Multiparticle production data on proton-nucleus collisions have been analyzed taking the number of ‘created’ charged particles instead of the observed number of shower particles as the variable. The mean normalized multiplicity,R_A, has been found to be independent of energy in the energy range (7–8000) GeV and its mass number dependence has been obtained. The modified analysis introduces some more regularities in the experimental results onp-nucleus collisions like the invariance with respect to energy of the relationshipR_A = α + βN_h and the KNO-like scaling of the multiplicity distributions of the created charged particles. The functional form of the scaling function has been calculated.

This article communicates the numerical consideration of 3D Carreau liquid flow under the impact of chemical responses over a stretched surface. Moreover, the heat transfer exploration is carried out with a view to improve the heat flux relation. This phenomenon is established upon the theory of Cattaneo–Christov heat flux relation that contributes by the thermal relaxation. On exploitation of an appropriate transformation a system of nonlinear ODEs is attained and then elucidated numerically by means of bvp4c scheme. The descriptions of temperature and concentration fields equivalent to the frequent somatic parameters are graphically scrutinised.Our analysis carries that the concentration of the Carreau liquid displays similar tendency and decline as theheterogeneous–homogeneous reaction parameters ($k_{2}, k_{1}$) augment. Furthermore, it is notable that for shear thinning ($n$ & lt; 1) liquid, the influence of local Weissenberg numbers ($We_{1}, We_{2}$) are absolutely conflicting compared with the instance of shear thickening ($n$ & gt; 1) liquid. Additionally, validation of numerical results is done via benchmarking with previously stated limiting cases with two different schemes namely, homotopy analysis method (HAM) and bvp4c scheme. These comparisons initiate a superb correspondence with these outcomes.

The pursuit of superior working liquids for heat/mass transfer mechanisms in engineering is on the rise, not only to maximise revenue but also to accommodate heat dissipation or chemical separation under extreme conditions. The addition of a small amount of nanoparticle, i.e. a product called nanofluid, has been initiated over the last decade. In this paper, we present a comprehensive study of unsteady three-dimensional (3D) flow of the Eyring–Powell nanofluid under convective and nanoparticles mass flux conditions. The effects of constructive/destructivechemical reactions and nonlinear thermal radiation are also considered in the Eyring–Powell nanofluid model. Additionally, suitable transformations are utilised to obtain coupled ordinary differential equations (ODEs) from the system of partial differential equations (PDEs) and the numerical solution of the system of the coupled ODEs is obtained by means of the bvp4c scheme. The obtained numerical data are plotted for the temperature and concentration profiles of nanofluids for various and converging values of physical parameters. Our findings demonstrate that the temperature of the Eyring–Powell nanofluid fall-off by changing the heat sink parameter. Furthermore, it is perceived from the sketches that the concentration of Eyring–Powell magneto-nanofluid decays at higher values of chemical reaction parameter.

A simple relation of chemical processes for three-dimensional flow of a cross magnetofluid over bidirectional stretched surface is constructed. The impact of convective heat transport in the manifestation of non-linear thermal radiation and features of heat source–sink are also considered for heat transfer mechanism.Furthermore, in this research paper, the innovative relation between heterogeneous and homogeneous responses with equivalent diffusivities for reactant and autocatalysis is exploited. Apposite alterations are guaranteed to obtain ordinary differential equations (ODEs) with high nonlinearity. Numerically, the bvp4c technique is exploited to the interpret the structure of ODEs. Portrayals of temperature and concentration for cross liquid equivalent to abundant somatic parameters are presented graphically as well as in tabular form. Our results reveal that the temperature of the cross fluid decreases with fluctuation in the heat sink parameter. Furthermore, it is perceived from the figuresthat the concentration of the cross fluid reduces for higher values of chemical reaction parameters.

Nanoliquids possess remarkable features that have fascinated numerous researchers because of their utilisation in nanoscience and nanotechnology. A mathematical relation for the two-dimensional flow of Maxwell nanoliquid over a stretching cylinder is established. Buongiorno’s relation is considered here to visualise the impact of Brownian moment and thermophoresis mechanisms on Maxwell liquid. The convective heat transport is deliberated for heat transfer mechanisms. Transformation procedure yields nonlinear differential system which is then computed through the homotopic approach. The results obtained are studied in detail in relation to somatic parameters. It is notable that the velocity of Maxwell liquid shows conflicting behaviour for curvature parameter $\alpha$ and Deborah number $\beta$. Moreover, the liquid temperature increases for increased values of Brownian motion $N_{b}$ and thermophoresis parameter $N_{t}$ . Additionally, the authentication of numerical consequences is prepared via benchmarking with formerly identified restrictive circumstances and we initiate a splendid communication with these results.

The current review proclaims the forced convective flow and heat–mass transfer characteristics of cross nanofluid past a bidirectional stretched surface. The most significant aim of the current review is to incorporatethe features of Buongiorno relation, activation energy, nonlinear thermal radiation and heat sink–source for a three-dimensional flow of the cross fluid. Appropriate transformations are employed to transform the modelledpartial differential equations (PDEs) of momentum, temperature and concentration into coupled nonlinear ordinary differential equations (ODEs). The governing boundary value problem is numerically integrated with the help of bvp4c scheme. The obtained numerical data are plotted for the temperature and concentration profiles of nanofluid for various converging values of physical parameters. The dependence of increasing thermophysical parameters on temperature and concentration profiles of the cross nanofluid is graphically demonstrated. Furthermore, detailedstudy reveals that the concentration of the cross nanofluid decreases for increasing values of Brownian motion parameter. It is also perceived from the sketches that the concentration of the cross nanofluid decreases for higher values of chemical reaction parameter. The validity of the achieved numerical outcomes is ensured by making a comparison with the existing work as special cases.

This research demonstrates the diverse characteristics of the cross fluid in the presence of Lorentz’s forces. Moreover, this work reviews the characteristics of variable diffusivity and variable conductivity. Mathematical modelling of the presented physical model is carried out in the Cartesian coordinate system and the formulated system of partial differential equations (PDEs) is simplified in ordinary differential equations (ODEs). Numerical algorithm leads to solution computations. Velocity, temperature and concentration are numerically analysed for the cross fluid. Outcomes of the current physical model are presented through graphical data and in tabular form. It is noted that variable conductivity and variable diffusivity significantly affect heat–mass transport mechanisms. Furthermore, graphical analysis reveals that the concentration of the cross nanofluid increase for increased values of variable diffusivity. Furthermore, this research reveals that concentration distribution is a reducing function of chemical reaction parameters.