Boundary Layer Phenomena Research Articles

On the enhancement of acceptable blockage of open jet wind tunnel by employing modifications based on concepts of boundary layer and jet flow for testing of aero-rotors

The conventional closed and open jet tunnels suffer from various adverse blockage effects that alter the flow conditions generating erroneous results when larger than allowable blockage models are tested. The maximum acceptable blockage is limited to 25 % for an open jet tunnel. The physics behind the inferior blockage capacity and their redressal is investigated. Hitherto, numerical models are developed to rectify the blockage troubles that introduce an additional complex case-based mathematical modeling to reckoning the results. In the current paper novel approach to redesigning the tunnel to solve blockage troubles is investigated that facilitates evading of mathematical techniques and securing accurate results straightforwardly. A hypothetical design incorporating the remedies is proposed, capable of conditioning the flow in a manner that debilitates the adverse effects of solid blockage, wake blockage, jet expansion, etc. The hypothesis is theoretically expounded by the boundary layer and jet flow phenomenon. The modified blower-type open jet wind tunnel constitutes a centrifugal blower, long inlet duct, plenum chamber around the test section, and collector. The modified wind tunnel is tested through experiments executed on four different-sized Savonius rotors creating 10 %, 30 %, 50 %, and 60 % blockage. The static pressure along the contours of the blades of rotors is obtained. The experimental results are juxtaposed with CFD results carried out at free-wind boundary conditions and standard classic experimental results performed in conventional wind tunnel at 11 % blockage. A blockage enhancement of 25 % is achieved by the approach of modification in the design of a conventional open jet wind tunnel.

Non-linear radiative second-grade nano fluid with sinusoidal magnetic force and arrhenius activation energy: A computational exploration

The goal of this work is to further improve our knowledge of the nonlinear radiative second-grade nano fluid flow boundary layer phenomena which is associated with an Arrhenius activation energy, a sinusoidal magnetic field, and a stretched peripheral with a heat source. The unsteady governing equations are transformed into a proper dimensionless arrangement, and then the explicit finite difference (EFD) method is applied to numerically calculate the equations. However, precise stability and convergence criteria have been developed to make the solution convergent. Along with the typical profile of other flow fields, the oscillatory forms of the velocity are shown. Tabular research has even demonstrated a relationship between the Nusselt number and other parameters, and graphical depiction has been used for regression and data prediction. The novel conclusions drawn from this research indicate that, in comparison to linear patterns, nonlinear radiative heat flux significantly raises (30.35 %) flow profiles with second-grade characteristics. Moreover, the heat transfer rates of second-grade Nano fluids are seen to be significantly influenced (35.14 %) by the sinusoidal magnetic component. When considering nonlinear thermal radiation, activation energy principles cause a major change (34.19 % more) in mass transmission, as high-temperature processes become an essential part of chemical reactions.

Flow over traveling and rotating cylinders using a hybrid Eulerian–Lagrangian solver

Hybrid Eulerian–Lagrangian solvers have gained increasing attention in the field of external aerodynamics, particularly when dealing with strong body–vortex interactions. This approach effectively combines the strengths of the Eulerian component, which accurately resolves boundary layer phenomena, and the Lagrangian component, which efficiently evolves the wake downstream. This study builds on our team’s previous work by enhancing the capabilities of a two-dimensional hybrid Eulerian–Lagrangian solver. We aim to upgrade our solver which was initially designed for static cases, to now also simulate cases involving moving objects. To ensure the reliability and applicability of a new solver, it is essential to validate its performance in complex cases. Here, the solver is validated across the case of a traveling cylinder and the case of a rotating cylinder in two different rotational speeds at low Reynolds numbers. In the realm of Eulerian solvers, such as OpenFOAM (utilized for the Eulerian component of this hybrid approach), traditional techniques include the use of morphing meshes, overset meshes, and Arbitrary Mesh Interfaces (AMI) to model body motion. The proposed methodology involves extending the Eulerian mesh up to a short distance from the solid boundary and moving it entirely as a solid entity. Then the Lagrangian solver is responsible for calculating the updated boundary conditions, thereby completing the hybrid solver’s functionality. This approach is very similar to the overset mesh technique. However, unlike the traditional method where an Eulerian mesh moves on top of a static one, our method involves the motion of an Eulerian mesh over a Lagrangian grid. We compared the results from our hybrid solver with those from a purely Eulerian solver, specifically OpenFOAM. The comparison demonstrates that our solver can replicate OpenFOAM’s results with high accuracy. Another interesting point highlighted in this study is the presence of high-frequency oscillations in the body forces in hybrid solvers that incorporate the redistribution of Lagrangian particles and do not utilize surface elements such as vortex panels, specifically when dealing with dynamic mesh simulations. When the Eulerian mesh travels on top of the Lagrangian grid of particles, the positions of the particles with respect to the Eulerian mesh continuously change. This results in a constant shift of particles near the solid body, where the highest vorticity is observed. Particles that are close to the solid boundary at one time step may find themselves inside the boundary at the next time step, leading to their removal. This pattern continuously changes during the simulation, causing fluctuations in the boundary conditions of the Eulerian solver and manifesting as oscillations in the forces acting on the body. It is shown that this issue can be alleviated either by increasing the spatial resolution of the Lagrangian solver or by synchronizing the movement of the Lagrangian grid with the motion of the Eulerian mesh. The results of the study make the solver trustworthy and pave the way for more demanding external aerodynamic simulations.

Thermally Driven Winds on Mars: A Review and a Slope Effect Numerical Study

AbstractRegional and local thermally driven winds are planetary boundary layer phenomena frequently observed on Mars over sloping and flat regions, when diurnal surface temperature variations are significant and large‐scale winds are sufficiently weak. In particular, slope flows are prevalent in many areas on Mars, where they can reach high speeds and have a substantial impact on the near‐surface wind patterns, pressure diurnal cycle, aeolian activity, etc. This work first reviews literature on Martian winds, listing wind speeds obtained either numerically or from in situ measurements, with special emphasis on slope flows. Second, a methodology is presented to perform numerical simulations of slope flows on Mars using the open‐source computational fluid dynamics code OpenFOAM. Slope winds are then studied in a simplified Martian mountain‐valley configuration, using realistic values for various parameters, such as the slope angle and temperature variation. The incompressible Navier‐Stokes equations with Boussinesq approximation are used, along with the k‐ɛ Reynolds‐averaged Navier‐Stokes turbulence model. Radiant heat transfer is included and proven to be paramount for a correct description of Martian slope winds. Results of velocity and temperature profiles are obtained for various slope angles. These simulations aim to provide further insight on the behavior of slope winds on Mars, to simplify the process of assessing numerous potential landing sites for upcoming missions, since these highly regular winds could be useful, for instance, for dust removal applications.

Temperature Gradient Effects on Stator Boundary-Layer Stability in a Rotor–Stator Cavity

A comprehensive approach, combining theoretical analysis and direct numerical simulation, is employed in this study to investigate the influence of temperature gradient on the stability phenomenon of the stator boundary layer in a rotor–stator cavity. In contrast to previous studies, a temperature term is introduced to account for centrifugal buoyancy within the cavity. The focus is on analyzing the transitional behavior and the effects of centrifugal buoyancy on the boundary layers of the stationary disk under operating conditions characterized by a Reynolds number of Re=2.5×104. The investigation reveals that this temperature gradient significantly affects the base flow and alters the instability governing the boundary-layer transition on the stationary disk. Specifically, the centrifugal buoyancy induced by the higher temperature on the stationary side weakens the spiral mode perturbations without inducing changes in the azimuthal wavenumber of the spiral mode. However, when the centrifugal buoyancy effect exceeds a certain threshold, it directly suppresses the generation of the spiral mode and induces the formation of low-radius circular waves, thereby promoting a more stable boundary layer. This research emphasizes the importance of considering temperature variations in the rotor–stator cavity for improved control of stability within the boundary-layer flow.

Numerical modeling of a MHD non-linear radiative Maxwell nano fluid with activation energy

The present research explores linear as well as nonlinear radiation patterns based on the MHD non-Newtonian (Maxwell) nanofluid flow having Arrhenius activation energy. This study's core focus is MHD properties in non-Newtonian fluid dynamics and boundary layer phenomena analysis. It initiates with time-dependent equations, employing boundary layer approximations. Extensive numerical computations, executed with custom Compact Visual Fortran code and the EFD method, provide profound insights into non-Newtonian fluid behavior, revealing intricate force interactions and fluid patterns. To check the stability of the solution, a convergence and stability analysis is performed. With the values of ΔY = 0.25, Δτ = 0.0005, and ΔX = 0.20; it is found that the model convergence occurs to the Lewis number, Le > 0.016 as well as the Prandtl number, Pr > 0.08. In this context, investigating non-dimensional results that depend on multiple physical factors. Explanation and visual representations of the effects of different physical characteristics and their resultant temperatures, concentrations, and velocity profiles are provided. As a result of the illustrations, the skin friction coefficient and Sherwood number, which are calculated, as well as Nusselt values, have all come up in discussion. Additionally, detailed representations of isothermal lines and streamlines are implemented, and it is pointed out that the development of these features occurs at the same time as Brownian motion. Furthermore, the temperature field for Maxwell fluid is modified due to the impression of chemical reaction as well as the Dufour number (Kr and Du). Our research demonstrates the superior performance of non-Newtonian solutions, notably in cases involving activation energy and nonlinear radiation. This paradigm shift carries significant implications. In another context, the interplay between Maxwell fluid and nonlinear radiation is notably affected by activation energy, offering promising applications in fields like medicine and industry, particularly in groundbreaking cancer treatment approaches.

Refractivity Observations from Radar Phase Measurements: The 22 May 2002 Dryline Case during IHOP Project

The dryline, often associated with the development of severe storms in the Southern Great Plains of the United States of America, is a boundary layer phenomenon that occurs when a warm and moist air mass from the Gulf of Mexico meets a hot and dry air mass from the southwest desert area. An accurate knowledge of the water vapor spatio-temporal variability in the lower part of the atmosphere is crucial for a better understanding of the evolution of the dryline. The tropospheric refractivity, directly related to water vapor content, is a proxy for the water vapor content of the troposphere. It has already been demonstrated that the refractivity and the refractivity vertical gradient can be jointly estimated from radar phase measurements. In fact, it has been shown that using kriging interpolation techniques, accurate refractivity maps within the coverage area of the radar can be obtained with high temporal resolution. In this paper, a detailed analysis of the time series of radar-based refractivity maps obtained during a dryline that occurred on the afternoon of 22 May 2002 during the International H2O Project (IHOP_2002) is presented. Comparisons between the time series of radar refractivity maps, obtained with the NCAR S-Pol radar, and the refractivity measurements derived from automatic ground-based weather stations and the AERI instrument, placed at different locations within the coverage area of the NCAR S-Pol radar, demonstrate the accuracy of radar refractivity estimates even for highly variable conditions, both in time and space, in the troposphere. Correlation coefficients higher than 0.95 are obtained in all weather station locations. Regarding the RMSE, errors less than 6 N-units are obtained for all cases, being even as low as 2.92 N-units at some locations.

Hiemenz stagnation point flow of a second-order micropolar slip flow with heat transfer

Purpose This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions. Design/methodology/approach The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions. Findings The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases. Originality/value The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up:•The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings.•Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm.•Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature.

On the Maxwell‐Bloch system in the sharp‐line limit without solitons

AbstractWe study the (characteristic) Cauchy problem for the Maxwell‐Bloch equations of light‐matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically‐motivated initial‐boundary conditions are satisfied. In particular, we present a proper Riemann‐Hilbert problem that generates the unique causal solution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading‐order asymptotics to self‐similar solutions that satisfy a system of ordinary differential equations related to the Painlevé‐III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schrödinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self‐similar solutions. We make this observation precise and for the first time we relate the PIII self‐similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially‐unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state.

Boundary layers for a fluid–particle interaction system with density-dependent viscosity and cylindrical symmetry

In this paper, we study the initial boundary value problem for a cylindrical symmetry fluid–particle interaction system in three dimensions. The boundary layer phenomena is investigated when the shear viscosity μ=κρβ goes to zero. Furthermore, we establish the boundary layer thickness of the order O(κα) for more general initial data when 0<α<12 and give the optimal boundary-layer thickness for the system with more general initial data. As a byproduct, this work improves the corresponding results in Yao et al. (2011) for isentropic compressible Navier–Stokes equations where 0<α<14.

Reliability analysis of floating wind turbine dynamic cables under realistic environmental loads

The need for more robust design of dynamic cables used for power export in floating wind turbine (FWT) systems is accentuated by the frequent occurrence of power cable failures. Such failures contribute considerably to the cost of global offshore wind farm losses. Fatigue is often times a critical consideration in the structural safety of these cables. This is understandably so, given that they experience numerous loading cycles–mainly induced by the combined action of wind and waves throughout their service life. It is therefore pertinent that the reliability level attainable by these cables is quantified and elaborated upon. In this paper, a probabilistic reliability analysis approach is developed to quantify the reliability levels of dynamic power cables. Uncertainties emanating from the randomness of realistic environmental scatter, geometric and material variables are all taken into account in this study. To facilitate the computation of the structural demands on the dynamic power cable, an efficient analytical model was developed. Key aspects such as cable–soil interaction and boundary-layer phenomenon are captured by the analytical model. Kriging metamodel is then employed to propagate relevant uncertainties into the reliability problem, making it possible to quantify the reliability levels of these cables.

Analysis of divergence free conforming virtual elements for the Brinkman problem

In this paper, we develop stability analysis, including inverse inequality, [Formula: see text] norm equivalence and interpolation error estimates, for divergence free conforming virtual elements in arbitrary dimension. A local energy projector based on the local Stokes problem is suggested, which commutes with the divergence operator. After defining a discrete bilinear form and a stabilization involving only the boundary degrees of freedom (DoF) and parts of the interior DoF, a new divergence free conforming virtual element method is advanced for the Brinkman problem, which can be reduced to a simpler method due to the divergence free discrete velocity. An optimal convergence rate is derived for the discrete method. Furthermore, we achieve a uniform half convergence rate of the discrete method in consideration of the boundary layer phenomenon. Finally, some numerical results are provided to validate the convergence of the discrete method.

The history of UK weather forecasting: the changing role of the central guidance forecaster. Part 6: the late twentieth century: forecasting smaller‐scale features

During the final two decades of the twentieth century, major strides had been made in forecasting synoptic-scale features such as fronts and cyclones. However, the forecasting of embedded smaller-scale features which could produce significant weather remained challenging. In addition, forecasting convection remained highly problematic. New conceptual models were developed, which when used in conjunction with satellite and radar imagery, assisted forecasters in both the understanding and short-period prediction of such features. A non-hydrostatic mesoscale computer model gradually became integrated into operational use by forecasters, providing some additional guidance on mesoscale precipitation distribution as well as boundary layer phenomena.

Combined impacts of thermoelectric and radiation on hydromagnetic nanofluid flow over a nonlinear stretching sheet

Controlling heat transmission in the polymer manufacturing business has always depended on the thermal radiation component and the development of a variety of novel technical processes, such as solar energy technology and combustion of fossil fuels occur at high temperatures. Therefore, it would appear significant to study heat radiation and because of that this research delivers a numerical evaluation of the cumulative influence of Hall current and radiation of heat on MHD fluid flow across stretchable surface which is non-linear. Nevertheless, it has been considered that the fluid conveys tiny (1-100 nm) particles. The model equations have been made non-dimensional using the boundary layer approach and non-dimensional components. Then, using the finite difference approach, the acquired non-dimensional equations are numerically constructed. The solution of the fluid flow is discussed to investigate the primary plus secondary velocity, temperature but also concentration ordination. Skin friction, coefficient of transmission of mass, and coefficient of transmission of heat are some of the significant boundary layer phenomena that are investigated. Streamlines, as well as isothermal lines, have also been employed to display an upgraded representation of the fluid flow. A closer examination reveals that the cumulative effects of the Hall current and radiation parameter have a larger influence on primary and secondary velocity than the individual impacts of the Hall current and radiation factor. It is also revealed that when both radiation and Hall current influences are included, the heat transfer capabilities of the fluid are considerably higher than when only one of them is examined.

Явление погранслоя в алгебро-дифференциальном уравнении первого порядка

The Cauchy problem for the first-order algebraic differential equation is considered A du/dt=(B+εC+ε^2 D)u(t,ε), u(t_0,ε)=u^0 (ε)∈E_1, where A,B,C,D are closed linear operators acting from a Banach space E_1 to a Banach space E_2 with domains everywhere dense in E_1, u^0 is a holomorphic function at the point ε=0, ε is a small parameter, t∈[t_0; t_max]. Such equations describe, in particular, the processes of filtration and moisture transfer, transverse vibrations of plates, vibrations in DNA molecules, phenomena in electromechanical systems, etc. The operator A is the Fredholm operator with zero index. The aim of the work is to study the boundary layer phenomenon caused by the presence of a small parameter. The necessary information and statements are given. A bifurcation equation is obtained. Two cases are considered: a) boundary layer functions of one type, b) boundary layer functions of two types. Newton’s diagram is used to solve the bifurcation equation. In both, the conditions under which boundary layer phenomenon arises are obtained — these are the conditions for the regularity of degeneracy. Case a) is illustrated by an example of the Cauchy problem with certain operator coefficients acting in the space R^4.

Impact of an MHD Cattaneo-Christov model for a Williamson fluid flow across a wedge with a convective boundary condition: Homotopy Analysis Method

The present study aims to scrutinise the influence of the Cattaneo-Christov (CC) heat flux model on magnetohydrodynamic (MHD) laminar boundary-layer flow through a wedge in a non-Newtonian Williamson fluid along with convective boundary condition embedded in a porous medium. Significantly, the presence of the CC model explores the heat transfer characteristics which forecast the effect of thermal relaxation time on the boundary-layer phenomenon. The governing equations are formulated and modified into non-linear, coupled ODEs by utilising similarity transformations and then tackled by Homotopy Analysis Method (HAM). The obtained results are discussed and presented graphically for various values of emerging parameters on velocity and temperature distributions. The results reveal that the velocity is an enhancing function of the magnetic parameter, permeability parameter and Hartree pressure gradient parameter whereas decreasing with larger values of the Williamson fluid parameter. It is evident that the temperature upsurges due to the Biot number and decays with the relaxation parameter. The numerical values of skin-friction and Nusselt number are tabulated. Comparative reviews between the formerly published literature and the current data are made for specific cases, which are inspected to be in tremendous agreement. Abbreviations: BLF: Boundary layer flow; MHD Magnetohydrodynamic; CC model: Cattaneo-Christov model; NNF: non-Newtonian fluid; ODEs: Ordinary differential equations; PDEs: Partial differential equations

Numerical Method to Generate and Evaluate Environmental Wind Over Hills: Comparison of Pedestrian Winds Over Hills and Plains

It is well known that the wind profile at altitudes below 10m from mean sea level (MSL) depends on the geometry of terrain, due to the boundary layer phenomenon. Hence, the profile of wind changes for hilly terrains and mountainous regions when compared with the plain regions. This phenomenon has become important to study due to the large-scale urbanisation taking place over hilly regions. The changing wind profile presents a challenge to evaluate the pedestrian winds, as depending on the aspect of the terrain an additional vertical velocity component is experienced due to the upwind climb of the winds. This creates a wind profile that is twisted in form. While wind tunnel studies have attempted to recreate this twisted wind profile (TWP), due to the inherent deficiency of wind tunnels to simultaneously map velocity and flow conditions, a lack of three-dimensional flow profile hinders pedestrian comfort evaluation. In the wind tunnel studies, it was also observed that small vertical eddies and wakes behind the interfering building were not identified which are an important factor to determine the pollution load dispersion. The authors have developed a numerical model to generate the twisted wind profile. The specialty of the numerical model lies in it’s unique boundary conditions that enable the visualization and quantification of the complete 3D wind profile, when the wind over a hilly terrain interacts with urban infrastructures. The developed model was validated with the wind tunnel experiments done previously by Tse and colleagues. The specialty of the model is that it ensures horizontal homogeneity while creating vertical heterogeneity. From the 3D flow profile hence generated the authors were able to deduce that the impact of twisted wind profile depends on the yaw angle of wind interacting with the structure and not on the wind attack angle. Also, the more the twist of the wind, more is the clockwise shifting of the far wakes behind the building. It was also seen that there are more low velocity zones in the pedestrian winds over a hill in comparison to that over the plains. The vertical eddies that aid in convective removal of pollutants were also missing in case of pedestrian winds over hilly terrains, which raises the risk of pollutant accumulation. The same was also observed in Hong-Kong during COVID 19, where due to the twisted nature of wind flow, the virus load increased and natural ventilation was inadequate in the removal of the viral load in the air near urban areas.

Entropy generation of MHD micropolar nanofluid past an exponentially stretching plate with higher order power-law slip model

Nonlinear power-law micropolar fluid (MF) distinguishes itself from other non-Newtonian fluids by the essential effects of microrotation characteristic, rheological properties and even electric conductive feature; among which microrotation as a vital factor for MF fluid flow can't be ignored. We highlight the contributions of the angle velocity and propose a mixed higher-order slip model based on it. Further, the established model is utilized to study boundary layer phenomena of MHD MF over an exponentially stretching plate. A set of numerical codes on account of collocation spectral method (CSM) is developed directly. The space derivatives are achieved with matrix multiplication (MM) and fast cosine transform (FCT) simultaneously. Entropy and Bejan number are analyzed through the second law of thermodynamics. This study reveals, for the first time, irrespective of power-law slip model may underestimate the actual temperature and concentration of the power-law MF. Consideration of the present model strengthens the attenuation effect on U distribution while diminishes the entropy generation. Viscous dissipation effect is suppressed by the increase of slip parameters. Dual directional heat and mass transfer are revealed due to the microrotation characteristic and the rheological properties of power law fluid. Nonlinear relation of Ns and the slip parameters is revealed.

Thermal energy and mass transport of shear thinning fluid under effects of low to high shear rate viscosity

The current investigation is deliberated the flow and heat transfer of a shear thinning fluid over a non-linear stretching sheet has variable thickness. All rheological aspects at low to high shear rates are accounted theoretical by using generalized Carreau model of viscosity. Theoretical flow model is formulated for boundary layer phenomena by applying boundary layer approximations and then convert it from partial differential equations to ordinary differential equations with the help of similarity transformations. The solution is obtained by numerical method and the result are displayed in the form of velocity and temperature profiles under impact of rheological and geometrical governing parameters. In addition, the results of skin friction coefficient and Nusselt number are obtained under effects of these parameters. It is found that Nusselt number is significantly decreased when stretching is increased by velocity index parameter.