# The Influences of Squeezed Inviscid Flow between Parallel Plates

Authors: Sobia Akbar; Azad Hussain
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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 [1–25], 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(1)T=−PI+μA1+α1A2+α2A12.We have restriction as(2)μ≥0,α1≥0,α1+α2=0,α1=−α2.So, equation (1) reduces to(3)T=−PI+μA1+α1A2−A12.A1 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 z=±l1−αt1/2, where characteristic parameter is α and length l at t=0 is considered. Furthermore, α>0 is relative to the motion of both squeezed plates till they connect each other at t=1/α, for when α<0, the plates are separated.The fields of the flows corresponding to Cartesian coordinates [16] are(4)V=ux,y,ti+vx,y.tj,(5)θ=θx,y,t.The momentum equations which govern for the problem become(6)ρ∂u∂t+u∂u∂x+v∂u∂y=−∂P∂x+2μ∂2u∂x2+2α1∂3u∂t∂x2−α1∂∂x∂v∂x2+μ∂2u∂y2+μ∂2v∂y∂x+α1∂3u∂t∂y2+α1∂3v∂t∂y2−α1∂2u∂y∂x∂u∂y−α1∂2u∂y2∂u∂x+α1∂2u∂y∂x∂v∂x+α1∂v∂x∂2v∂y∂x+α1∂2u∂y2∂v∂y+α1∂u∂y∂2v∂y2−α1∂v∂x∂2v∂y2−α1∂v∂y∂2v∂y∂x,(7)ρ∂v∂t+u∂v∂x+v∂v∂y=−∂P∂y+2μ∂2v∂x2+2α1∂3v∂t∂y2−α1∂∂y∂u∂y2+μ∂2v∂x2+μ∂2u∂y∂x+α1∂3u∂t∂x∂y+α1∂2u∂x2∂v∂x+α1∂2v∂x2∂u∂x−α1∂u∂y∂2u∂x2−α1∂2u∂y∂x∂u∂x−α1∂2u∂y∂x∂v∂y−α1∂u∂y∂2v∂y∂x−α1∂v∂x∂2v∂y∂x−α1∂v∂y∂2v∂x2.The obtained energy equation is(8)ρCP∂∂t+u∂∂x+v∂∂yT=k∂2∂x2+∂2∂y2T−p∂u∂x+2μ∂u∂x2+2α1∂u∂x∂2u∂t∂x−α1∂u∂x∂v∂x2+μ∂u∂y2+μ∂v∂x∂u∂y+α1∂u∂y∂2u∂t∂y+α1∂u∂y∂2v∂t∂y−α1∂u∂x∂u∂y2+α1∂u∂x∂u∂y∂v∂x+α1∂v∂y∂u∂y2+μ∂u∂y∂v∂x+μ∂v∂x2+α1∂v∂x∂2u∂t∂y+α1∂v∂x∂2v∂t∂x−α1∂v∂x∂u∂y∂u∂x+α1∂u∂x∂v∂x2−−α1∂v∂y∂v∂x2−p∂v∂y+2μ∂v∂y2+2α1∂v∂y∂2v∂t∂y−α1∂v∂y∂u∂y2,where specific heat is CP.By using boundary layer flow equations, (3)–(9) become(9)ρ∂u∂t+u∂u∂x+v∂u∂y=μρ∂2u∂y2+α1ρ∂3u∂t∂y2−α1ρ∂u∂y∂2u∂x∂y+α1ρ∂u∂x∂2u∂y2+α1ρ∂v∂y∂2u∂y2,(10)∂T∂t+u∂T∂x=kρCP∂2T∂x2+∂2T∂y2+μρ∂u∂y2−α1ρ∂u∂x∂u∂y2.Conforming to layout of the inquisition in Figure1, the boundary conditions’ suits for the flow are(11)u=0,v=vw=dhdt,T=THaty=ht,(12)∂u∂y=0,∂T∂t,v=0aty=0.Figure 1
Physical geometry of the problem.


### 2.1. Nondimensional Equations

Using transformation [11],(13)u=αx21−αtF′η,v=−αl21−αt1/2Fη,where(14)η=yl1−αt1/2,(15)θη=TTH.Applying equations (15)–(17) in equations (10)–(12), we obtain(16)2λSFIV+δ2F‴+4λ2F‴−SBF″−Sδ4ηF″−2F′2+FF″−2F′=0,(17)δ2λ5θ″−2ηλS2Prδθ′+δEcPrλ5F″2−2δS2F′F″=0,where(18)δ=l1−αtα,S=αl22μ,Pr=CPμk,Ec=1CPTHαx21−αt,Here, nondimensional length isδ,Ec is Eckert number, S is squeezing number, and Pr is Prandtl number is dimensionless.The nondimensional boundary conditions are(19)F0=0,F″0=0,(20)F1=1,F′1=0,(21)θ′0=0,(22)θ1=1.


## 2.1. Nondimensional Equations

Using transformation [11],(13)u=αx21−αtF′η,v=−αl21−αt1/2Fη,where(14)η=yl1−αt1/2,(15)θη=TTH.Applying equations (15)–(17) in equations (10)–(12), we obtain(16)2λSFIV+δ2F‴+4λ2F‴−SBF″−Sδ4ηF″−2F′2+FF″−2F′=0,(17)δ2λ5θ″−2ηλS2Prδθ′+δEcPrλ5F″2−2δS2F′F″=0,where(18)δ=l1−αtα,S=αl22μ,Pr=CPμk,Ec=1CPTHαx21−αt,Here, nondimensional length isδ,Ec is Eckert number, S is squeezing number, and Pr is Prandtl number is dimensionless.The nondimensional boundary conditions are(19)F0=0,F″0=0,(20)F1=1,F′1=0,(21)θ′0=0,(22)θ1=1.


## 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.,(23)F=w1,F′=w2,F″=w3,F‴=w4,FIV=w4′,θ=w5,θ′=w6,θ″=w6′.New system of ODEs by using equation (23) is formed, i.e.,(24)w1′=w2,w2′=w3,w3′=w4,w4′=w5,(25)w4′=12λS−δ2F‴−4λ2F‴+SBF″+Sδ4ηF″−2F′2+FF″−2F′,(26)w5′=1δ2λ52ηλS2Prδθ′−δEcPrλ5F″2−2δS2F′F″,along with boundary conditions(27)a10=0,a1∞=1,a20=1,a2∞=0,a40=1,a4∞=0.


## 4. Graphical Discussion

The graphical discussion of the problem described is as given below. Figure2 shows the results of δ for velocity by assuming the values of δ as δ=0.1,1.5,1.8,2.0 which give a decline in the graph. Figures 3–5 give the temperature profile for the Eckert number, Prandtl number, and squeezing number by using different values of all numbers as Ec=11.5,11.8,12.1,12.4 and Pr=7.3,7.5,7.7,7.9, and we have used such values of the Prandtl number due to non-Newtonian fluid flow and similarly for S=0.1,0.11,0.12,0.13. The graph decreases for Ec and Pr while increases for the squeezing number. Figure 6 shows skin friction graph for λ and δ for the values λ=0.1,0.3,0.5 with positively increasing values. Figure 7 explicates the graphical representations of the Nusselt number for λ and δ with λ=1.1,2.1,3.1 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 8–10 illustrate the stream lines for the values of δ=0.1,0.3,0.5. Figures 11–13 demonstrate the three-dimensional graph for δ. Table 1 gives different values for skin friction for δ,λ,S,andB.Figure 2
Impact of decreasing effects ofδ for velocity portray.Figure 3
Illustration of decreasing effect ofEc on the temperature distribution.Figure 4
Decreasing effect ofPr on temperature portray.Figure 5
Influence of increasing effects of the squeezing number on temperature.Figure 6
Increasing effects ofλ on the skin friction factor.Figure 7
Impact of increasing effects ofλ for the Nusselt number.Figure 8
Stream lines forδ=0.1.Figure 9
Stream lines forδ=0.3.Figure 10
Stream lines forδ=0.5.Figure 11
3D graph for δ=0.1.Figure 12
3D graph for δ=0.3.Figure 13
3D graph for δ=0.5.Table 1
Values ofδ,λ,S,andB for the skin friction coefficient.
δSλB1/2CfRe[13]0.120.47.00.50.99999−2.0480521.50.84691−2.2272791.80.71694−2.4235372.00.585360−2.04805220.80.99966−0.93824921.20.99669−0.63087621.60.99696−2.0480527.10.78506−2.2385067.20.51946−2.4519467.30.480520−2.0480520.60.852435−2.0563360.70.75236−2.0574560.80.625413−2.048052


## 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 forEc and Pr 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



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