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Journal articles on the topic 'Shrinking Sheet'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Nur Hazirah Adilla Norzawary, Norfifah Bachok, Fadzilah Md Ali, and Norihan Md Arifin. "Slip Flow Over an Exponentially Stretching/Shrinking Sheet in a Carbon Nanotubes with Heat Generation: Stability Analysis." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 108, no. 1 (2023): 28–38. http://dx.doi.org/10.37934/arfmts.108.1.2838.

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This study is to analyse the problem of slip flow via exponentially stretching/shrinking sheet in carbon nanotubes (CNTs) with heat generation effects. The governing partial differential equations are transformed into nonlinear ordinary differential equations via transformation of similarity. The bvp4c solver in Matlab is then used to resolve them numerically. Water is used as the base fluid together with single wall and multi wall CNTs. The flow parameters effect is investigated, shown in the graphs form, and physically evaluated for the dimensionless velocity, temperature, skin friction, and

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2

Hafidzuddin, Mohd Ezad Hafidz, Roslinda Nazar, Norihan M. Arifin, and Ioan Pop. "Unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 6 (2018): 1457–70. http://dx.doi.org/10.1108/hff-11-2016-0440.

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Purpose This study aims to investigate the unsteady two-dimensional viscous flow and heat transfer over an unsteady permeable stretching/shrinking sheet (surface) with generalized slip velocity condition. Design/methodology/approach Similarity transformation is used to reduce the system of partial differential equations into a system of nonlinear ordinary differential equations. The resulting equations are then solved numerically using “bvp4c” function in MATLAB software. Findings Dual solutions are found for a certain range of the unsteady, suction and stretching/shrinking parameters. Stabili

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3

Mansur, Syahira, and Anuar Ishak. "The Magnetohydrodynamic Boundary Layer Flow of a Nanofluid past a Stretching/Shrinking Sheet with Slip Boundary Conditions." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/907152.

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The magnetohydrodynamic (MHD) boundary layer flow of a nanofluid past a stretching/shrinking sheet with velocity, thermal, and solutal slip boundary conditions is studied. Numerical solutions to the governing equations were obtained using a shooting method. The skin friction coefficient and the local Sherwood number increase as the stretching/shrinking parameter increases. However, the local Nusselt number decreases with increasing the stretching/shrinking parameter. The range of the stretching/shrinking parameter for which the solution exists increases as the velocity slip parameter and the m

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4

Yang, Zong Ru, Daniel Scherer, Matthias Golle, and Hartmut Hoffmann. "Geometrical Modeling of the Sheet Metal Parts in the Incremental Shrinking Process." Key Engineering Materials 473 (March 2011): 509–15. http://dx.doi.org/10.4028/www.scientific.net/kem.473.509.

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Shrinking is an incremental forming process and can be carried out using a driving machine, so called “Kraftformer”. It needs an upper and a lower shrinking tool, each of which has two moveable jaws as contact and force transform units. During every forming operation the tools clamp the metal sheet, so that the vertical forces from the upper tool are switched by the leverages inside the tools into the horizontal directions. The moveable jaws are practiced by the horizontal forces to compress the metal sheet. The shrinking of the metal sheet brings out the different three-dimensional forms. As

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5

Nadeem, Sohail, Anwar Hussain, and Majid K. "Stagnation Flow of a Jeffrey Fluid over a Shrinking Sheet." Zeitschrift für Naturforschung A 65, no. 6-7 (2010): 540–48. http://dx.doi.org/10.1515/zna-2010-6-709.

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January 22, 2009 The present paper describes the analytical solutions for the steady boundary layer flow of a Jeffrey fluid over a shrinking sheet. The governing equations of motions are reduced into a set of nonlinear ordinary differential equations by using similarity transformations. Two types of problems, namely, (1) two-dimensional stagnation flow towards a shrinking sheet and (2) axisymmetric stagnation flow towards an axisymmetric shrinking sheet, have been discussed. The series solutions of the problems are obtained by using the homotopy analysis method (HAM). The convergence of the ob

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6

Wang, C. Y. "Stagnation flow towards a shrinking sheet." International Journal of Non-Linear Mechanics 43, no. 5 (2008): 377–82. http://dx.doi.org/10.1016/j.ijnonlinmec.2007.12.021.

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7

Lok, Yian Yian, Anuar Ishak, and Ioan Pop. "Stagnation-Point Flow and Heat Transfer Towards a Shrinking Sheet with Suction in an Upper Convected Maxwell Fluid." Zeitschrift für Naturforschung A 68, no. 10-11 (2013): 693–700. http://dx.doi.org/10.5560/zna.2013-0047.

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A steady two-dimensional boundary-layer flow and heat transfer of an upper convected Maxwell fluid near a stagnation-point of a permeable shrinking sheet is studied numerically. The effects of elasticity, shrinking, and suction parameters on the flow and heat transfer characteristics are investigated. A similarity transformation reduces the governing equations to third-order nonlinear ordinary differential equations which are then solved numerically. For a fixed value of elastic parameter, it is found that dual solutions exist for some values of shrinking and suction parameters. The plotted st

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8

Ali, Azhar, Aziz Khan, Aftab Alam, Khan Marwat, K. Kamran, and Thabet Abdeljawad. "Multiple series solutions of viscous flows over a stretching/shrinking and porous sheet." Thermal Science 27, Spec. issue 1 (2023): 185–94. http://dx.doi.org/10.2298/tsci23s1185a.

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The viscous fluid-flow over a stretching (shrinking) and porous sheets of non-uniform thickness is investigated in this paper. The modeled problem is presented by utilizing the stretching/shrinking and porous velocities and variable thickness of the sheet. Consequently, the new problem reproduces the different available forms of flow motion maintained over a stretching/shrinking and porous sheet of variable thickness in one go. As a result, the governing equations are embedded in several parameters which can be transformed into classical cases of stretched/shrunk flows over porous sheets. A se

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9

Chauhan, Dileep Singh, and Rashmi Agrawal. "MHD Flow and Heat Transfer in a Channel Bounded by a Shrinking Sheet and a Porous Medium Bed: Homotopy Analysis Method." ISRN Thermodynamics 2013 (January 17, 2013): 1–10. http://dx.doi.org/10.1155/2013/291270.

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MHD flow of viscous conducting fluid is considered between a shrinking sheet and a porous medium bed. Suction is applied at the upper shrinking sheet and its surface temperature is always maintained higher than the temperature of the lower porous bed surface. Similarity transformations and HAM are used to solve the governing equations for velocity and temperature fields. The effects of various pertinent parameters on the results are discussed graphically.

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10

Layek, G. C., Bidyut Mandal, Krishnendu Bhattacharyya, and Astick Banerjee. "Lie Symmetry Analysis of Boundary Layer Stagnation-Point Flow and Heat Transfer of Non-Newtonian Power-Law Fluids Over a Nonlinearly Shrinking/Stretching Sheet with Thermal Radiation." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 3-4 (2018): 415–26. http://dx.doi.org/10.1515/ijnsns-2017-0211.

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AbstractA symmetry analysis of steady two-dimensional boundary layer stagnation-point flow and heat transfer of viscous incompressible non-Newtonian power-law fluids over a nonlinearly shrinking/stretching sheet with thermal radiation effect is presented. Lie group of continuous symmetry transformations is employed to the boundary layer flow and heat transfer equations, that gives scaling laws and self-similar equations for a special type of shrinking/stretching velocity ($c{x^{1/3}}$) and free-stream straining velocity ($a{x^{1/3}}$) along the axial direction to the sheet. The self-similar eq

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11

Malvandi, A., F. Hedayati, DD Ganji, and Y. Rostamiyan. "Unsteady boundary layer flow of nanofluid past a permeable stretching/shrinking sheet with convective heat transfer." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 7 (2013): 1175–84. http://dx.doi.org/10.1177/0954406213503259.

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This paper deals with the unsteady two-dimensional boundary layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet in the presence of heat generation/absorption. A convective heating boundary condition instead of a conventional prescribed temperature at the surface was employed to probe the variations of heat and concentration rates for different parameters including unsteadiness parameter, Brownian motion parameter, thermophoresis parameter and suction/injection. Results indicate that a dual solution exists for both stretching and shrinking sheets and that the

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12

Thomas, R. H. "Remote sensing reveals shrinking Greenland ice sheet." Eos, Transactions American Geophysical Union 82, no. 34 (2001): 369. http://dx.doi.org/10.1029/01eo00226.

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13

Yacob, Nor Azizah, and Anuar Ishak. "Micropolar fluid flow over a shrinking sheet." Meccanica 47, no. 2 (2011): 293–99. http://dx.doi.org/10.1007/s11012-011-9439-8.

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14

Miklavčič, M., and C. Y. Wang. "Viscous flow due to a shrinking sheet." Quarterly of Applied Mathematics 64, no. 2 (2006): 283–90. http://dx.doi.org/10.1090/s0033-569x-06-01002-5.

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15

Chaudhary, Santosh, and Pradeep Kumar. "MHD Slip Flow past a Shrinking Sheet." Applied Mathematics 04, no. 03 (2013): 574–81. http://dx.doi.org/10.4236/am.2013.43081.

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16

Wong, Sin Wei, Abu Omar Awang, and Anuar Ishak. "Stagnation-Point Flow over an Exponentially Shrinking/Stretching Sheet." Zeitschrift für Naturforschung A 66, no. 12 (2011): 705–11. http://dx.doi.org/10.5560/zna.2011-0037.

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The steady two-dimensional stagnation-point flow of an incompressible viscous fluid over an exponentially shrinking/stretching sheet is studied. The shrinking/stretching velocity, the free stream velocity, and the surface temperature are assumed to vary in a power-law form with the distance from the stagnation point. The governing partial differential equations are transformed into a system of ordinary differential equations before being solved numerically by a finite difference scheme known as the Keller-box method. The features of the flow and heat transfer characteristics for different valu

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17

Abd Rahman, Nor Hathirah, Norfifah Bachok, and Haliza Rosali. "MHD Stagnation Point Flow over a Nonlinear Stretching/Shrinking Sheet in Nanofluids." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 76, no. 3 (2020): 139–52. http://dx.doi.org/10.37934/arfmts.76.3.139152.

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In this study, an investigation of the steady 2-D magnetohydrodynamiic (MHD) flow of stagnation point past a nonlinear sheet of stretching/shrinking within of a non-uniform transverse magnetic intensity in nanofluids had been analysed. Considered material of nanoparticles such as copper (Cu) in water base fluid with Pr = 6.2 to analyze the influence of volume fraction parameter of nanoparticles and the stretching/shrinking sheet parameter. The governing nonlinear partial differential equations (PDEs) are converted in to the nonlinear ordinary differential equations (ODEs) and use the boundary

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18

Ahlam Aljabali, Abdul Rahman Mohd Kasim, Najiyah Safwa Khashi’ie, Iskandar Waini, and Noor Amalina Nisa Ariffin. "Eyring-Powell Hybrid Nanofluid with Radiative Heat Flux: Case Over a Shrinking Sheet." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 111, no. 1 (2023): 109–21. http://dx.doi.org/10.37934/arfmts.111.1.109121.

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This study investigates the radiative heat flux effect on Eyring-Powell hybrid nanofluid together with analysis thermal passing towards a shrinking sheet containing Ag-MgO nanoparticles. The proposed mathematical model respected to the boundary condition is converted to the similarity equations by adopting the suitable transformations. In order to reduce the complexity of model, the similarity equations are then computed by applying the bvp4c function in MATLAB software. Outcomes reveal the thermal performance is upgraded in dispersing the nanoparticle to the base fluid, and it is marked under

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19

Yahaya, Rusya Iryanti, Mohd Shafie Mustafa, Norihan Md Arifin, Ioan Pop, Fadzilah Md Ali, and Siti Suzilliana Putri Mohamed Isa. "Heat Transfer Optimization using RSM for Hybrid Nanofluid Flow Impinging Obliquely on a Permeable Shrinking Sheet." Journal of Advanced Research in Numerical Heat Transfer 29, no. 1 (2025): 1–15. https://doi.org/10.37934/arnht.29.1.115.

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Fluid flow may strike a surface at an angle due to the physical limitations of the nozzle or contouring of the surface. The heat transfer optimization for the Al2O3-Cu/water hybrid nanofluid flow impinging obliquely on a permeable shrinking sheet is analyzed in this study. Flow over a shrinking sheet may occur during polymer and metal sheet extraction, wire drawing, and glass-fiber production. The first step in this study involves reducing the governing partial differential equations and boundary conditions into non-linear ordinary differential equations via similarity transformation. Subseque

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20

Khan, Najeeb Alam, and Sidra Khan. "Dual solution of Casson fluid over a porous medium: Exact solutions with extra boundary condition." ACTA Universitatis Cibiniensis 68, no. 1 (2016): 35–49. http://dx.doi.org/10.1515/aucts-2016-0008.

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Abstract In this article we calculate the exact solution of the steady flow of non-Newtonian Casson fluid, over a stretching/shrinking sheet. The governing partial differential equations (PDEs) are transformed into ordinary differential equation (ODE) by using similarity transformation and then solved analytically by utilizing the exact solution. The closed form unique solution is obtained in the case of stretching sheet whereas for shrinking sheet unique and dual solutions are obtained. Influences of Casson fluid and suction/injection parameter on dimensionless velocity function are discussed

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21

Nur Hazirah Adilla Norzawary and Norfifah Bachok. "Slip Flow Via Nonlinearly Stretching/Shrinking Sheet in A Carbon Nanotubes." Journal of Advanced Research in Micro and Nano Engieering 14, no. 1 (2024): 8–19. http://dx.doi.org/10.37934/armne.14.1.819.

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This study is to analyze the problem of slip flow via nonlinearly stretching/shrinking sheet in carbon nanotubes (CNTs) with suction/injection. The governing partial differential equations are transformed into nonlinear ordinary differential equations via the transformation of similarity. The bvp4c solver in Matlab is then used to resolve them numerically. Water is used as the base fluid together with single-wall and multi-wall CNTs. Both velocity slip and thermal slip are considered in this study. The flow parameters effect is investigated, shown in the form of a graph, and physically evaluat

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22

Mansur, Syahira, Anuar Ishak, and Ioan Pop. "Three-Dimensional Flow and Heat Transfer Past a Permeable Exponentially Stretching/Shrinking Sheet in a Nanofluid." Journal of Applied Mathematics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/517273.

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The three-dimensional flow and heat transfer of a nanofluid over a stretching/shrinking sheet is investigated. Numerical results are obtained using bvp4c in MATLAB. The results show nonunique solutions for the shrinking case. The effects of the stretching/shrinking parameter, suction parameter, Brownian motion parameter, thermophoresis parameter, and Lewis number on the local skin friction coefficient and the local Nusselt number are studied. Suction increases the solution domain. Furthermore, as the sheet is shrunk in thex-direction, suction increases the skin friction coefficient in the same

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23

Deng, Qi, Weiguo Xiao, and Huan Yan. "The Spillover Effects of U.S. Monetary Policy Normalization on the BRICS Based on Panel VAR Model." Journal of Mathematics 2022 (January 27, 2022): 1–9. http://dx.doi.org/10.1155/2022/3844128.

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Using the monthly data from December 2008 to March 2018 and a panel vector autoregression (PVAR) model, this paper empirically analyzes the spillover effects of U.S. monetary policy normalization on the total output, inflation, trade balance, and exchange rates in BRICS. The results show that the Fed’s interest rate hike and balance sheet shrinking will both lead to a decrease in BRICS’ output, a decline in inflation, a deterioration in the trade balance, and a depreciation of the exchange rate. In addition, the spillover effects of the Fed’s interest rate hike and shrinking of a balance sheet

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24

Alharbi, Ahmad Ayyad, and Ali Rashash R. Alzahrani. "A COMSOL-Based Numerical Simulation of Heat Transfer in a Hybrid Nanofluid Flow at the Stagnant Point across a Stretching/Shrinking Sheet: Implementation for Understanding and Improving Solar Systems." Mathematics 12, no. 16 (2024): 2493. http://dx.doi.org/10.3390/math12162493.

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The present study investigates hybrid nanofluid (HNF) behavior at the stagnation point near a stretching/shrinking sheet using the Tiwari and Das model. The governing equations were transformed into a boundary layer flow model and simulated using COMSOL Multiphysics 6.0. This research examines flow characteristics, temperature profiles, and distributions by varying parameters: stretching/shrinking (λ, −2 to 2), slip flow (δ, 0 to 1 m), suction (γ, 0 to 1), and similarity variables (η, 0 to 5). The HNF comprised equal ratios of copper and alumina with total concentrations ranging from 0.01 to 0

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Pop, Ioan, Siti Suzilliana Putri Mohamed Isa, Norihan M. Arifin, Roslinda Nazar, Norfifah Bachok, and Fadzilah M. Ali. "Unsteady viscous MHD flow over a permeable curved stretching/shrinking sheet." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 8 (2016): 2370–92. http://dx.doi.org/10.1108/hff-07-2015-0301.

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Purpose The purpose of this paper is to theoretically study the problem of the unsteady boundary layer flow past a permeable curved stretching/shrinking surface in the presence of a uniform magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically. Design/methodology/approach The transformed system of ordinary differential equations was solved using a fourth-order Runge-Kutta integration scheme. Results for the reduced skin friction coefficient and velocity profiles a

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26

Douglas, Bruce C., Robert E. Cheney, Laury Miller, Russell W. Agreen, William E. Carter, and Douglas S. Robertson. "Greenland Ice Sheet: Is It Growing or Shrinking?" Science 248, no. 4953 (1990): 288. http://dx.doi.org/10.1126/science.248.4953.288.a.

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27

Lok, Y. Y., A. Ishak, and I. Pop. "MHD stagnation‐point flow towards a shrinking sheet." International Journal of Numerical Methods for Heat & Fluid Flow 21, no. 1 (2011): 61–72. http://dx.doi.org/10.1108/09615531111095076.

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28

Miklavčič, M. "Instability of viscous flows over a shrinking sheet." Quarterly of Applied Mathematics 72, no. 2 (2013): 363–71. http://dx.doi.org/10.1090/s0033-569x-2013-01340-3.

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29

Douglas, B. C., R. E. Cheney, L. Miller, R. W. Agreen, W. E. Carter, and D. S. Robertson. "Greenland Ice Sheet: Is It Growing or Shrinking?" Science 248, no. 4953 (1990): 288. http://dx.doi.org/10.1126/science.248.4953.288.

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30

Krishnamurthy, Arvind, Sydney C. Ludvigson, and Jonathan H. Wright. "Panel on Shrinking the Federal Reserve Balance Sheet." Brookings Papers on Economic Activity 2022, no. 2 (2022): 231–80. http://dx.doi.org/10.1353/eca.2022.a901273.

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31

Zaimi, Khairy, and Anuar Ishak. "Stagnation-Point Flow and Heat Transfer over a Nonlinearly Stretching/Shrinking Sheet in a Micropolar Fluid." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/261630.

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This paper considers the problem of a steady two-dimensional stagnation-point flow and heat transfer of an incompressible micropolar fluid over a nonlinearly stretching/shrinking sheet. A similarity transformation is employed to convert the partial differential equations into nonlinear ordinary ones which are then solved numerically using a shooting method. Numerical results obtained are presented graphically, showing the effects of the micropolar or material parameter and the stretching/shrinking parameter on the flow field and heat transfer characteristics. The dual solutions are found to ex

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32

Motsa, S. S., Y. Khan, and S. Shateyi. "A New Numerical Solution of Maxwell Fluid over a Shrinking Sheet in the Region of a Stagnation Point." Mathematical Problems in Engineering 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/290615.

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The mathematical model for the incompressible two-dimensional stagnation flow of a Maxwell fluid towards a shrinking sheet is proposed. The developed equations are used to discuss the problem of being two dimensional in the region of stagnation point over a shrinking sheet. The nonlinear partial differential equations are transformed to ordinary differential equations by first-taking boundary-layer approximations into account and then using the similarity transformations. The obtained equations are then solved by using a successive linearisation method. The influence of the pertinent fluid par

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Mohammad, Shafique, Nazir Atif, and Abbas Fatima. "Numerical solution of two dimensional stagnation flows of Micropolar fluids towards a shrinking sheet by using SOR Iterative Procedure." Journal of Progressive Research in Mathematics 3, no. 1 (2015): 131–39. https://doi.org/10.5281/zenodo.3980688.

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In this paper, the problem of two dimensional stagnation flows of micropolar fluids towards a shrinking sheet has been solved numerically by SOR iterative procedure. The similarity transformations have been used to reduce the highly nonlinear partial differential equations of motion to ordinary differential equations. The resulting equations are then integrated by using appropriate numerical techniques. The results have been calculated on three different grid sizes to check the accuracy of the results. The numerical results have been obtained for various values of the parameter. For   0 , t

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Nor Fadhilah Dzulkifli, Norfifah Bachok, Nor Azizah Yacob, Ioan Pop, Norihan Arifin, and Haliza Rosali. "Stability Solution of Unsteady Stagnation-Point Flow and Heat Transfer over a Stretching/Shrinking Sheet in Nanofluid with Slip Velocity Effect." CFD Letters 14, no. 1 (2022): 66–86. http://dx.doi.org/10.37934/cfdl.14.1.6686.

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Computational of unsteady flow with slip condition is essential since physically the heat transfer process is time-dependent and there may exist slip between fluid and surface. Therefore, this study aims to investigate the unsteady stagnation-point flow and heat transfer over a stretching/shrinking sheet immersed in nanofluid in the presence of slip velocity. By applying boundary layer theory and Tiwari-Das model, the governing equations are developed and transformed into a system of ordinary differential equations using similarity transformation, which are then solved numerically using bvp4c

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Rashidi, Mohammad, Muhammad Ashraf, Behnam Rostami, Taher Rastegari, and Sumra Bashir. "Mixed convection boundary-layer flow of a micro polar fluid towards a heated shrinking sheet by homotopy analysis method." Thermal Science 20, no. 1 (2016): 21–34. http://dx.doi.org/10.2298/tsci130212096r.

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A comprehensive study of two dimensional stagnation flow of an incompressible micro polar fluid with heat transfer characteristics towards a heated shrinking sheet is analyzed analytically. The main goal of this paper is to find the analytic solutions using a powerful technique namely the Homotopy Analysis Method (HAM) for the velocity and the temperature distributions and to study the steady mixed convection in two-dimensional stagnation flows of a micro polar fluid around a vertical shrinking sheet. The governing equations of motion together with the associated boundary conditions are first

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Nadeem, Sohail, Anwar Hussain, and Kuppalapalle Vajravelu. "Effects of Heat Transfer on the Stagnation Flow of a Third-Order Fluid over a Shrinking Sheet." Zeitschrift für Naturforschung A 65, no. 11 (2010): 969–94. http://dx.doi.org/10.1515/zna-2010-1109.

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This paper is devoted to the study of a stagnation point flow of an incompressible third-order fluid towards a shrinking sheet (with heat transfer). The governing nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by means of a similarity transformation and then solved by the homotopy analysis method. Two types of flow problems, namely, (i) two dimensional stagnation flow toward a shrinking sheet and (ii) axisymmetric stagnation flow towards an axisymmetric shrinking surface have been discussed. Also, two types of boundary conditions are taken i

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Alias, I. A., and M. E. H. Hafidzuddin. "Effect of suction and MHD induced Navier slip flow due to a non-linear stretching/shrinking sheet." Mathematical Modeling and Computing 9, no. 1 (2022): 83–91. http://dx.doi.org/10.23939/mmc2022.01.083.

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In this study, a problem of a magnetohydrodynamic (MHD) induced Navier slip flow over a non-linear stretching and shrinking sheet with the existence of suction is considered. Similarity transformation is used to transform the governing nonlinear partial differential equations into a system of nonlinear ordinary equations. The transformed ordinary differential equations are then solved by using the Shooting Method in Maple software. Dual solutions are obtained for certain governing parameters. The results obtained show that suction improves skin friction, while the slip parameter reduces shear

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Asghar, Adnan, Mallika Vasugi Govindarajoo, Hussan Ara, Khairy Zaimi, Teh Yuan Ying, and Liaquat Ali Lund. "Effect of Water-based Alumina-Copper MHD Hybrid Nanofluid on a Power-Law Form Stretching/Shrinking Sheet with Joule Heating and Slip condition: Dual Solutions Study." CFD Letters 17, no. 4 (2024): 119–35. http://dx.doi.org/10.37934/cfdl.17.4.119135.

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The application of hybrid nanofluid is now being employed to augment the efficiency of heat transfer rates. A numerical study was conducted to investigate the flow characteristics of water-based-alumina copper hybrid nanofluids towards a power-law form stretching/shrinking sheet. This study also considered the influence of magnetic, Joule heating, and thermal slip parameters. This study is significant because it advances our understanding of hybrid nanofluids in the presence of magnetic fields, power-law form stretching/shrinking sheet, and heat transfer mechanisms, providing valuable insights

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Prasad, K. V., K. Vajravelu, and I. Pop. "Flow and Heat Transfer at a Nonlinearly Shrinking Porous Sheet:The Case of Asymptotically Large Powerlaw Shrinking Rates." International Journal of Applied Mechanics and Engineering 18, no. 3 (2013): 779–91. http://dx.doi.org/10.2478/ijame-2013-0047.

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Abstract The boundary layer flow and heat transfer of a viscous fluid over a nonlinear permeable shrinking sheet in a thermally stratified environment is considered. The sheet is assumed to shrink in its own plane with an arbitrary power-law velocity proportional to the distance from the stagnation point. The governing differential equations are first transformed into ordinary differential equations by introducing a new similarity transformation. This is different from the transform commonly used in the literature in that it permits numerical solutions even for asymptotically large values of t

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Eldabe, Nabil, and Mahmoud Abu Zeid. "Thermal Diffusion and Diffusion Thermo Effects on the Viscous Fluid Flow with Heat and Mass Transfer through Porous Medium over a Shrinking Sheet." Journal of Applied Mathematics 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/584534.

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The motion of the viscous, incompressible fluid through a porous medium with heat and mass transfer over a shrinking sheet is investigated. The cross-diffusion effect between temperature and concentration is considered. This phenomenon is modulated mathematically by a set of partial differential equations which govern the continuity, momentum, heat, and mass. These equations are transformed to a set of ordinary differential equations by using similarity solutions. The analytical solutions of these equations are obtained. The velocity, temperature, and concentration of the fluid as well as the

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Hsu, Cheng-Hsing, Te-Hui Tsai, Ching-Chuan Chang, and Wen-Han Huang. "A boundary layer flow analysis of a magnetohydrodynamic fluid over a shrinking sheet." Advances in Mechanical Engineering 11, no. 3 (2019): 168781401983506. http://dx.doi.org/10.1177/1687814019835069.

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A steady-state boundary layer flow analysis of a non-Newtonian magnetic fluid over a shrinking sheet was studied. The boundary layer thickness and the velocity distribution in the layer were studied under the conditions of a uniform magnetic field normal to the shrinking sheet and/or a vertical uniform mass suction across the sheet. The similarity transformation method was used to transform the governing partial differential equations to ordinary differential equations. The shooting method with Newton’s algorithm and Runge–Kutta integration method were used to obtain the solutions of the equat

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Hamid, Rohana Abdul, Roslinda Nazar, and Ioan Pop. "Boundary layer flow of a dusty fluid over a permeable shrinking surface." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 4 (2017): 758–72. http://dx.doi.org/10.1108/hff-01-2016-0030.

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Purpose The purpose of this paper is to numerically study the boundary layer problem for the case of two-dimensional flow of dusty fluid over a shrinking surface in the presence of the fluid suction at the surface. Design/methodology/approach The governing equations of the problem are reduced to the system of ordinary differential equations using the similarity transformation and then solved using the bvp4c method in the Matlab software. Findings The effects of the drag coefficient parameter L, the fluid–particle interaction parameter δ, the suction parameter s and the particle loading paramet

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A. F. S. Raman, M. E. H. Hafidzuddin, N. S. Wahid, N. M. Arifin, R. Nazar, and I. Pop. "Three‐dimensional Rotating Hybrid Nanofluid over a Shrinking Sheet with Velocity Slip." Applied Mathematics and Computational Intelligence (AMCI) 14, no. 1 (2025): 79–95. https://doi.org/10.58915/amci.v14i1.1361.

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This study deals with three dimensional rotating nanofluid over a shrinking sheet with velocity slip. Similarity transformations have been used for reducing the partial differential equations into a system of ordinary differential equations. The transformed ordinary diffential equations are solved numerically using BVP4C. The effects of Prandtl number Pr, suction parameter S, shrinking parameter λ, rotation parameter ω and slip parameter K on the velocity and temperature fields are presented and discussed in detail. The change in Prandtl number only affects the temperature profile while changi

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Khan, Umair, Aurang Zaib, Sakhinah Abu Bakar, Anuar Ishak, Dumitru Baleanu, and El-Sayed M. Sherif. "Computational simulation of cross-flow of Williamson fluid over a porous shrinking/stretching surface comprising hybrid nanofluid and thermal radiation." AIMS Mathematics 7, no. 4 (2022): 6489–515. http://dx.doi.org/10.3934/math.2022362.

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<abstract> <p>Recent nanotechnology advancements have created a remarkable platform for the development of a better performance of ultrahigh coolant acknowledged as nanofluid for numerous industrial and engineering technologies. The current study aims to examine the boundary-layer cross-flow of Williamson fluid through a rotational stagnation point towards either a shrinking or stretching permeable wall incorporated by a hybrid nanofluid. The shape factors along with the radiation effect are also taken into account. The contained boundary layers are the type of stream-wise by shrin

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Faraz, N., and Y. Khan. "Thin film flow of an unsteady Maxwell fluid over a shrinking/stretching sheet with variable fluid properties." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 7 (2018): 1596–612. http://dx.doi.org/10.1108/hff-12-2017-0498.

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Purpose This paper aims to explore the variable properties of a flow inside the thin film of a unsteady Maxwell fluid and to analyze the effects of shrinking and stretching sheet. Design/methodology/approach The governing mathematical model has been developed by considering the boundary layer limitations. As a result of boundary layer assumption, a nonlinear partial differential equation is obtained. Later on, similarity transformations have been adopted to convert partial differential equation into an ordinary differential equation. A well-known homotopy analysis method is implemented to solv

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Nasir, Nor Ain Azeany Mohd, Anuar Ishak, and Ioan Pop. "Stagnation-Point Flow Past a Permeable Stretching/Shrinking Sheet." Advanced Science Letters 23, no. 11 (2017): 11040–43. http://dx.doi.org/10.1166/asl.2017.10215.

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Rosali, Haliza, Anuar Ishak, Roslinda Nazar, and Ioan Pop. "Rotating flow over an exponentially shrinking sheet with suction." Journal of Molecular Liquids 211 (November 2015): 965–69. http://dx.doi.org/10.1016/j.molliq.2015.08.026.

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Waini, Iskandar, Anuar Ishak, and Ioan Pop. "Hybrid nanofluid flow induced by an exponentially shrinking sheet." Chinese Journal of Physics 68 (December 2020): 468–82. http://dx.doi.org/10.1016/j.cjph.2019.12.015.

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Fang, Tie-Gang, Ji Zhang, and Shan-Shan Yao. "Slip Magnetohydrodynamic Viscous Flow over a Permeable Shrinking Sheet." Chinese Physics Letters 27, no. 12 (2010): 124702. http://dx.doi.org/10.1088/0256-307x/27/12/124702.

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Saleh, Siti Hidayah M., Norihan M. Arifin, Roslinda Nazar, Fadzilah M. Ali, and Ioan Pop. "Mixed convection stagnation flow towards a vertical shrinking sheet." International Journal of Heat and Mass Transfer 73 (June 2014): 839–48. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.060.

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