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Nanoparticles have gained recognition for significantly improving convective heat transfer efficiency near boundary layer flows. The characteristics of both momentum and thermal boundary layers are significantly influenced by the Prandtl number, which holds a crucial role. In this vein, the current study conducted a detailed computational analysis of the mixed convection flow of |$\gamma$| Al |$_2$| O |$_3$| -H |$_2$| O and |$\gamma$| Al |$_2$| O |$_3$| -C |$_2$| H |$_6$| O |$_2$| nanofluids over a stretching surface. This research integrates an effective Prandtl number, utilizing viscosity and thermal conductivity models based on empirical findings. Additionally, a unique double-fractional constitutive model is debuted to accurately evaluate the effective Prandtl number's function in the boundary layer. The equations were solved using a numerical technique that combined the finite-difference method with the L |$_1$| algorithm. This investigation presents numerical findings related to the velocity, temperature distributions, wall shear stress coefficient, and heat transfer coefficient, contrasting scenarios with and without the effective Prandtl number. The research shows that integrating nanoparticles into the base fluids reduces the temperature of the nanofluid with an effective Prandtl number while enhancing the heat transfer rate irrespective of its presence. Nonetheless, the introduction of a fractional parameter reduced the heat transfer efficiency within the system. Notably, the |$\gamma$| Al |$_2$| O |$_3$| -C |$_2$| H |$_6$| O |$_2$| nanofluid demonstrates superior heat transfer enhancement capabilities compared to its |$\gamma$| Al |$_2$| O |$_3$| -H |$_2$| O counterpart but also exacerbates the drag coefficient more significantly. Many practical applications of this study include electronics cooling, industrial process heat exchangers, and rotating and stationary gas turbines in power plants, and efficient heat exchangers in aircraft. [ABSTRACT FROM AUTHOR]

Numerical investigation on the influence of effective Prandtl number on the steady MHD Williamson nanofluid flow over an inclined stretching surface in the presence of aligned magnetic field, heat generation and chemical reaction is carried out. By applying a suitable similarity transformation, the system of non-linear coupled partial differential equations are converted to a system of non-linear coupled ordinary differential equations. Numerical solutions of these equations are obtained by using MATLAB package, bvp4c. The impact of various parameters on the velocity, temperature and nanoparticle volume fraction, skin friction, Nusselt number and Sherwood number are discussed through the aid of graphs and tabulated data. The numerical computations reveal that the escalating values of the effective Prandtl number suppresses the concentration profile closer to the wall, the temperature profile and the velocity profile. Furthermore, the velocity is largest when both the channel slope and the magnetic field inclination angles are lowest. The inclination angles have opposite effect on the temperature and concentration profiles. Moreover, results show that varying the Eckert number and the magnetic parameter will have no effect on the skin friction coefficient, Nusselt number and Sherwood number whenever the Brownian motion parameter and the thermophoresis parameter have the same value. [ABSTRACT FROM AUTHOR]

The study examines the impact of inclined stretching sheet, aligned magnetic field and effective Prandtl number on MHD Maxwell fluid in the presence of thermophoresis and heat generation. By employing similarity transformations the given partial differential equations are converted into nonlinear ordinary differential equations which are then solved numerically using MATLAB bvp4c technique. The investigations reveal that enlargement of the effective Prandtl number enhances the temperature and concentration of the fluid and suppresses the magnitude of the fluid velocity. It is observed that the fluid concentration is highest when both the channel slope and magnetic field inclination angles assume least values provided the effective Prandtl number is greater or equal to unity or when both inclination angles attain maximum values when the effective Prandtl number is less than unity. Additionally, the fluid temperature profile is greatest when channel slope is maximal and magnetic field angle is minimal. It is also observed that enlarged inclination angles enhances the temperature profile for larger Epr values and suppresses the temperature profile for smaller Epr values. Finally, it is deduced that raising the value of the thermophoretic parameter reduces the skin friction and mass transfer rate, whereas the heat transfer rate is improved. [ABSTRACT FROM AUTHOR]

Numerical study of steady MHD convective flow of viscous incompressible electrically conducting fluid over an inclined stretching surface with aligned magnetic field, Hall effect, effective Prandtl number and heat generation is carried out. The partial differential equations are transformed to a system of non-linear ordinary differential equations which are then solved numerically by MATLAB bvp4c solver. The profiles of the velocity, temperature and concentration are analysed and duscussed with the numerical results presented graphically. The calculated values of skin-friction, the heat transfer rate and mass transfer rate at the surface are discussed numerically for various values of the physical parameters and tabulated. Results reveal that for attainment of optimal velocity profile, the amount of inclination of the magnetic field and the stretching surface need to be relatively small and that for a given effective Prandtl number the value of the Prandtl number should be greater than that of the radiation parameter. [ABSTRACT FROM AUTHOR]

The paper reports on the influence of effective Prandtl number, aligned magnetic field, slope of the stretching channel and heat generation on unsteady MHD Casson fluid flow. The partial differential equations of momentum, temperature and concentration are converted to non-linear ordinary differential equations by introducing appropriate similarity variables. The solution of the ODE's are calculated numerically using MATLAB bvp4c solver. The role of the different parameters on the temperature, velocity, and concentration profiles are analyzed and presented in the form of graphs and tables. Finally the impact of the different parameters on the mass transfer rate, heat transfer rate and skin friction are discussed. It is revealed that lowering the inclination angle enhances the velocity and temperature profiles with opposite effect on the concentration distribution. The concentration, temperature and velocity profiles decline with enlarged values of the unsteadiness parameter. Furthermore, the velocity profile, the temperature profile at free stream, skin friction coefficient, mass transfer rate and heat transfer rate diminishes with magnified effective Prandtl number. [ABSTRACT FROM AUTHOR]

The stretched surface's convective heat transfer capability can be improved by using nanoparticles. There is a significant role of the Prandtl number in determining the thermal and momentum stretching layer surfaces. It is proposed in this study that an effective Prandtl number model be used to explore the two-dimensional oblique stagnation point flow of γAl2O3−H2O and γAl2O3−C2H6O2 nanofluids moving over a convective stretching surface. The fluid in question is subjected to a thorough investigation. It is necessary to apply non-linear ordinary differential equations in order to connect the controlling partial differential equations with the boundary conditions. To solve these equations, an efficient and reliable numerical technique is used. Shooting Method with Runge Kutta-IV in Mathematica software. Visual representations of normal and tangential velocity and temperature as well as streamlines as a function of many physical parameters are shown. The results show that as the volume fraction of nanoparticles increases, the fluid flow f(y), h(y) and velocity f′(y), h′(y) all increase, whereas the flow f(y) and velocity f′(y) both increase against the stretching ratio parameter, while the flow h(y) and velocity h′(y) both decrease. When the volume percentage of nanoparticles and the Biot number are both increased, the temperature rises. However, when the stretching ratio parameter is increased, the temperature falls. Physical attributes like the local skin friction coefficient and the heat flow may be characterized in many ways. A nanofluid comprised of γAl2O3−C2H6O2 outperformed a γAl2O3−H2O nanofluid in terms of heat transfer rate. The source of zero skin friction may be observed to move to the left or right depending on the balance of obliqueness and straining motion at point xs. The computed numerical results of the current research correspond well with those accessible in the literature for the limiting scenario.

A mathematical model of micropolar squeezing flow of nanofluids between parallel planes is taken into consideration under the influence of the effective Prandtl number using ethyl glycol (C 2 H 6 O 2) and water (H 2 O) as base fluids along with nanoparticles of gamma alumina (γ A l 2 O 3) . The governing nonlinear PDEs are changed into a system of ODEs via suitable transformations. The RKF (Range–Kutta–Fehlberg) technique is used to solve the system of nonlinear equations deriving from the governing equation. The velocity, temperature, and concentration profiles are depicted graphically for emerging parameters such as Hartmann number M , micronation parameter K , squeeze number R , Brownian motion parameter N b , and thermophoresis parameter N t . However, physical parameters such as skin friction coefficient, Nusselt number, and Sherwood number are portrayed in tabulated form. The inclusion of the effective Prandtl number model indicated that the effect of the micropolar parameter K on angular velocity h(ξ) in both suction and injection cases is opposite for both nanofluids. It is observed that the increase in angular velocity is rapid for γ A l 2 O 3 − C 2 H 6 O 2 throughout the study. [ABSTRACT FROM AUTHOR]

In the present communication, we have model peristaltic transport of γAl2O3/H2O and γAl2O3/C2H6O2 in an asymmetric channel by using small Reynolds number (Re) and long wavelength (δ ≪ 1) assumptions. Two dimensional equations of peristaltic nanofluid flow are modeled and then solved by using analytical regular perturbation technique. We have formulated the peristaltic motion equations of nanofluids by applied magnetic field effect on asymmetric channel and derive energy equation by using heat generation/absorption effect. By using numerical integration the solution of pressure behavior and solution of resulting governing equations are calculated by Mathematica software. By using Matlab software the graphical behavior of pressure rise, trapping phenomena, temperature profile and pressure gradient are illustrated. We found that the narrow region of channel requires large pressure gradient, also the pressure gradient decreases with increase of channel width d. The pressure gradient of nanofluids increases due to increase in nanoparticle volume fraction, amplitudes and Hartmann number. Moreover, in trap phenomena the bolus size decreases by increase of Hartmann number and bolus size increases by increase of volume flow rate.

The current article depicts the impact of the effective Prandtl number on thermally developed flow of magnetized tangent hyperbolic fluid containing nanoparticles over a porous surface with shear rate viscosity. Zero normal mass flux condition for nano particles is used with second order slip features. In order to prop the suspended nanoparticles by associate effect of buoyancy force and magnetic field is implemented. Bio-convective effect with Nield boundaries due to gyrotactic motile microorganisms is the concerned factor of current communication. The implemented appropriate transformation function is used to transform the system of self-determining partial differential equations (PDEs) into the sets of non-linear ordinary differential equations (ODEs). These transmuted expression computed numerically by bvp4c function on MATLAB using mathematical scheme shooting technique. Furthermore, the interesting phenomena of controlling parameters, is inspected on velocity profile, temperature distribution, concentration profile of nano particles and motile microorganisms by using graphical and numerical techniques. The first order and the second order slip parameters decay the momentum boundry layer, and boosted up the velocity profile as enhance in the value of mixed convection parameter. It is deduced that the Peclet number and the bio-convection Lewis number results delay in the density of the motile microorganism. For the efficient of the thermal field, the bioconvection is induced by the gyrotactic microorganisms to stabilized the nanoparticles.

In the present article, radiation effect on mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder with constant heat flux has been numerically analyzed. The governing boundary layer equations are transformed to dimensionless nonlinear partial differential equations. The equations are solved numerically by using Keller-box method. The computed results are in excellent agreement with the previous studies. Skin friction coefficient and Nusselt number are emphasized specifically. These quantities are displayed against the curvature parameter. The effects of pertinent parameters involved in the problem namely effective Prandtl number and mixed convection parameter on skin friction coefficient and Nusselt number are shown through graphs and table. Boundary layer separation points are also calculated with and without radiation and a comparison is shown. The presence of radiation helps to decrease or increase the skin friction coefficient for the negative or positive values of the mixed convection parameter accordingly. The decrease in value of effective Prandtl number helps to increase the value of skin friction coefficient and Nusselt number for viscoelastic fluids.

The flow squeezed between parallel flat plates that placed horizontally in nonporous media is of great interest because of its uses in many industries and engineering side as well. In the light of this fact, we have considered the flow of nanofluids between two plates. The dimensional flow is successfully reduced into a set of self-similar form with the help of feasible similarity variables. In the next step, the system is solved analytically (Homotopy Analysis method) and to ensure the analytical computation, numerical solutions (by means of Runge-Kutta method) are also performed. Influences of the effective Prandtl number are investigated on thermal filed and local rate of heat transfer along with ingrained physical quantities. Under certain assumptions, present results are compared with already existing ones in the literature and found to be agree well. The major outcomes of the presented letter are described in the last section. [ABSTRACT FROM AUTHOR]

Nanoparticles provide potentials in augmenting the performance of convective heat transfer in the boundary layer flow region. Prandtl number plays a vital role in controlling the momentum and thermal boundary layers. In view of this, the influences of an effective Prandtl number model which is derived from experimental data (Pop et al., 2007) on the nano boundary layer, steady, two-dimensional and laminar flow of an incompressible γ Al 2 O 3 –H 2 O and γ Al 2 O 3 –C 2 H 6 O 2 nanofluids over a vertical stretching sheet are investigated for the first time in the present article. The models which are used for viscosity and thermal conductivity also derived from experimental data (Maiga et al., 2004a; Maiga et al., 2005). The second law of thermodynamics also analysed for the present problem. The transformed governing nonlinear boundary layer equations are solved numerically using fourth order Runge–Kutta method with shooting technique and analytical solutions are presented for a special case. The numerical results obtained for the temperature profile, skin friction coefficient and reduced Nusselt number are presented through plots for two different cases such as with and without effective Prandtl number. It is found that the increasing values of nanoparticle volume fraction of γ Al 2 O 3 nanoparticles decrease the temperature of the nanofluids in the presence of effective Prandtl number and increase in the absence of effective Prandtl number. The entropy generation number is higher for ethylene glycol based nanofluids compared to water based nanofluids. [ABSTRACT FROM AUTHOR]

In the present article, radiation effect on mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder with constant heat flux has been numerically analyzed. The governing boundary layer equations are transformed to dimensionless nonlinear partial differential equations. The equations are solved numerically by using Keller-box method. The computed results are in excellent agreement with the previous studies. Skin friction coefficient and Nusselt number are emphasized specifically. These quantities are displayed against the curvature parameter. The effects of pertinent parameters involved in the problem namely effective Prandtl number and mixed convection parameter on skin friction coefficient and Nusselt number are shown through graphs and table. Boundary layer separation points are also calculated with and without radiation and a comparison is shown. The presence of radiation helps to decrease or increase the skin friction coefficient for the negative or positive values of the mixed convection parameter accordingly. The decrease in value of effective Prandtl number helps to increase the value of skin friction coefficient and Nusselt number for viscoelastic fluids. [ABSTRACT FROM AUTHOR]

The stretched surface's convective heat transfer capability can be improved by using nanoparticles. There is a significant role of the Prandtl number in determining the thermal and momentum stretching layer surfaces. It is proposed in this study that an effective Prandtl number model be used to explore the two-dimensional oblique stagnation point flow of γ A l 2 O 3 - H 2 O and γ A l 2 O 3 - C 2 H 6 O 2 nanofluids moving over a convective stretching surface. The fluid in question is subjected to a thorough investigation. It is necessary to apply non-linear ordinary differential equations in order to connect the controlling partial differential equations with the boundary conditions. To solve these equations, an efficient and reliable numerical technique is used. Shooting Method with Runge Kutta-IV in Mathematica software. Visual representations of normal and tangential velocity and temperature as well as streamlines as a function of many physical parameters are shown. The results show that as the volume fraction of nanoparticles increases, the fluid flow f ( y ) , h ( y ) and velocity f ' ( y ) , h ' ( y ) all increase, whereas the flow f ( y ) and velocity f ' ( y ) both increase against the stretching ratio parameter, while the flow h ( y ) and velocity h ' ( y ) both decrease. When the volume percentage of nanoparticles and the Biot number are both increased, the temperature rises. However, when the stretching ratio parameter is increased, the temperature falls. Physical attributes like the local skin friction coefficient and the heat flow may be characterized in many ways. A nanofluid comprised of γ A l 2 O 3 - C 2 H 6 O 2 outperformed a γ A l 2 O 3 - H 2 O nanofluid in terms of heat transfer rate. The source of zero skin friction may be observed to move to the left or right depending on the balance of obliqueness and straining motion at point x s . The computed numerical results of the current research correspond well with those accessible in the literature for the limiting scenario., Competing Interests: The authors declare no conflict of interest., (© 2023 The Authors.)

The heat transfer characteristics of a viscous liquid film flow over an unsteady stretching sheet subject to variable heat flux are investigated numerically. The effect of thermal radiation applying to an optically thick medium is also considered. The governing boundary layer equations are transformed into a set of nonlinear ordinary differential equations using an efficient fifth-order step-adapted Runge-Kutta integration scheme together with Newton-Raphson method. The dimensionless temperature is plotted for various governing parameters; say, unsteadiness parameter, effective Prandtl number, distance index, as well as time index. It is found that the heat transfer aspects are strongly influenced by the relevant parameters. [ABSTRACT FROM AUTHOR]

Abstract: In the latter years the title problem has been examined in a large number of research papers. The present Note emphasizes, however, that the effect of thermal radiation in the linearized Rosseland approximation is quite trivial, both physically and computationally. Namely, it always reduces to a simple rescaling of the Prandtl number by a factor involving the radiation parameter. This implies that a comprehensive study of the Prandtl-number dependence without thermal radiation effects represents per se a detailed study of the radiation effects, too. In other words, the solution of the radiation problem for optically thick media in the linearized Rosseland approximation does not require any additional numerical or analytical effort compared to the same problem without radiation, making in this respect dozens of papers superfluous. [Copyright &y& Elsevier]