plotted. The results show that the micropolar fluids contain microstructure having non-symmetric stress tensor and are useful in lubrication theory. Moreover, the thermal and concentration waves in CattaneoChristov models have a significance role in the laser heating and enhancement in thermal conductivity.     

Cattaneo-Christov heat flux model for third-grade fluid flow towards exponentially stretching sheet

The Cattaneo-Christov heat flux in the two-dimensional (2D) flow of a third-grade fluid towards an exponentially stretching sheet is investigated. The energy equation is considered through thermal relaxation. Similarity transformations are accounted to obtain the ordinary differential systems. The converted non-dimensional equations are solved for the series solutions. The convergence analysis of the computed solutions is reported. The graphical results of the velocity and temperature profiles are plotted and elaborated in detail. The results show that the thermal relaxation enhances the temperature gradient while reduces the temperature profile.     

Dynamics of Sutterby fluid flow due to a spinning stretching disk with non-Fourier/Fick heat and mass flux models

The magnetohydrodynamic Sutterby fluid flow instigated by a spinning stretchable disk is modeled in this study. The Stefan blowing and heat and mass flux aspects are incorporated in the thermal phenomenon. The conventional models for heat and mass flux, i.e., Fourier and Fick models, are modified using the Cattaneo-Christov(CC)model for the more accurate modeling of the process. The boundary layer equations that govern this problem are solved using the apt similarity variables. The subsequent system of equations is tackled by the Runge-Kutta-Fehlberg(RKF) scheme. The graphical visualizations of the results are discussed with the physical significance. The rates of mass and heat transmission are evaluated for the augmentation in the pertinent parameters. The Stefan blowing leads to more species diffusion which in turn increases the concentration field of the fluid. The external magnetism is observed to decrease the velocity field. Also,more thermal relaxation leads to a lower thermal field which is due to the increased time required to transfer the heat among fluid particles. The heat transport is enhanced by the stretching of the rotating disk.     

Darcy-Forchheimer flow of variable conductivity Jeffrey liquid with Cattaneo-Christov heat flux theory

The role of the Cattaneo-Christov heat flux theory in the two-dimensional laminar flow of the Jeffrey liquid is discussed with a vertical sheet. The salient feature in the energy equation is accounted due to the implementation of the Cattaneo-Christov heat flux. A liquid with variable thermal conductivity is considered in the Darcy-Forchheimer porous space. The mathematical expressions of momentum and energy are coupled due to the presence of mixed convection. A highly nonlinear coupled system of equations is tackled with the homotopic algorithm. The convergence of the homotopy expressions is calculated graphically and numerically. The solutions of the velocity and temperature are expressed for various values of the Deborah number, the ratio of the relaxation time to the retardation time, the porosity parameter, the mixed convective parameter, the Darcy-Forchheimer parameter, and the conductivity parameter. The results show that the velocity and temperature are higher in Fourier's law of heat conduction cases in comparison with the Cattaneo-Christov heat flux model.     

Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.     

Magnetohydrodynamic and heat transfer effects on the stagnation-point flow of an electrically conducting nanofluid past a porous vertical shrinking/stretching sheet in the presence of variable stream conditions

A steady two-dimensional magnetohydrodynamic stagnation-point flow of an electrically conducting fluid and heat transfer with thermal radiation of a nanofluid past a shrinking and stretching sheet is investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and thermophoresis. A similarity transformation is used to convert the governing nonlinear boundary-layer equations into coupled higher-order nonlinear ordinary differential equations. The result shows that the velocity, temperature, and concentration profiles are significantly influenced by the Brownian motion, heat radiation, and thermophoresis particle deposition.     

Influence of temperature dependent conductivity of a nanofluid in a vertical rectangular duct

Natural convective heat transfer and fluid flow in a vertical rectangular duct filled with a nanofluid is studied numerically assuming the thermal conductivity to be dependent on the fluid temperature. The transport equations for mass, momentum and energy formulated in dimensionless form are solved numerically using finite difference method. Particular efforts have been focused on the effects of the thermal conductivity variation parameter, Grashof number, Brinkman number, nanoparticles volume fraction, aspect ratio and type of nanoparticles on the fluid flow and heat transfer inside the cavity. It is found that the flow was enhanced for the increase in Grashof number, Brinkman number and aspect ratio for any values of conductivity variation parameter and for regular fluid and nanofluid. The heat transfer rate for regular fluid is less than that for the nanofluid for all governing parameters.     

Active and passive controls of nanoparticles in Maxwell stagnation point flow over a slipped stretched surface

A steady stagnation-point flow of an incompressible Maxwell fluid towards a linearly stretching sheet with active and passive controls of nanoparticles is studied numerically. The momentum equation of the Maxwell nanofluid is inserted with an external velocity term as a result of the flow approaches the stagnation point. Conventional energy equation is modified by incorporation of nanofluid Brownian and thermophoresis effects. The condition of zero normal flux of nanoparticles at the stretching surface is defined to impulse the particles away from the surface in combination with nonzero normal flux condition. A hydrodynamic slip velocity is also added to the initial condition as a component of the entrenched stretching velocity. The governing partial differential equations are then reduced into a system of ordinary differential equations by using similarity transformation. A classical shooting method is applied to solve the nonlinear coupled differential equations. The velocity, temperature and nanoparticle volume fraction profiles together with the reduced skin friction coefficient, Nusselt number and Sherwood number are graphically presented to visualize the effects of particular parameters. Temperature distributions in passive control model are consistently lower than in the active control model. The magnitude of the reduced skin friction coefficient, Nusselt number and Sherwood number decrease as the hydrodynamic slip parameter increases while the Brownian parameter has negligible effect on the reduced heat transfer rate when nanoparticles are passively controlled at the surface. It is also found that the stagnation parameter contributes better heat transfer performance of the nanofluid under both active and passive controls of normal mass flux.     

Molybdenum disulfide and magnesium oxide nanoparticle performance on micropolar Cattaneo-Christov heat flux model

This article intends to illustrate the Darcy flow and melting heat transmission in micropolar liquid.The major advantage of micropolar fluid is the liquid particle rotation through an independent kinematic vector named the microrotation vector.The novel aspects of the Cattaneo-Christov(C-C)heat flux and Joule heating are incorporated in the energy transport expression.Two different nanoparticles,namely,MoS2 and MgO,are suspended into the base-fluid.The governing partial differential equations(PDEs)of the prevailing problem are slackening into ordinary differential expressions(ODEs)via similarity transformations.The resulting mathematical phenomenon is illustrated by the implication of fourth-fifth order Runge-Kutta-Fehlberg(RKF)scheme.The fluid velocity and temperature distributions are deliberated by using graphical phenomena for multiple values of physical constraints.The results are displayed for both molybdenum disulphide and magnesium oxide nanoparticles.A comparative benchmark in the limiting approach is reported for the validation of the present technique.It is revealed that the incrementing material constraint results in a higher fluid velocity for both molybdenum disulphide and magnesium oxide nanoparticle situations.     

Extended thermodynamics of molecular ideal gases

The objective of extended thermodynamics of molecular ideal gases is the determination of the 17 fields ofmass density, velocity, energy density, pressure deviator, heat flux, intrinsic energy density and intrinsic heat flux. The intrinsic energy represents the rotational or the vibrational energy of the molecules. The necessary field equations are based upon balance laws and the system of equations is closed by constitutive relations which are characteristic for the gas under consideration. The generality of the constitutive relations is restricted by theprinciple of material frame indifference, and by the entropy principle. These principles reduce the constitutive coefficients of all fluxes to the thermal and caloric equation of state of the gas and provide inequalities for the transport coefficients. The transport coefficients can be related to the shear viscosity, the heat conductivity, and the coefficients of self-diffusion and attenuation of sound waves, so that the field equations become quite specific. The theory is in perfect agreement with the kinetic theory of molecular gases. It is shown that in non-equilibrium the temperature is discontinuous at thermometric walls. The dynamic pressure and the volume viscosity, are discussed and it is shown how extended thermodynamics and ordinary thermodynamics are related.     

Dual solutions of time-dependent magnetohydrodynamic stagnation point boundary layer micropolar nanofluid flow over shrinking/stretching surface

Time-dependent, two-dimensional(2 D) magnetohydrodynamic(MHD)micropolar nanomaterial flow over a shrinking/stretching surface near the stagnant point is considered. Mass and heat transfer characteristics are incorporated in the problem. A model of the partial differential expressions is altered into the forms of the ordinary differential equations via similarity transformations. The obtained equations are numerically solved by a shooting scheme in the MAPLE software. Dual solutions are observed at different values of the specified physical parameters. The stability of first and second solutions is examined through the stability analysis process. This analysis interprets that the first solution is stabilized and physically feasible while the second one is un-stable and not feasible. Furthermore, the natures of various physical factors on the drag force, skin-friction factor, and rate of mass and heat transfer are determined and interpreted. The micropolar nanofluid velocity declines with a rise in the suction and magnetic parameters, whereas it increases by increasing the unsteadiness parameter.The temperature of the micropolar nanofluid rises with increase in the Brownian motion,radiation, thermophoresis, unsteady and magnetic parameters, but it decreases against an increment in the thermal slip constraint and Prandtl number. The concentration of nanoparticles reduces against the augmented Schmidt number and Brownian movement values but rises for incremented thermophoresis parameter values.     

Dynamics of bioconvection flow of micropolar nanoparticles with Cattaneo-Christov expressions

A numerical analysis is performed to analyze the bioconvective double diffusive micropolar non-Newtonian nanofluid flow caused by stationary porous disks.The consequences of the current flow problem are further extended by incorporating the Brownian and thermophoresis aspects. The energy and mass species equations are developed by utilizing the Cattaneo and Christov model of heat-mass fluxes. The flow equations are converted into an ordinary differential model by employing the appropriate variables. The numerical solution is reported by using the MATLAB builtin bvp4c method. The consequences of engineering parameters on the flow velocity,the concentration, the microorganisms, and the temperature profiles are evaluated graphically. The numerical data for fascinating physical quantities, namely, the motile density number, the local Sherwood number, and the local Nusselt number, are calculated and executed against various parametric values. The microrotation magnitude reduces for increasing magnetic parameters. The intensity of the applied magnetic field may be utilized to reduce the angular rotation which occurs in the lubrication processes, especially in the suspension of flows. On the account of industrial applications, the constituted output can be useful to enhance the energy transport efficacy and microbial fuel cells.