Research Article | Open Access

Volume 2021 |Article ID 6647708 | https://doi.org/10.1155/2021/6647708

Sobia Akbar, Azad Hussain, "The Influences of Squeezed Inviscid Flow between Parallel Plates", Mathematical Problems in Engineering, vol. 2021, Article ID 6647708, 9 pages, 2021. https://doi.org/10.1155/2021/6647708

# The Influences of Squeezed Inviscid Flow between Parallel Plates

Academic Editor: Francesco Aggogeri
Received11 Nov 2020
Revised18 Feb 2021
Accepted30 Mar 2021
Published22 Apr 2021

#### Abstract

Purpose. The main purpose of this study is to investigate the unsteady flow behavior of second-grade inviscid fluid between parallel plates. The effects on the flow are explored through modeling of continuity, momentum, and energy equations. Graphical and tabular exploration has been made to analyze the impact of several influential variables on the dimensionless temperature and velocity profiles. Three-dimensional graphs and stream lines are also mentioned. Design/Approach/Methodology. The governing equations have been metamorphosed into nonlinear ordinary differential equations by using suitable transformation which is the main focus of the study. To approach the solution of the problem numerically, we have used the numerical method such as shooting technique along with Runge–Kutta method is implemented. Findings. The graphs for the squeezing number, Prandtl number, and Eckert number are decreasing by increasing the values of these parameters. The graphs of skin friction coefficient and Nusselt number are increasing by changing the values of both parameters. Originality/Value. The significances of an unsteady squeezed flow of a nonviscous second-grade fluid between parallel plates by using boundary layer phenomenon are discussed.

#### 1. Introduction

Some important applications of non-Newtonian fluids are introduced to enhance the research interest in food preservation, polymeric substitutions, nuclear fuels, liquid metals, paints, and blood flow. In non-Newtonian fluids, mixed convection phenomenon has enchanted scientific experts because of its momentousness real-world applications, for example, solar energy, electronic appliances cooled with fans, and cooling of nuclear reactors. Complicated comparison of viscous fluids is formed due to strange nature between shear stress and strain rate in such fluids. The criterion of viscoelasticity contributed further complexities in the governing equations at the time of comparison with Navier–Stokes equations. Most of the investigations [125], worked on such problems by assuming different types of flows and effects on various fluids. The flow squeezed between parallel walls happens in many biological and industrial systems. The nonsteady viscous flow fluid squeezed between parallel plates is a great subject of interest in hydrodynamic machines due to their motion normal to their own surfaces. The initiate work and the fundamental formulation of under lubrication squeezing flows were assumed by Stefan [14]. In previous literature, over few decades, the flow squeezed by elliptic plates was discussed by Reynolds [15] while Archibald [16] suggested the same inquisition for rectangular plates.

The evaluation of boundary layer squeezed flow is an interesting research matter due to its wide range of applications in industry and engineering. The most common scientific and engineering applications are in the drawing of plastic wires and films, extrusion of a polymer in a melt-spinning process, manufacturing of foods, crystal growing, liquid film in condensation process, electrochemical process, paper and glass fiber production, thermal energy storage, electronic chips, flow through filtering devices, food processing, cooling towers, marine engineering, hydro towers, distillation columns, and so on. The viscosity and thermic conductivity are presumed as a function of temperature. Unsteadiness is the loss of equilibrium with environment, usually with an affection of almost falling, or the consequences of bumping into objects. There are numerous reasons for unsteadiness, together with the problems in the cerebral or cerebellar sections of the spinal cord, brain, inner ear, or vestibular system. Unsteady flow of the fluid is the one where properties of the fluid vary with time. It is worthless to say that any beginning procedure is unsteady.

Now, the development in industries has motivated the researchers to discover non-Newtonian fluids properties in a more organized way. In nature, a number of fluids show non-Newtonian behavior, i.e., slurries, honey, glue, gels, toothpaste, ketchup, etc. Various paradigms of unsteadiness can be found from our daily life like the flow of water out from a tap which has been just opened. This is unsteady flow in the start, but it becomes steady with time. In the current problem, the boundary layer approximation is utilized to construct an unsteady second-grade fluid flow model. The obtained coupled partial differential equations are simplified by using suitable mathematical techniques. The dimensionless equations are being solved by using numerical techniques, i.e., shooting technique. A comprehensive graphical and tabular study is constructed to check the convergence of the obtained results.

#### 2. Mathematical Description of the Flow Phenomenon

The stress tensor [14] for the current problem is given by

We have restriction as

So, equation (1) reduces to

is Rivlin–Ericksen material expansion of the strain rate tensor as the derivative rotates and translates with flow. The nonviscous squeezed flow of an unsteady second grade fluid between parallel plates segregated by a distance where characteristic parameter is and length at is considered. Furthermore, is relative to the motion of both squeezed plates till they connect each other at , for when , the plates are separated.

The fields of the flows corresponding to Cartesian coordinates [16] are

The momentum equations which govern for the problem become

The obtained energy equation iswhere specific heat is .

By using boundary layer flow equations, (3)–(9) become

Conforming to layout of the inquisition in Figure 1, the boundary conditions’ suits for the flow are

##### 2.1. Nondimensional Equations

Using transformation [11],where

Applying equations (15)–(17) in equations (10)–(12), we obtainwhere

Here, nondimensional length is is Eckert number, is squeezing number, and is Prandtl number is dimensionless.

The nondimensional boundary conditions are

#### 3. Numerical Solution

The highly nonlinear partial differential equations are changed to ordinary differential equations by using transformations by shooting technique along with Runge–Kutta scheme are numerically solved with the aid of Maple software equations (34)–(45).

New variables are used to lessen the higher order ODEs into 1st order equations, i.e.,

New system of ODEs by using equation (23) is formed, i.e.,along with boundary conditions

#### 4. Graphical Discussion

The graphical discussion of the problem described is as given below. Figure 2 shows the results of for velocity by assuming the values of as which give a decline in the graph. Figures 35 give the temperature profile for the Eckert number, Prandtl number, and squeezing number by using different values of all numbers as and , and we have used such values of the Prandtl number due to non-Newtonian fluid flow and similarly for . The graph decreases for and while increases for the squeezing number. Figure 6 shows skin friction graph for and for the values with positively increasing values. Figure 7 explicates the graphical representations of the Nusselt number for and with with increment in the graph, as the Nusselt number is the ratio of convective to conductive heat transfer at the boundary in a fluid. Figures 810 illustrate the stream lines for the values of . Figures 1113 demonstrate the three-dimensional graph for . Table 1 gives different values for skin friction for .

 [13]

#### 5. Concluding Remarks

Non-Newtonian fluid flow is very persuasive topic since many years because it has an extensive use in many applications such as mining industry, chemical engineering, petroleum engineering, and plastic processing industry. Now, the development in industries has motivated the researchers to discover non-Newtonian fluids’ properties in a more organized way. We have discussed the unsteady boundary layer flow of a second-grade fluid. The discussion is significantly influenced by the fluid which is constructed to check the obtained results which are given below:(1)The velocity with respect to delta and temperature for and are decreasing(2)We have increasing graphs of the skin friction, Nusselt number, and squeezing number of the fluid by increasing the values of their parameters

#### Abbreviations

 : Specific heat at constant pressure : Time : Squeezed strength : Body force : Pressure field : Viscosity ratio : Temperature : Density : Temperature : Prandtl number : Stress tensor : Velocity of the fluid : Viscosity of the fluid : Dimensionless number : Distribution function : Rivlin–Ericksen tensor : Eckert number : Velocity components along direction : Identity tensor : Spatial coordinates .

#### Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### References

1. M. M. Rashidi, A. M. Siddiqui, and M. Asadi, “Application of homotopy analysis method to the unsteady squeezing flow of a second-grade fluid between circular plates,” Mathematical Problems in Engineering, vol. 2010, Article ID 706840, 18 pages, 2010. View at: Publisher Site | Google Scholar
2. A. Hussain, S. Akbar, L. Sarwar, and S. Nadeem, “Probe of radiant flow on temperature-dependent viscosity models of differential type MHD fluid,” Mathematical Problems in Engineering, vol. 2020, Article ID 2927013, 16 pages, 2020. View at: Publisher Site | Google Scholar
3. M. Sheikholeslami, D. Domiri Ganji, M. Younus Javed, and R. Ellahi, “Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model,” Journal of Magnetism and Magnetic Materials, vol. 374, pp. 36–43, 2015. View at: Publisher Site | Google Scholar
4. A. Hussain, L. Sarwar, S. Akbar, M. Y. Malik, and S. Ghafoor, “Model for MHD viscoelastic nanofluid flow with prominence effects of radiation,” Heat Transfer Asian Research, vol. 48, no. 2, 2018. View at: Publisher Site | Google Scholar
5. R. Ellahi and A. Riaz, “Analytical solutions for MHD flow in a third-grade fluid with variable viscosity,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1783–1793, 2010. View at: Publisher Site | Google Scholar
6. A. Hussain, S. Akbar, L. Sarwar, S. Nadeem, and Z. Iqbal, “Effect of time dependent viscosity and radiation effcacy on a non-Newtonian fluid,” Heliyon, vol. 5, no. 2, 2019. View at: Publisher Site | Google Scholar
7. R. Ellahi and S. Afzal, “Effects of variable viscosity in a third grade fluid with porous medium: an analytic solution,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2056–2072, 2009. View at: Publisher Site | Google Scholar
8. A. Wakif, I. L. Animasaun, P. V. Satya Narayana, and G. Sarojamma, “Meta-analysis on thermo-migration of tiny/nano-sized particles in the motion of various fluids,” Chinese Journal of Physics, vol. 68, pp. 293–307, 2020. View at: Publisher Site | Google Scholar
9. F. Mabood, A. Shafiq, T. Hayat, and S. Abelman, “Radiation effects on stagnation point flow with melting heat transfer and second order slip,” Results in Physics, vol. 7, pp. 31–42, 2017. View at: Publisher Site | Google Scholar
10. A. Hussain, L. Sarwar, S. Akbar, and S. Nadeem, “Mathematical model for blood flow through the stenosed channel,” Physica Scripta, vol. 95, no. 2, 2019. View at: Publisher Site | Google Scholar
11. S. Saleem, M. Qasim, A. A. Alderremy, and S. Noreen, “Heat transfer enhancement using different shapes of Cu nanoparticles in the flow of water based nanofluid,” Physica Scripta, vol. 95, no. 5, 2020. View at: Google Scholar
12. A. Hussain, S. Akbar, L. Sarwar, and M. Y. Malik, “Numerical investigation of squeezing flow of Walters' B fluid through parallel plates,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 41, no. 11, 2019. View at: Publisher Site | Google Scholar
13. O. Reynolds, “On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments including an experimental determination of the viscosity of olive oil,” Philosophical Transactions of the Royal Society A, vol. 177, pp. 157–234, 1886. View at: Google Scholar
14. F. R. Archibald, “Load capacity and time relations for squeeze films,” Journal of Lubrication Technology, vol. 78, pp. A231–A245, 1956. View at: Google Scholar
15. S. Saleem and N. Sandeep, “MHD flow and heat transfer of a dusty nano fluid over a stretching surface in porous medium,” Jordan Journal of Civil Engineering, vol. 4, no. 11, pp. 23–35, 2015. View at: Google Scholar
16. S. Saleem, S. Nadeem, M. M. Rashidi, and C. S. K. Raju, “An optimal analysis of radiated nanomaterial flow with viscous dissipation and heat source,” Microsystem Technologies, vol. 25, no. 2, pp. 683–689, 2019. View at: Publisher Site | Google Scholar
17. A. Naseem, A. Shafiq, L. Zhao, and M. U. Farooq, “Analytical investigation of third grade nanofluidic flow over a riga plate using Cattaneo-Christov model,” Results in Physics, vol. 9, pp. 961–969, 2018. View at: Publisher Site | Google Scholar
18. A. Shafiq, S. Jabeen, T. Hayat, and A. Alsaedi, “Cattaneo-Christov heat flux model for squeezed flow of third grade fluid,” Surface Review and Letters, vol. 24, no. 7, Article ID 1750098, 2017. View at: Publisher Site | Google Scholar
19. A. Shafiq, Z. Hammouch, and T. N. Sindhu, “Bioconvective MHD flow of tangent hyperbolic nanofluid with Newtonian heating,” International Journal of Mechanical Sciences, vol. 133, pp. 759–766, 2017. View at: Publisher Site | Google Scholar
20. G. Rasool, T. Zhang, A. J. Chamkha, A. Shafiq, I. Tlili, and G. Shahzadi, “Entropy generation and consequences of binary chemical reaction on MHD Darcy-Forchheimer Williamson nanofluid flow over non-linearly stretching surface,” Entropy, vol. 22, no. 1, p. 18, 2020. View at: Publisher Site | Google Scholar
21. A. Shafiq, I. Zari, G. Rasool, I. Tlili, and T. S. Khan, “On the MHD Casson axisymmetric Marangoni forced convective flow of nanofluids,” Mathematics, vol. 7, no. 11, p. 1087, 2019. View at: Publisher Site | Google Scholar
22. A. Shafiq, I. Khan, G. Rasool, A. Seikh, and E.-S. Sherif, “Significance of double stratification in stagnation point flow of third-grade fluid towards a radiative stretching cylinder,” Mathematics, vol. 7, no. 11, p. 1103, 2019. View at: Publisher Site | Google Scholar
23. G. Rasool, A. Shafiq, C. M. Khalique, and T. Zhang, “Magnetohydrodynamic Darcy-Forchheimer nanofluid flow over a nonlinear stretching sheet,” Physica Scripta, vol. 94, no. 10, Article ID 105221, 2019. View at: Publisher Site | Google Scholar
24. A. Shafiq, Z. Hammouch, and A. Turab, “Impact of radiation in a stagnation point flow of Walters’ B fluid towards a Riga plate,” Thermal Science and Engineering Progress, vol. 6, pp. 27–33, 2018. View at: Publisher Site | Google Scholar
25. A. Shafiq and T. N. Sindhu, “Statistical study of hydromagnetic boundary layer flow of Williamson fluid regarding a radiative surface,” Results in Physics, vol. 7, pp. 3059–3067, 2017. View at: Publisher Site | Google Scholar

Copyright © 2021 Sobia Akbar and Azad Hussain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Related articles

No related content is available yet for this article.
Download other formatsMore
Order printed copiesOrder
Views280
Downloads302
Citations

#### Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.