Permeable Sheet Research Articles

Comparison of Adomian Decomposition Method with Differential Transformation Method for Unsteady MHD Flow and Heat Transfer Over a Stretching/Shrinking Permeable Sheet with Ohmic Heating

In this paper, two semi-analytical techniques were implemented to solve a two-dimensional unsteady MHD fluid flow and heat transfer through a stretching/shrinking permeable sheet with ohmic heating and viscous dissipation. The governing flow equations in PDE form were reduced to ordinary differential equations using appropriate similarity transformation. We obtained approximate expressions for the velocity and temperature profiles. Comparative results obtained employing Adomian decomposition method and differential transformation method were benchmarked against a numerical solution using Keller box scheme. Our findings revealed that the approximate analytical solution obtained using DTM is more dependable with fast convergence, highly accurate with minimal calculations and computationally convenient. However, the requirement of Adomian polynomials to tackle the nonlinear terms in ADM makes its execution sometimes cumbersome and difficult.

Multiple slips and double stratification in MHD flow of hybrid nanofluid past a permeable sheet: triple solutions and stability analysis

Hybrid nanofluid flow past a stretching/shrinking sheet has various applications in industrial and engineering processes, e.g. in glass blowing, the extrusion of polymer sheets, and paper production. Motivated by these numerous uses of hybrid nanofluid in diverse geometries and conditions, the present study analyzes the solutions for MHD flow of Ag-CuO/water hybrid nanofluid past a shrinking sheet. The governing equations and boundary conditions are formulated together with the effects of Brownian motion, double stratification, porous medium, suction, slips, and thermophoresis. Then, similarity transformations are employed to form non-linear ordinary differential equations and boundary conditions for numerical computation in Matlab using the bvp4c solver. A significant finding of triple solutions in the shrinking sheet case prompted a stability analysis to be carried out, and the results show that only the first solution is stable. The effects of controlling parameters on the physical quantities of interest, velocity, temperature, and concentration profiles are analyzed and discussed. The heat and mass transfer rates are noted to improve, with an average of 7.91% and 258.36%, by increasing the Darcy number related to the permeability of the porous medium. Meanwhile, augmenting the nanoparticle volume fraction of Ag from 0.03 to 0.05 enhances the skin friction, heat transfer, and mass transfer rates by 9.4%, 7.36%, and 150.31%, respectively. However, the heat and mass transfer performances of the hybrid nanofluid are inhibited by the double stratification parameters.

Soret and Dufour effects on MHD mixed convection flow of Casson hybrid nanofluid over a permeable stretching sheet

The use of hybrid nanofluids can improve the engine operation and fuel efficiency in automotive engineering, as well as the efficiency of cooling systems used in food processing and preservation. This study provides insight for the development of efficient heat exchangers, thermal energy storage systems and cooling systems for various industrial applications, which can improve product quality and safety. The study aims to investigate the Soret and Dufour effects in hybrid nanofluid flow, with the goal of understanding their behaviour and potential applications in various industries. The flow is over a vertically stretched permeable sheet with porous material under convective boundary conditions and a non-uniform heat source/sink. The reduced governing equations are represented by non-linear-coupled ordinary differential equations solved by means of the shooting method based on the fourth-order Runge–Kutta method. Enhancing the Casson parameter reduces the fluid velocity while enhancing the mass and thermal Grashof numbers increases the flow velocity. The Soret and Dufour effects improve temperature and concentration, and a higher Biot number increases the temperature. The novelty of this investigation lies in its incorporation of Soret and Dufour effects, porosity, heat source/sink and a more refined definition of nanoflow, which is compared to previous results.

Numerical Solution of Radiative Boundary Layer Flow in Porous Medium Due to Exponentially Shrinking Permeable Sheet Under Fuzzy Environment

In this paper we are considering a fluid flows problem that contains two equation of motions and more than two parameters in the governing equation of motion. Which is namely Radiative Boundary Layer Flow in Porous Medium due to Exponentially Shrinking Permeable Sheet. The parameter are K=ck0/Lθ, Pr=μcp/κ∞, N=4σ1(T∞)3/(3κ1κ∞), and ε denote the permeability parameter, Prandtl number, and radiation parameter and is the thermal conductivity variation parameter, respectively. The governing differential equation can be obtained by using the similarity variable technique, and then the governing equation of motion can be Fuzzified by the help of Zadeh extension theorem. The technique is used for the validation of the uncertainty of the equation of the motion. The effect of the K, Pr, N, and ε are discussed with the fuzzified governing equation of motion under fuzzy environment. It is observed none of the parameters are directly involved in the occurrence of the uncertainty of the solutions. The uncertainty occurs in the problem is due to the assumption and the numerical computation. Finally, the solution is being carried out under fuzzy environment. It is found that the increasing values of permeability parameter, the values of both the numbers Skin friction coefficient as well as Nusselt number are increases.

Scrutinization of MHD stagnation point flow in hybrid nanofluid based on the extended version of Yamada-Ota and Xue models

This article exhibits a brief comparative study of magnetohydrodynamics stagnant point flow in hybrid nanofluid based on the extended version of two models, known as the Yamada-Ota model and Xue model. The time-dependent and thermally radiative two-dimensional flow is processed with the involvement of a stretchable/shrinking permeable sheet. The suspension of two nanoparticles Aluminum Oxide Al2O3 and Copper Cu into the base fluid water is considered for the formulation of hybrid nanofluid. With the execution of the significant similarity transformation, the nonlinear dimensionless system in the form of ordinary differential equations is developed. The Yamada-Ota and Xue models have been implemented on the dimensionless setup of equations that further are numerically worked out via the bvp4c methodology. The velocity and temperature fields corresponding to the pertinent parameters are figured out and discussed for both concerned models. From this study, we have come up with the result that the heat transfer rate involving the Yamada-Ota model is prominent as contrasted with the Xue model. The larger strength of the stretching parameter boosts up the field of the velocity but develops the deterioration in the temperature distribution. Moreover, the field of the velocity diminishes with the accelerating quantities of the solid nanoparticle.

Double solutions of unsteady stagnation-point of Carbon Nanotubes across a permeable exponential stretching/shrinking sheet

Carbon nanotube (CNTs) has an extraordinary interest due to its escalating applications in industry and technological processes. Thus, the analysis of unsteady stagnation-point of carbon nanotubes through exponential permeable (extending/shrinking) sheet with slip effects is inspected in this work. Comprising SWCNTs/MWCNTs which suspended into a carrier fluid (Water (H2O) or kerosene (oil)). The similarity transformation technique is adapted to attain the dimensionless system of differential equations from the envisioned mathematical model in (pde). Later, the finite difference method is utilized in MATLAB software to attain the approximate solutions. The outcomes of quantities of practical interest versus crucial parameters are discussed graphically and interpreted physically. According to the finding, the double solution discovered for both extending and shrinking, the addition of SWCNTs volume fraction will speed up the separation of flow at the critical point, stability analysis performed the rational of first solution as time evolves. Finally, this analysis supports the adoption of controlling parameters in abroad range of distinct industries including processing and biomedical.

PRESSURE DISTRIBUTION IN THE CONCRETE ROOF SLAB MEMBER UNDER THE WATERPROOFING MEMBRANE INSTALLED BY THE RE-COVERING METHOD DUE TO CHANGE IN TEMPERATURE

The temperature and pressure changes by a solar radiation in the rooftop slab were measured. When the heat insulation was installed, the member located outside of it became hotter and the pressure increased. And the pressure was transferred to other adjacent low pressure members. However, even in such a state, the air vent method by waterproofing membrane through an air permeable sheet was effective in ventilating the pressure inside the slab. A method of ventilation from the lower layer of roof protect concrete, assuming the release of pressure from the joint parts, was proposed and its effectiveness was also confirmed.

A design of neuro-computational approach for double‐diffusive natural convection nanofluid flow

The artificial intelligence based neural networking with Back Propagated Levenberg-Marquardt method (NN-BPLMM) is developed to explore the modeling of double‐diffusive free convection nanofluid flow considering suction/injection, Brownian motion and thermophoresis effects past an inclined permeable sheet implanted in a porous medium. By applying suitable transformations, the PDEs presenting the proposed problem are transformed into ordinary ones. A reference dataset of NN-BPLMM is fabricated for multiple influential variants of the model representing scenarios by applying Lobatto III-A numerical technique. The reference data is trained through testing, training and validation operations to optimize and compare the approximated solution with desired (standard) results. The reliability, steadiness, capability and robustness of NN-BPLMM is authenticated through MSE based fitness curves, error through histograms, regression illustrations and absolute errors. The investigations suggest that the temperature enhances with the upsurge in thermophoresis impact during suction and decays for injection, whereas increasing Brownian effect decreases the temperature in the presence of wall suction and reverse behavior is seen for injection. The best measures of performance in form of mean square errors are attained as 7.1058×10−10,2.9262×10−10,1.1652×10−08,1.5657×10−10 and 5.5652×10−10 against 969, 824, 467, 277 and 650 iterations. The comparative study signifies the authenticity of proposed solver with the absolute errors about 10−7 to 10−3 for all influential parameters results.

Unsteady MHD Flow of Casson Fluid Past Vertical Surface Using Laplace Transform Solution

The study explores the velocity of Casson fluid that is time-independent over an exponentially infinite isotherm vertical permeable sheet. The impact of magnetohydrodynamic (MHD) with Casson flow over the permeability sheet is examined. In contrast, thermal radiation and heat sink parameters have been incorporated. This study’s primary goal is to determine the significance of thermal radiation on Casson flow with MHD using an analytical solution over a permeable sheet. The flow of the fluid occurs above the sheet when [Formula: see text] is greater than zero, and the sheet extends far away in the [Formula: see text]-direction. The model of governing equations is reduced by applying a suitable set of dimensionless parameters. These dimensionless systems of equations are solved through the Laplace transformation method. The impacts of various variables over velocity, temperature, concentration, skin friction, and Nuseelt number are scrutinized. These variables contain magnetic field [Formula: see text], Casson fluid parameter [Formula: see text], Ghroshof number [Formula: see text], modified Grashof number [Formula: see text], Prandtl number [Formula: see text], thermal radiation [Formula: see text] and Scimdth number [Formula: see text]. These plots are sketched for the considerable magnitude of these variables through the Mathematica Software, and these plots are discussed in detail. Results show that the increasing value of [Formula: see text] reduces the fluid velocity but velocity of fluid is enhanced with larger values of [Formula: see text], [Formula: see text] and [Formula: see text]. The impact of Skin friction and Nusselt number is elaborated by tabular outlined.

Stability analysis of radiative-magnetic hybrid nanofluid slip flow due to an exponentially stretching/shrinking permeable sheet with heat generation

The present article aims to find the duality of solutions and investigate the magneto-radiative hybrid nanofluid flow over an exponentially stretching/shrinking permeable sheet with heat generation and multiple slip boundary conditions. The governing nonlinear partial differential equations are transformed into dimensionless nonlinear ordinary differential equations which are computed by bvp4c in Matlab platform. The hybridisation of alumina and copper nanoparticles enhances the system's heat transfer performance. The dual nature solution is obtained only if the suction is applied to the stretching/shrinking surface. From stability analysis, we can conclude that the first solution is stable, but the other is unstable. It is realistic to have the solution stable with the smallest positive eigenvalues, while the lower solution branch is unstable with the smallest negative eigenvalues. It is found that the upsurges of solid volume fraction of the copper nanoparticle enhanced the skin friction coefficient and the local Nusselt number.

Thermal transport in nanofluid across a radiated permeable sheet with irreversible effects based on the shape of the particles

Purpose The purpose of this study is to examine the flow and heat transfer performance of titanium oxide/water and copper/water nanofluids with varying nanoparticle morphologies by considering magnetic, Joule heating and viscous dissipation effects. Furthermore, it studies the irreversibility caused by the flow of a hydromagnetic nanofluid past a radiated stretching sheet by considering different shapes of TiO2 and Cu nanoparticles with water as the base fluid. Design/methodology/approach In this study, the authors investigated entropy production in an unsteady two-dimensional magneto-hydrodynamic nanofluid regime using water as the base fluid and five unique TiO2 and Cu nanoparticle morphologies. Using appropriate similarity transformations, the controlling nonlinear system of partial differential equations is transformed into a system of ordinary differential equations. The shooting technique with Runge–Kutta method was then used to solve these equations quantitatively. The findings of this study are depicted graphically, and the skin friction corresponding to various nanoparticle geometries and physical parameter variations is tabulated. Findings To assess the reliability of the current findings, a tabular representation of the data was compared to that of previously published studies. It is noted that a reduction in thermal energy was detected as a result of the higher levels of Prandtl number (Pr). It is further analysed that the highest heat energy generation of TiO2 nanoparticles was larger than that of Cu nanoparticles. The most important finding was that the sphere-shaped Cu/H2O nanofluid had the lowest velocity and greatest temperature. Also, Cu nanoparticles in the shape of platelets generate the most entropy, while TiO2 nanoparticles in the shape of spheres generate the least. Originality/value To the best of the knowledge of the authors, the attempt to investigate the previously unexplored shape effects of TiO2 and Cu nanoparticles on the heat transfer enhancement and inherent irreversibility caused by hydromagnetic nanofluid flow past a radiated stretching sheet with magnetic, Joule heating and viscous dissipation effects. This study fills this gap in the existing literature and encourages scientists, engineers and businesses to do more research in this area. This model can be used to improve heat transfer in systems that use renewable energy, thermal management in industry and the processing of materials.

On the Role of Bioconvection and Activation Energy for MHD-Stretched Flow of Williamson and Casson Nanofluid Transportation across a Porous Medium Past a Permeable Sheet

The goal of the current study is to theoretically evaluate the heat and mass transfer behavior of magnetohydrodynamic Casson and Williamson fluids as they pass through a stretched sheet with Brownian and thermophoresis effects. The evolution of high-density heat devices necessitates efficient thermal transportation. For these requirements, the concept of nanofluid plays an active role. This article describes the effects of Cattaneo–Christov. This study introduces new concepts such as motile microorganism bioconvection, non-Fourier heat flux, and activation energy. A Runge–Kutta-based shooting procedure is used to solve the problem. The effects of the relevant parameters on velocity, thermal, concentration, and motile microorganisms are depicted graphically. The computed reduced Nusselt, Sherwood, and motile density numbers are displayed in tables in this study. When compared to the Williamson fluid, the thermal and concentration fields of the Casson fluid are heavily influenced by the parameters.

Numerical calculation of thermally radiative fluid flow with the energy and mass transmission across a stretching wavy surface

AbstractThe current study revealed a comprehensive assessment of three‐dimensional Jeffery fluid flow under the significances of thermal radiation and magnetohydrodynamics (MHD) across a stretching irregular permeable sheet. The flow phenomena have been studied with an additional effect of Darcy media, Arrhenius activation energy, slip conditions, heat source, and chemical reaction. The modeled have been expressed in the form of system of nonlinear partial differential equations (PDEs), which are degraded and dimensionalized by the similarity replacement to the set of ordinary differential equations (ODEs). The Matlab package bvp4c (boundary value solver) is employed to estimate the numerical solution of the problems. The influence of several physical factor on velocity, mass, and energy outlines is revealed through tables and figures. It is perceived that the Jeffery fluid motion drops with the growing values of porosity term, while amplified with the rising influence of wall thickness factor. The rising values of Darcy–Forchheimer and magnetic field factor diminish the velocity curve in both x and y directions. Furthermore, the temperature curve intensifies with the addition of porosity factor and Deborah number while lessening with the rising influence of power index.

Thermal analysis for radiative flow of Darcy–Forchheimer nanomaterials subject to entropy generation

Abstract Background and objective Flow due to the Darcy–Forchheimer medium is an important perspective in various geophysics, industrial processes, geothermal energy, and thermodynamic processes. The importance of the Darcy–Forchheimer medium is noticed in technical, mechanical, industrial, and scientific fields including atomic waste archive, artificial dialysis, catalytic converters, gas turbine, improved oil recuperation, atherosclerosis, grain stockpiling, geo-energy production, and warm protection designing, etc. In view of such industrial and geothermal applications, the objective of this paper is to highlight the influence of entropy generation in chemical reactive MHD (magnetohydrodynamic) Darcy–Forchheimer nanoliquid flow with radiation. Flow by an exponentially stretching permeable sheet is taken. Thermal radiation, heat source, magnetic force, and dissipation impacts are considered in thermal expression. Additionally, Buongiorno’s model with random and thermophoresis diffusions is explained. Physical features of entropy are deliberated. The first-order isothermal reaction is discussed. Methodology Non-linear expressions are reduced to the dimensionless non-linear system through the implementation of non-similar transformations. The resultant non-linear systems are solved subject to local non-similarity via the ND-solve technique Results Graphical results for entropy rate, concentration, velocity, and thermal field versus emerging variables are studied. The reverse trend holds for entropy and velocity through the magnetic variable. A larger approximation of the Eckert number intensifies the thermal field. Conclusions A higher Forchheimer number reduces the fluid flow. A reverse impact for concentration and thermal field is seen through random motion variable. Similar behavior for thermal distribution is seen by thermophoresis and radiation effects. A larger porosity variable declines the entropy rate, while the reverse effect holds for the Brinkman number. A larger diffusion variable increases the entropy generation.

Significance of Darcy-Brinkman porous drag forces and Cattaneo-Christov thermal transportation on the features of second-grade fluid flows conveying copper nanomaterials past an unsteady stretching permeable sheet heated convectively

The main thermo-hydro-rheological behaviors featuring the unsteady convective laminar motion of copper-methanol second-grade nanofluid over a linearly stretching permeable sheet is explored numerically in the framework of Cattaneo-Christov theory to provide a better understanding of the heat transfer mechanism within a Darcy-Brinkman porous medium. By adopting the single-phase nanofluid model along with the boundary layer approximations, the leading partial differential equations (PDEs) are derived properly and then converted mathematically into a differential structure of ordinary differential equations (ODEs) via appropriate similarity alterations. Accordingly, the resulting dimensionless boundary layer equations are tacked numerically for convective and suction boundary conditions by utilizing the generalized differential quadrature method (GDQM). To reveal the effect of such a pertinent flow parameter, extensive GDQM numerical data are outputted graphically for the velocity and temperature fields as well as tabularly for skin friction coefficient and the Nusselt number. As the main findings, it is noted that both the velocity profile and the wall temperature lessen with the suction parameter and the nanoparticle’s loading. However, an enhancement is depicted for the wall heat transfer rate with the reinforcement in those influences. In addition, it is witnessed that the effect of the surface drag forces is strengthened with the nanoparticle’s loading and weakened with a higher estimation of the suction effect.

Mixed convection MHD boundary layer flow, heat, and mass transfer past an exponential stretching sheet in porous medium with temperature-dependent fluid properties

The mixed convection MHD boundary layer flow, heat and mass transfer of a Newtonian fluid mixture across an exponentially stretched permeable sheet in porous medium with Soret and Dufour effects subject to temperature-dependent fluid properties have been investigated numerically. The distribution of temperature and concentration at the surface is supposed to grow exponentially. Temperature is considered to affect fluid viscosity and thermal conductivity. The guiding partial differential equations with appropriate boundary conditions are converted into coupled, nonlinear ordinary differential equations with variable coefficients via a similarity transformation. Matlab's built in solver bvp4c is used to find numerical solutions. Using a graphical approach, the effects of several emerging physical parameters on velocity, temperature and concentration distributions on a flow field of a chemically nonreacting fluid mixture are examined and discussed. Results are compared to earlier literature available and found to be in good agreement. Furthermore, this research demonstrates that flow, heat and mass transmission are greatly influenced by mixed convection parameters, magnetic field, Prandtl number, viscosity, conductivity, permeability, and Schmidt number. Skin friction grows for mixed convection parameter, magnetic, viscosity. Highest growths of wall temperature and wall concentration gradients are observed for Prandtl number and Schmidt number, respectively. HIGHLIGHTS Heat and mass transfer rates increase with higher values of mixed convection parameters and Darcy parameter. To reduce heat transfer rate magnetic field may be used. Fluid with lower thermal conductivity and with higher Prandtl number may be employed to boost the cooling rate in conducting flows. Higher Schmidt enhances mass transfer rates. Suction stabilizes boundary layer growth. Soret effect can be used to separate light and medium molecular weight components from a fluid mixture and hence can be used for control of air pollution.

Unsteady magneto-convective heat-mass transport passing in a vertical permeable sheet with internal heat generation effect

The unsteady magneto-convective heat-mass transport passing in a vertical permeable sheet has been discussed with the internal heat generation/absorption effect. The governing partial differential equations (PDEs) have been transferred into ordinary differential equations (ODEs) by applying the local similarity transformation. The dimensionless ODEs are solved numerically by exerting the Quasilinearization Method with the help of ‘MATLAB ODE45’software. The influence of emerging dimensionless parameters/numbers for example the internal generation/absorption (Q), the suction parameter (v0), Prandtl number (Pr), Soret number (So), Dufour number (Df), Schmidt number (Sc), magnetic force parameter (M), local modified Grashof number (Gm) and local Grashof number (Gr) on the flow, temperature, and concentration profiles are presented and discussed. The fluid velocity and temperature enhance rising values of the internal heat generation and reduces the concentration but the reverse tends to be found for absorption. The velocity field reduces for increasing values of the suction parameter, Magnetic parameter, Prandtl number, and Schmidt number. The local skin friction increases by about 18% and reduces by 6% due to an increase in the internal heat generation parameter (1.0-3.0) and heat absorption parameter (3.0-10.0), respectively. Uplifting values of the heat absorption parameter (3.0-10.0) and internal heat parameter (1.0-3.0) enhance the heat transfer rate by 56% and decrease by about 42%, respectively. The mass transfer rate enhances by about 59% and decreases by 85% due to an increase in the internal heat generation parameter (1.0-3.0) and heat absorption parameter (3.0-10.0), respectively. Finally, we compared our numerical results with previously published paper and observed them be a excellent agreement.

Mathematical modelling of boundary layer flow over a permeable and time-dependent shrinking sheet – A stability analysis

Micropolar fluid is one type of non-Newtonian fluid which consists of non-deformable spherical particles that suspended in viscous medium. In this paper, the problem of two-dimensional boundary layer flow over a permeable shrinking sheet with time dependent velocity in strong concentration micropolar fluid is studied theoretically. The mathematical model is governed by continuity, momentum and microrotation equations. Similarity variables are introduced so that, after performing the similarity transformation on the governing equations, the resulting system of nonlinear ordinary differential equations is then numerically solved using the program bvp4c in Matlab software. The effects of the micropolar material parameter, the unsteadiness parameter, the shrinking parameter and the mass suction parameter to the skin friction coefficient, velocity profiles and microrotation profiles are investigated. It is found that triple solutions exist for some values of the parameters that were considered. Based on the stability analysis that was performed, it showed that only two branches of solutions are categorized as stable, whereas one solution branch is unstable.

Thermal Radiation Effect on Unsteady Magneto-Convective Heat-Mass Transport Passing in a Vertical Permeable Sheet with Chemical Reaction.

The unsteady magneto-convective heat-mass transport passing in a vertical porous sheet with the thermal radiation and the chemical reaction effects has been examined numerically. The governing PDEs have been transferred into ODEs by applying the local similarity transformation. The nondimensional governing equations including the boundary conditions are solved by applying the superposition method with the help of the “MATLAB ODE45” software numerically. The influence of emerging nondimensional numbers/parameters, for example, the Prandtl number (Pr), thermal radiation parameter (R), Schmidt number (Sc), and chemical reaction parameter (Kr), on fluid velocity, concentration, and thermal radiation within the boundary layer has been examined. The outcomes indicate that enhancing values of the Soret and Dufour numbers reduce the thermal boundary layer thickness. Uplifting values of the thermal radiation (0.5-3.5) enhance the local skin friction coefficient and mass transfer rate by approximately 15% and 78% but decrease the heat transfer rate by 47%. The local skin friction coefficient enhances about 21%, and the mass transfer rate reduces about 64% due to an increase in the chemical reaction parameter (0.5-2.0). Finally, we compared our numerical results with previously published literature and observed them to have a good agreement.

Regression analysis on MHD Darcy-Forchheimer hybrid nanoliquid flow over an elongated permeable sheet in a porous medium with hydrodynamic slip constraint: a realistic two-phase modified Buongiorno model

Injection and suction effects play a crucial role in astronomical disciplines, fuel injection, solar collector plate, thermal protection, and aerodynamics. Therefore, the dynamics of hydromagnetic Darcy-Forchheimer hybrid nanoliquid flow over an elongated permeable sheet in the presence of hydrodynamic slip and Newtonian boundary constraints is investigated. A realistic model considering the passive control of nanoparticles and the modified Buongiorno nanoliquid model has been utilised. The modelled partial differential equations are transmuted into a system of nonlinear ordinary differential equations with the aid of apposite similarity transformations which are then resolved numerically using the finite-difference based bvp5c routine. Regression analysis is employed to statistically scrutinise the relationship between the pertinent parameters and the drag coefficients. It is observed that the -directional velocity descends with the Forchheimer number and hydrodynamic slip parameter whereas the -directional velocity ascends with the stretching parameter. It is also noted that the Biot number and the volume fraction of copper nanoparticles have a constructive effect on the hybrid nanoliquid temperature. Furthermore, lower temperature and velocity profiles were exhibited by the suction case when compared with the injection case. An improved heat transfer rate is observed for higher values of Biot number and volume fraction of copper nanoparticles.