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This work is devoted to the effects of heat transfer study on peristaltically induced flow of an electrically conducting viscous fluid in an asymmetric channel. An incompressible fluid fills the porous space in a channel. Analysis is made in the presence of Hall and ion-slip effects. The effects of velocity and thermal slips on the boundaries are considered. The problems which govern the fluid flow and heat transfer are modeled in the wave frame of reference. Exact solutions have been constructed under realistic assumptions. Important features of peristaltic motion and heat transfer are studied with the varying values of Hall parameter, ion-slip parameter, Hartman number, permeability parameter, velocity slip parameter, and thermal slip parameter. The graphical results concerning the field quantities are given in detail. [ABSTRACT FROM AUTHOR]

This investigation looks at the influence of an inclined magnetic field on peristaltic transport of hyperbolic tangent nanofluid in inclined channel having flexible walls. Nanofluid consisting of Brownian motion and thermophoresis effects is employed in the definition of problem. Thermal radiation and Joule heating are present. Formulation is further completed by consideration of slip conditions in terms of velocity, temperature and concentration. Lubrication approach has been followed for the development of problem formulation. The key role of various involved parameters on the flow phenomenon are sketched. Slip effect causes the velocity and temperature to increase while reduces the concentration. Reduction in temperature is noticed with thermal radiation whereas temperature enhances for larger Hartman number in responce to the Joule heating effect. [ABSTRACT FROM AUTHOR]

This article examines the magneto-hydrodynamic (MHD) flow of nanofluid bounded by a stretching surface. The slip conditions are utilized in the present analysis. The involved differential systems are solved for the velocity, temperature and mass fraction. Graphical and numerical results are reported for the analysis of various parameters of interest entering into the modeled problems. Combined effects of thermal and concentration jump are analyzed. Various tables are constructed to show the rheological effects of different physical parameters. Streamlines are plotted showing the rheology for the slip and no slip flow regime. Plots of skin friction are also prepared for the slip and magnetic field effects. [ABSTRACT FROM AUTHOR]

This paper presents the analysis of the process of blade coating through a computational intensive method for an incompressible Newtonian fluid along with Magnetohydrodynamics (MHD). The slip between the substrate and the fluid is also taken into account. The nature of the existing steady solutions has been investigated with the help of exact and numerical methods. Those obtained exact solutions include the solutions for the velocity profiles, volumetric flow rate and pressure gradient. The important engineering quantities like maximum pressure, pressure distribution and load are also computed. It is assumed that the relative velocity between the plate and the fluid is proportional to the shear rate at the plate. An external magnetic field is applied normal to the plates. It is observed and concluded that both slip parameter and the magnetic field parameter serve as the controlling parameters in the industrial blade coating process. [ABSTRACT FROM AUTHOR]

Objective The objective of this paper is to study the influences of slip and Cu-blood nanofluid in a physiological study of cilia. Design/methodology/approach The right wall and the left wall possess metachronal wave that is traveling along the outer boundary of the channel. The features of ciliary structures are determined by the dominance of viscous effects over inertial effects using the long wavelength approximation. The flow possessions for the viscous nano fluid are solved as a function of the cilia and metachronal wave velocity. Mathematica numerical simulation is used to calculate pressure rise. Exact solutions are calculated for the temperature and for the velocity profile. Findings Numerical integration has been implemented to get the expression of pressure rise. Graphical results have been offered for pressure rise, temperature and stream function for various physical parameters of interest. Symmetry of the curved channel is recovered for larger values of the curvature parameter. It is found from the graphs of the pressure rise that reflux case occurs for increasing values of curvature and velocity slip parameter. It is also found that temperature decreases for increasing values of nanoparticle volume fraction. Basically, higher thermal conductivity of the nanoparticles plays a key role for quick heat dissipation, and this justifies the use of the copper nanoparticles in different situations as a coolant. Conclusions Velocity graph near the right wall of the channel decreases when we add nanoparticles into our base fluid, whereas an opposite behavior is depicted near the left wall due to ciliated tips. Pressure gradient increases for increasing values of G r (Grashof number), L (velocity slip parameter) ϕ (nanoparticle volume fraction) and k (curvature parameter). Pressure rise shows increasing behavior for the increasing values of ε , G r and ϕ throughout the region. Shear stress graphs show the increasing behavior for increasing values of curvature parameter and volume fraction of the nanoparticle. Temperature profile decreases when we add nanoparticles into our base fluid. Temperature profile also increases with an increase in thermal slip parameter γ and heat absorption parameter. It is observed that the number of trapped bolus increases when we increase the value of the cilia length parameter. [ABSTRACT FROM AUTHOR]

The peristaltic motion of a third order fluid due to asymmetric waves propagating on the sidewalls of a inclined asymmetric channel is discussed. The key features of the problem includes long- wavelength and low-Reynolds number assumptions. A mathematical analysis has been carried out to investigate the effect of slip condition, variable viscosity and magnetohydrodynamics (MHD). Followed by the nondimensionalization of the nonlinear governing equations along with the non- linear boundary conditions, a perturbation analysis is made. For the validity of the approximate solution, a numerical solution is obtained using the iterative collocation technique. [ABSTRACT FROM AUTHOR]

The area of the exit and throat region of a nozzle play a crucial role in its design. This paper is a report of numerical simulations carried out to investigate the influence of these parameters on the performance factors of an axisymmetric cold gas nozzle of micron-size throat diameter. It was assumed that the deviations of the flow behavior from that of a continuum flow can be taken care of by applying the first-order slip boundary conditions at the wall. The solution methodology includes a finite-volume-based numerical procedure based on structured quadrilateral grids. A parametric study reveals that to reach the highest values of the thrust and specific impulse, one should choose a nozzle with the highest possible throat diameter. However, by increasing the outlet diameter, the thrust initially reaches a maximum and then decreases. In these conditions, the specific impulse is always a decreasing function of the outlet diameter of the micronozzle. It is also observed that the mass flow rate is an increasing function of both the throat and outlet diameters. In addition, the comparison of the results with and without slip velocity shows that the amounts of the mass flow rate, thrust force, and specific impulse are higher when the rarefaction effects are taken into account. Nevertheless, no fundamental difference is observed in flow physics with and without slip velocity. [ABSTRACT FROM AUTHOR]

The effects of heat and mass transfer on the flow of an oscillatory convective MHD channelflow of an electrically conducting viscous incompressible fluid in a slip flow regime in the presence of Hall current and thermal radiation are studied. A magnetic field of uniform strength is applied normal to the plates of channel. The two vertical porous plates of the channel are subjected to a constant injection/suction. The entire system rotates about an axis normal to the plates with a uniform angular velocity Ω. A closed form analytical solutions to the equations governing the fluid motion are obtained for the velocity, temperature and concentration fields. The numerical calculations for the velocity, temperature and concentration fields, and the coefficient of skin friction, rate of heat and mass transfer at the plates are performed. The results are presented graphically for different values of the physical parameters involved. It is seen that primary fluid flow is retarded due to angular velocity of the system. [ABSTRACT FROM AUTHOR]

Casson flow ferromagnetic liquid blood flow over stretching region is studied numerically. The domain is influence by radiation and blood flow velocity and thermal slip conditions. Blood acts an impenetrable magneto-dynamic liquid yields governing equations. The conservative governing nonlinear partial differential equations, reduced to ODEs by the help of similarity translation technique. The transport equations were transformed into first order ODEs and the resultant system are solved with help of 4th order R-K scheme. Performing a magnetic dipole with a Casson flow across a stretched region with Brownian motion and Thermophoresis is novelty of the problem. Significant applications of the study in some spheres are metallurgy, extrusion of polymers, production in papers and rubber manufactured sheets. Electronics, analytical instruments, medicine, friction reduction, angular momentum shift, heat transmission, etc. are only few of the many uses for ferromagnetic fluids. As ferromagnetic interaction parameter value improves, the skin-friction, Sherwood and Nusselt numbers depreciates. A comparative study of the present numerical scheme for specific situations reveals a splendid correlation with earlier published work. A change in blood flow velocity magnitude has been noted due to Casson parameter. Increasing change in blood flow temperature noted due to Casson parameter. Skin-friction strengthened and Nusselt number is declined with Casson parameter. The limitation of current work is a non-invasive magnetic blood flow collection system using commercially available magnetic sensors instead of SQUID or electrodes., Competing Interests: The authors declare no conflict of interest., (© 2023 The Authors.)

In this paper, we investigate the effects of wall slip conditions, elasticity wall properties and heat transfer on the peristaltic transport of conducting Bingham fluid in a non-uniform channel under the assumptions of long wavelength and low-Reynolds number. The expressions for velocity, stream function, temperature and heat transfer coefficient are obtained. The effects of the physical parameters on the velocity and temperature fields are presented through graphs. The effect of increase in the two ( E 1 & E 2 ) of the three wall property parameters is to enhance the velocity and temperature in the channel whereas opposite behaviour is noticed regarding the other parameter E 3 . Also the size of the trapped bolus gets reduced due to the increase in the magnetic parameter M . [ABSTRACT FROM AUTHOR]

We perform a study for the boundary layer flow of viscous fluid about a stagnation point on an unsteady stretching sheet in the presence of variable free stream. Both the velocity and thermal slip conditions are taken into consideration. The stretching velocity and the surface temperature are time-dependent. In the solutions we adopt the homotopy analysis method (HAM). The analysis related to convergence of the solutions is explicitly discussed. The dependence of velocity and temperature profiles on various quantities is discussed by displaying graphs. Comparison of the present solution yields an excellent agreement with an exact solution in a limiting sense.

In the present study, hydromagnetic convective flow and heat transfer of an absorbing and electrically conducting Cu-water nanofluid over a semi-infinite, ideally transparent, permeable stretching sheet due to solar radiation is considered. The flow considered is under both surface and thermal slip conditions. The governing equations are transformed into a nonlinear ordinary differential equations using classical Lie group approach which are solved numerically by means of the efficient numerical shooting technique with fourth-order Runge-Kutta scheme. The effects of involved parameters on the velocity and temperature profiles, skin friction and Nusselt number are examined and discussed through graphs and tables. Comparisons with previously published works are performed, and excellent agreement between the results is obtained. [ABSTRACT FROM AUTHOR]

In hydrodynamic lubrication, the slip conditions have been studied previously from the model of the slip length and the wall shear stress, but recently, in the literature on the slip at the interface fluid-solid, it has been shown that many questions will remain unsolved for long time to come. In fact, the modeling of the back-flow near the wall was not really taken into account. In this paper, we propose a more rigorous writing of the Reynolds equation (modified), taking into account the phenomena of slip and back-flow at wall. Subsequently, this equation is applied to the case of an inclined pad bearing and the results show that there is a significant influence of those phenomena on the operating characteristics. [ABSTRACT FROM AUTHOR]

[Display omitted] • Hydromagnetic flow of Reiner-Rivlin nanofluid over a rotating disk is addressed. • Thermophoresis and random diffusions are addressed. • Joule heating, thermal radiation and heat generation in heat expression are discussed. • Slip conditions are analyzed. • Physical aspect of entropy optimization is addressed. Non-Newtonian (Reiner-Rivlin) nanofluid is novel prospective in various biology and biomedical (medicine) engineering processes and in many other biological science. In addition, various biological liquids, such as saliva, synovial liquid, blood have non-Newtonian behavior and can demonstrate important viscoelastic characteristics. This treatise elucidates broad feature of cavitation in non-Newtonian liquids and bubble dynamics and applies them to the fields of bioengineering and biomedicine. In view of such biomedical and bioengineering applications the aim of this paper is to discuss the hydromagnetic flow of Reiner-Rivlin nanomaterial subject to a rotating disk is addressed. Disk rotates with constant angular frequency about vertical axis. Reiner's equation of a general viscous fluid differs from the Navier's Stokes equation by a quadratic expression describing cross-viscosity coefficient. Energy equation is obtained using thermal radiation and heat generation. Thermodynamics second law investigates the entropy. Multiple slip conditions are analyzed. Thermophoresis and random diffusion behaviors are addressed. First order reaction is also taken into account. Nonlinear systems are reduced to dimensionless system by employing similarity transformation. ND-solve procedure is implemented on Mathematics software for the computation of nonlinear differential system. Graphical representation of velocity, temperature and concentration are obtained. Entropy generation has been emphasized. Velocity has decaying trends for magnetic variable. Random diffusion variable have opposite trends for concentration and temperature. An intensification in thermal slip variable reduces temperature. [ABSTRACT FROM AUTHOR]

In this article, the effects of Newtonian heating along with wall slip condition on temperature is critically examined on unsteady magnetohydrodynamic (MHD) flows of Prabhakar-like non integer Maxwell fluid near an infinitely vertical plate under constant concentration. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on a newly introduced Prabhakar fractional operator with generalized Fourier's law and Fick's law. This fractional model has been solved analytically and exact solutions for dimensionless velocity, concentration, and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. Physical impacts of different parameters such as α , P r , β , S c , G r , γ , and G m are studied and demonstrated graphically by Mathcad software. Furthermore, to validate our current results, some limiting models such as classical Maxwell model, classical Newtonian model, and fractional Newtonian model are recovered from Prabhakar fractional Maxwell fluid. Moreover, we compare the results between Maxwell and Newtonian fluids for both fractional and classical cases with and without slip conditions, showing that the movement of the Maxwell fluid is faster than viscous fluid. Additionally, it is visualized that both classical Maxwell and viscous fluid have relatively higher velocity as compared to fractional Maxwell and viscous fluid. [ABSTRACT FROM AUTHOR]

Highlights: [•] We studied flow and mixing in a T-type micromixer at high Reynolds numbers. [•] The Reynolds number was varied from 1 to 1000. [•] Investigated the slip and non-slip boundary conditions on the channel walls. [•] It was shown that by using the hydrophobic coating can be controlled by mixing. [ABSTRACT FROM AUTHOR]

We perform a study for the boundary layer flow of viscous fluid about a stagnation point on an unsteady stretching sheet in the presence of variable free stream. Both the velocity and thermal slip conditions are taken into consideration. The stretching velocity and the surface temperature are time-dependent. In the solutions we adopt the homotopy analysis method (HAM). The analysis related to convergence of the solutions is explicitly discussed. The dependence of velocity and temperature profiles on various quantities is discussed by displaying graphs. Comparison of the present solution yields an excellent agreement with an exact solution in a limiting sense. [ABSTRACT FROM AUTHOR]

Abstract: This research is concerned with the peristaltic motion of a Johnson–Segalman fluid in an inclined asymmetric channel. The equations for a magnetohydrodynamic fluid in an inclined asymmetric channel are developed. Both the thermal and velocity slip conditions are used. Series solutions for stream function, temperature, pressure gradient and heat transfer coefficients are derived. The solutions are compared with the previous published work. Numerical integration has been performed for pressure rise per wavelength. Graphical results are presented and discussed for some embedded parameters. A comparative study with the existing available results is shown. [Copyright &y& Elsevier]

The peristaltic flow of a magnetohydrodynamic (MHD) Tangent hyperbolic fluid in an inclined asymmetric channel is investigated. The governing equations for the proposed fluid model are derived in Cartesian coordinates system. Simultaneous effects of slip and heat transfer have been taken into account for the present analysis. Regular perturbation method is utilised to get the solutions for stream function, temperature, pressure gradient and heat transfer coefficients. The present solutions are compared with the existing work and comparison is shown through table. The presented graphical results analyse the variations of embedded parameters into the flow system. [ABSTRACT FROM AUTHOR]

Abstract: In the present article, peristaltic flow of an incompressible six constant Jeffrey’s fluid model have been discussed. The flow have been analyzed in a wave frame of reference moving with the wave speed c. We have modeled the governing equations of a two dimensional and two directional flow under long wave length and low Reynold’s number approximation. Slip effects have been utilized for both velocity and temperature profile. The analytical solution for the proposed problem have been evaluated. The expression for the pressure rise is calculated using numerical integration. The graphical results are presented to interpret various physical parameters of interest for velocity, pressure gradient and temperature profile. [Copyright &y& Elsevier]

The problem of peristaltic flow of a nanofluid in an asymmetric channel is analyzed by taking into account the slip effects. The relevant equations for the nanofluid are presented and simplified by the long wavelength and small Reynolds number. Closed form solutions for stream function and pressure gradient are developed. However series expressions for temperature and Nanoparticle profiles are constructed. Finally, the influence of several parameters on the physical quantities of interest is discussed. [ABSTRACT FROM AUTHOR]

Abstract: The present article deals with the peristaltic flow of a Williamson fluid in an inclined asymmetric channel. The relevant equations have been modeled. Analysis has been carried out in the presence of velocity and thermal slip conditions. Expressions for stream function, temperature, pressure gradient and heat transfer coefficients are derived. The solutions are compared with the existing available results in a limiting sense. Numerical integration has been performed for pressure rise per wavelength. Plots are presented and analyzed for various embedded parameters into the problem. Comparison between the solutions is also shown. [Copyright &y& Elsevier]

Analysis has been carried out to examine the heat and mass transfer effects on the magnetohydrodynamics (MHD) peristaltic flow in a channel with compliant walls. An incompressible Maxwell fluid occupies the porous space. Modified Darcy's law and slip conditions are used in the problem formulation. Solutions for small wave number are derived. The effects of emerging parameters in the obtained solutions are displayed and discussed. [ABSTRACT FROM AUTHOR]

Investigation of the hydrodynamics and of the process of averaging a highly concentrated granulated loose medium in a vertical rectangular channel with a sudden contraction has been carried out on the basis of the model of a viscous nonlinear fluid. The characteristic features of the model of motion of such a medium consists in taking the slipping of particles on solid surfaces into account by introducing an independent empirical coefficient. The validity of numerical results was checked by comparing with the well-known experimental data. Based on the hydrodynamics obtained, the modeling of the process of averaging a granulated medium is presented, the intensity of which was analyzed depending on the regime parameters of flow. [ABSTRACT FROM AUTHOR]

Abstract: The present investigation is concerned with the peristaltic flow of an incompressible and magnetohydrodynamic MHD third order fluid in an inclined asymmetric channel. Both thermal and velocity slip conditions have been taken into account. The channel walls are maintained at different temperatures. The governing differential systems subjected to their boundary conditions have been solved for small Deborah number. In the derived solution expressions, the long wavelength and low Reynolds number assumptions are utilized. Graphical results are presented for the pressure rise, longitudinal velocity and temperature. Comparison with the previous published work regarding viscous fluid and no-slip case is performed. Pumping and trapping phenomena are displayed graphically just to examine the various interesting aspects of the present work. [ABSTRACT FROM AUTHOR]

Micro shock tube flows were simulated using unsteady 2D Navier–Stokes equations combined with boundary slip velocities and temperature jumps conditions. These simulations were performed using the parallel version of a multi-block finite-volume home code. Different initial low pressures and shock tube diameters allow to have the scaling ratio ReD/4L vary. The numerical results show a strong attenuation of the shock wave strength with a decrease of the hot flow values along the tube. When the scaling ratio decreases the shock waves can transform into compression waves. Comparison to the existing 1D models also shows the limit of these models. [ABSTRACT FROM AUTHOR]

Abstract: A set of slip-flow boundary conditions for the flow due to a lubricated disk rotating in a Newtonian fluid is derived. Similarity solutions are generally prohibited by the replacement of the conventional no-slip conditions by the new slip-flow conditions, except in the particular case when the power-law index of the non-Newtonian lubricant equals 1/3. The amount of velocity slip is controlled by a single dimensionless slip coefficient. Numerical solutions are presented for this case, showing that the three-dimensional flow field is dramatically affected by accentuated velocity slip. In particular, the axial flow towards the disk, i.e. the pumping efficiency, and the torque required to maintain steady rotation of the disk, decrease monotonically with increasing slip. [Copyright &y& Elsevier]

Here, direct noninvasive neutron reflectivity measurements reveal the presence of a reduced (deuterated) water density region, with a sigmoidal density profile at the hydrophobic silane-water interface that depends on the type and concentration of dissolved gases in the water. Removal of dissolved gases decreases the width of the reduced water density region, and their reintroduction leads to its increase. When compared with recent computer simulations, a locally fluctuating density profile is proposed, whereas preexisting nanobubbles are excluded. The presence of a fluctuating reduced water density region between two hydrophobic surfaces and the attractive "depletion force" to which it leads may help explain the hydrophobic force and its reported diminution in deaerated water. Our results are also quantitatively consistent with recent dynamic surface force apparatus results that drastically revise previous estimates of the slip length of water flowing past hydrophobic surfaces from microns to ≈20 nm. Our observations, therefore, go a long way toward reconciling three quite different types of experiments and phenomena: water depletion at hydrophobic surfaces, water slip at hydrophobic surfaces, and the hydrophobic interaction. [ABSTRACT FROM AUTHOR]

Kinetic boundary conditions (in particular, slip conditions) are used in the theory of rarefied gases; these boundary conditions are a corollary of the Boltzmann equation and make it possible to use the Navier-Stokes equations for describing flows of moderately rarefied gases. It is demonstrated that the slip conditions may be derived as a result of solution of generalized hydrodynamic equations (GHE). In other words, the GHE may be used for an adequate description of the kinetic wall layer. [ABSTRACT FROM AUTHOR]

This study presents the magnetized hybrid nanofluid flow with heat source/sink over an exponentially stretching/shrinking sheet. Slip conditions are implemented to analyze the hybrid nanofluid flow for both slip and no-slip conditions. Additionally, the hybrid nanofluid of alumina and copper (hybrid nanoparticles) with blood (base fluid) has been considered and discussed with both suction and injection parameters. The appropriate similarity variables are used to convert partial differential equations (PDEs) into ordinary differential equations (ODEs) and solved analytically with the help of the homotopy analysis method (HAM). The impact of different embedded parameters has been shown in the form of graphs and tables. The numerical values of skin friction and Nusselt number are presented in the form of Tables for both slip and no-slip cases. It is summarized that the upsurge of the velocity slip parameter and magnetic parameter increases the skin friction, while the rising of the thermal slip parameter and heat generation parameter decreases the Nusselt number. [ABSTRACT FROM AUTHOR]

Time-dependent and incompressible squeezing flow of Casson and micropolar nanofluids is analyzed. The non-Newtonian liquids are confined through parallel disks. The bottom disk is saturated in the plane z = 0 and the fluid is squeezed by the movement of the upper disk along axial direction. Nanofluid theory (Buongiorno model) is accomplished in the flow phenomenon. A uniform injection/suction is act upon at the lower disk. Velocity, thermal and concentration slip effects are also incorporated at the bottom fixed disk. The similarity functions are first accomplished in order to achieve the governing system of ordinary differential equations and then the system is solved in numerical manner using Runge-Kutta-Fehlberg fourth fifth (RKF-45) order numerical procedure. Results are disclosed in tabular and graphical forms against the various physical quantities. For the validation of our numerical technique, the results are compared well with the literature work under limiting case. It is perceived that increased squeezing Reynolds number pushed the radial velocity profiles towards the upper disk. The micropolar parameter tends to the fluid rotation in opposite way due to which microrotational field first enhanced and then reduced by the vortex viscosity parameter. [Display omitted] [ABSTRACT FROM AUTHOR]

This article highlights the influence of mixed convection, Hall current and magnetic field on peristaltic motion of Powell-Eyring nanofluid in the symmetric channel. Energy equation include the viscous dissipation, Ohmic heating and thermal radiation effects. Brownian motion and thermophoresis are considered to explore the nanofluid characteristics. Velocity slip, thermal jump and zero mass flux conditions are considered on the boundary wall. Further temperature dependent viscosity is taken into account. Lubrication approach is used to simplify the dimensionless form of governing equations. Final form of equations are solved numerically. Velocity, temperature, concentration and entropy generation behaviors are studied with the help of graphs. Temperature depicts increasing behavior for higher magnetic field whereas the opposite trend is noticed for Hall effects. [ABSTRACT FROM AUTHOR]

In present study the impacts of thermal radiation and Joule heating for MHD peristaltic motion of fourth-grade nanoliquid in a channel are addressed. Elastic characteristics of channel walls are studied. Velocity, concentration and thermal slip effects are imposed. Furthermore the influences of viscous dissipation, heat source /sink and chemical reaction in modelling are retained. Nanofluid model consisting of thermophoresis and Brownian motion is employed. Mathematical formulation is developed by adopting long wave length and low Reynolds number. Numerical technique is used for finding the solutions of the governing problem. Comprehensive study of emerging flow variables for velocity, temperature, concentration of nanoparticles, heat transfer coefficient and trapping is addressed graphically. Moreover the results for skin friction coefficient and Sherwood number are also examined. [ABSTRACT FROM AUTHOR]

• These comments are increasing the awareness among the journal readers. • Comments are presented on the paper by Hayat et al. [1]. • The damping parameter (E 3) is dimensional. • The Hartmann number (M) is dimensional. • The dimensionless velocity slip parameter (β 1 *) is dimensional. This discussion is presented to increase the awareness among the readers of Results in Physics. Comments are presented on the paper by Hayat et al. (2018), where the authors investigated MHD Williamson fluid using modified Darcy's law. These comments make obvious that the damping parameter (E 3) is dimensional. Also, the Hartmann number (M) is dimensional. In addition, the dimensionless velocity slip parameter (β 1 *) is dimensional in the above paper. [ABSTRACT FROM AUTHOR]

This paper investigates the steady, two dimensional, and magnetohydrodynamic flow of copper and alumina/water hybrid nanofluid on a permeable exponentially shrinking surface in the presence of Joule heating, velocity slip, and thermal slip parameters. Adopting the model of Tiwari and Das, the mathematical formulation of governing partial differential equations was constructed, which was then transformed into the equivalent system of non-linear ordinary differential equations by employing exponential similarity transformation variables. The resultant system was solved numerically using the BVP4C solver in the MATLAB software. For validation purposes, the obtained numerical results were compared graphically with those in previous studies, and found to be in good agreement, as the critical points are the same up to three decimal points. Based on the numerical results, it was revealed that dual solutions exist within specific ranges of the suction and magnetic parameters. Stability analysis was performed on both solutions in order to determine which solution(s) is/are stable. The analysis indicated that only the first solution is stable. Furthermore, it was also found that the temperature increases in both solutions when the magnetic parameter and Eckert number are increased, while it reduces as the thermal slip parameter rises. Furthermore, the coefficient of skin friction and the heat transfer rate increase for the first solution when the magnetic and the suction parameters are increased. Meanwhile, no change is noticed in the boundary layer separation for the various values of the Eckert number in the heat transfer rate. [ABSTRACT FROM AUTHOR]