# Nonviscous Oblique Stagnation Point Flow towards Riga Plate

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

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## Abstract

Purpose. The flow of nonviscous Casson fluid is examined in this study over an oscillating surface. The model of the fluid flow has been inspected in the presence of oblique stagnation point flow. The scrutiny is subsumed for the Riga plate by considering the effects of magnetohydrodynamics. The Riga plate is considered as an electromagnetic lever which carries eternal magnets and a stretching line up of alternating electrodes coupled on a plane surface. We have considered nonboundary layer two-dimensional incompressible flow of the fluid. The fluid flow model is analyzed in the fixed frame of reference. Motivation. The motivation of achieving more suitable results has always been a quest of life for scientists; the capability of determining the boundary layer of flow on aircraft which either stays laminar or turns turbulent has encouraged the researcher to study compressible flow in depth. The compressible fluid with boundary layer flow has been utilized by numerous researchers to reduce skin friction and enhance thermal and convectional heat exchange. Design/Approach/Methodology. The attained partial differential equations will be critically inspected by using suitable similarity transformation to transform these flows thrived equations into higher nonlinear ordinary differential equations (ODE). Then, these equations of motion are intercepted by mathematical techniques such as the bvp4c method in Maple and Matlab. The graphical and tabular representation of different parameters is also given. Findings. The behavior of β and modified Hartmann number M increases by positively increasing the values of both parameters for Fη, while ω decreases with increasing the values of ω for Fη. The graph of β shows upward behavior for distinct values for both Gη and G′η for velocity portray. Prandtl number and β for the temperature profile of θη and θ1η goes downward with increasing parameters.


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## Body

## 1. Introduction

The idea of stagnation point flow to find the exact solution of the Navier–Stokes equations was earlier discussed by Hiemenz [1] The applications of heat flow of stagnation point flows are impartially manifested in paper manufacture, melt spinning process, revolving fibers, continuous molding, and crystal puffing, given by Mehmood et al. [2] In the divination of skin friction in addition to mass or heat transfer near to stagnation zone of bodies in transpiration cooling, great speed movements, project of thrust compartment, radial diffusers, thermal oil repossession, and drag decrease along with several substantial control have considerable physical consequences [3].The flows of magnetohydrodynamic electrical conducting fluids have apprehended the notice of scientists and researchers as they have numeral utilizations in industrial and engineering fields. The skin friction and heat transfer coefficient have higher results for turbulent flow as compared to laminar flow; any numerical technique that can be employed to enhance the stability of laminar flow is worth investigating. However, the properties of high electrical conductivity are not effective in all fluids. Ibrahim [4] analyzed the properties of nonlinear MHD flow with thermal radiations of nanofluid regulated with a stretching sheet in the zone of stagnation point flow with the effects of convective heat transfer. Li et al. [5] studied mannerisms of MHD flow of non-Fourier nanofluid with heat conduction determined by the stretched surface. The radiative flow of MHD nanofluid with the effects of heat transfer regarding a stretching surface was analyzed by Ghasemi et al. [6]. The micropolar boundary layer flow of nanofluid with stretching sheet was described by Turkyilmazoglu [7].Wall-parallel Lorentz force is processed by using the Riga plate, as Riga plate is a compound of alternating electrodes and persistent magnets connected in a flat surface. In this regulation, first strive was taken into account to produce a wall-parallel Lorentz force which was discussed by Gailitis et al. [8]. In this study, it is initiated that, by suing electromagnetic body forces, the electrically conducting fluid flows can be handled to repay the momentum scarcity in the boundary layer. It can be worn as a sufficient factor for decline of skin friction and pressure drag of submarines circumvents the segregation of the boundary layer.


## 2. Problem Statement

Consider the unsteady incompressible flow model for non-Newtonian Casson fluid with the Riga plate. The unsteadiness in the fluid has been induced by the time-dependent flow. The model geometry is shown in Figure1(a). In the rectangular coordinate system, u and v are velocity components along x- and y-axis. The plate vacillating in its own plan and the flow of the nanofluid distract obliquely on it. The fluid is in the upper half of the plane y≥0. The stress tensor for the given flow field is defined as(1)T=−PI+μβ1+1βA1.Figure 1
Physical geometry of the problem.


## 3. Mathematical Analysis

The continuity equation for the given problem is(2)∂u∂x+∂v∂y=0.The momentum equations for the given flow field is expressed as(3)ρ∂u∂t+u∂u∂x+v∂u∂y=−∂P∂x+μβ1+1β∂2u∂x2+∂2v∂y2+πJ0M08e−π/ay,(4)ρ∂v∂t+u∂v∂x+v∂v∂y=−∂P∂y+μβ1+1β∂2u∂x2+∂2v∂y2.Similarly, the energy equation is given as(5)ρCP∂∂t+u∂∂x+v∂∂yT=kμβ1+1β∂2∂x2+∂2∂y2T,where the current density width of magnet is J0, specific heat is CP, and the magnetization of permanent magnets is M0.Now, we are going to introduce the occurrence of the stream function as(6)u=∂Ψ∂y,v=−∂Ψ∂x.By using stream function in equations (3)–(5), we have(7)∂∂t∂2Ψ∂x2+∂2Ψ∂y2+∂Ψ∂y∂3Ψ∂x∂y2+∂3Ψ∂x3−∂Ψ∂x∂3Ψ∂y3+∂3Ψ∂y∂x2−μβ1+1/βρ∂4Ψ∂x4+2∂4Ψ∂x2∂y2+∂4Ψ∂y4=0,(8)∂∂t∇4Ψ−∂Ψ∂x∂∂y∇2Ψ−∂Ψ∂y∂∂x∇2Ψ−μβ1+1/βρ∇2Ψ=0,(9)kμβ1+1/βρCP∂2T∂x2+∂2T∂y2−∂T∂t−∂Ψ∂y∂T∂x−∂Ψ∂x∂T∂y=0.


## 4. Fixed Frame of Reference for the Fluid Flow Model

Conferring to [9], we consider that(10)Ψ=kxfy+gy,where k is a parameter. We examine that the plate is oscillating at the point y=0, and the fluid engaged at the upper half plane y>0. The flow function is given by [41] Ψ=1/2γy2+xy.The above equations can also be written as(11)μβ1+1/βρd3fdy3+afd2fdy2−dfdydfdy=−a,(12)μβ1+1/βρd3gdg3+af∂2g∂y2−dgdydfdy−∂2g∂y∂t=−a.The boundary conditions are given below:(13)fy=0,f'y=0,gy=0,g′y=0,aty=0,f′y=1,g′y=γy,aty⟶∞,where γ is the dimensionless number.We introduce(14)fy=υaFη,gt,y=υaG0η+εG1ηeiτ,where(15)τ=ωt,ε=U√υk,θy+εθ1y=T−T∞Tw−T∞.By using the nondimensional form of equations (10) and (11), we get the following form of equations (9), (11), and (12) as follows:(16)1+1βF‴+FF″−F′2+1+Me−ωη=0,(17)1+1βG‴+FG″−G′F′=0,(18)1+1βG1‴+G1″F−G1′F′−iσG1′=0,(19)1+1/βPrθ″+Fθ′=0,(20)1+1/βPrθ1″+Fθ1′−iσθ1=0.The nondimensional boundary conditions for the above all equations (16)–(20) are given below:(21)F0=0,F′0=0,F′∞=1,G0=0,G′0=0,G″∞=γ,G10=0,G1′0=1,G1″∞=γ,(22)θ∞=0,θ0=1,θ1∞=0,θ10=1.


## 5. Numerical Solution

The equations which we obtained from the basic governing equations are highly nonlinear which are changed to ODEs numerically. Then, we have solved these equations with the help of Maple software (equations (4) and (11)).Now, we are going to introduce new scheme of variables to solve these higher order ODEs into 1st order equations, i.e.,(23)F=q1,F′=q2,F″=q3,F‴=q3′,G=q4,G′=q5,G″=q6,G‴=q6′,G1=q7,G1′=q8,G1″=q9,G1″′=q9′,θ=q10,θ′=q11,θ″=q11′,θ1=q12,θ1′=q13,θ1″=q13′.By using (23), we have the following form of equations (16)–(20) as(24)q1′=q2,q2′=q3,q3′=q4,q4′=q5,q5′=q6,q6′=q7,q7′=q8,q8′=q9,q9′=q10,q10′=q11,q11′=q12,q12′=q13,q3′=11+1/βF′2−FF″−1−Me−ωη,q6′=11+1/βG′F′−FG″,q9′=11+1/βG1′F′−G1″F+iσG1′,q11′=Pr1+1/βFθ′,q13′=Pr1+1/β−Fθ1′+iσθ1,under appropriate boundary conditions:(25)w10=0,w1∞=1,w10=0,w40=0,w50=0,w5∞=γ,w70=0,w80=0,w8∞=γ,w100=1,w11∞=0,w120=1,w13∞=0.


## 6. Graphical Discussion

Newtonian and non-Newtonian fluids are widely used in chemical engineering, bio-chemical engineering, food processing, oil exploration, and medical engineering. In particular, what has been studied most intensely for obvious practical reasons is how heat and momentum are transferred to moving Newtonian and non-Newtonian fluids. The inquisition which we have considered for investigation can be represented graphically by different distinct parameters as stated below. Figure1 gives the geometry of the given problem for nonviscous behavior of the flow model. Figure 2 represents the Riga plate that persuades Lorentz force opposite to the external free stream. The portray of β for the different values of β, which gives the increasing behavior for Fη, is shown in Figure 3. The increasing graph of modified Hartmann number M for Fη is given in Figure 4. Figure 5 illustrates the velocity profile for β with different values for Gη. The graph increases positively by increasing the values of β. Figure 6 shows the decreasing graph of ω for Fη. The upward graph of β for G1η is demonstrated for distinct values of β in Figure 7. The illustration of β for the temperature profile of θ1η for positive values of β is given in Figure 8. The Prandtl number for high values of Pr for θ1η is given in Figure 9, which goes downwards with increasing Pr. Similarly, the decreasing graphs of Pr for θη with positive values of the Prandtl number is shown in Figure 10. The portray for temperature of β decreasing with increasing the values of β for θη represented in Figure 11 The Nusselt number graph of β=0.1,0.3,0.5 is shown in Figure 12, which demonstrates the upward lines in the graph. Figure 13 describes the unsteady flow for the given fluid model. Figures 14–16 express the three-dimensional portray for the distinct values of β. Similarly, different values of M show the stream lines graphs in Figures 17–19. Different values of the Nusselt number for θη are represented in Table 1. In Table 2, distinct values of the Nusselt number for θ1η are given. Table 3 shows the skin friction coefficient.Figure 2
Riga plate that persuades Lorentz force opposite to the external free stream.Figure 3
Effects ofβ for fη.Figure 4
Impact ofM for fη.Figure 5
Effects ofβ for gη.Figure 6
Influence ofω for fη.Figure 7
Implementation ofβ for gη.Figure 8
Impact ofβ for θ1.Figure 9
Impact ofPr for θ1.Figure 10
Effects ofPr for θ.Figure 11
Impact ofβ for θ.Figure 12
Graph showing upward behavior forβ for the Nusselt number.Figure 13
Impact of time-dependent flow on the fluid model.Figure 14
Three-dimensional graph forM=0.1.Figure 15
Three-dimensional graph forM=0.5.Figure 16
Three-dimensional graph forM=0.9.Figure 17
Stream lines forβ=0.1.Figure 18
Stream lines forβ=0.5.Figure 19
Stream lines forβ=0.9.Table 1
Values for the Nusselt number forθη distinct parameters which are compared with the previous study [10].
Prβ−θ′0[10]7.80.11.2399580.6649147.91.2459070.8218948.01.2518071.0819427.81.2399580.6649140.20.4712560.8562560.30.6200500.8879560.11.2399580.856256Table 2
Values for the Nusselt number forθ1η distinct parameters.
Prσβ−θ′0[11]7.81.81.80.3869200.6659358.80.3342430.8572609.80.3256990.8889697.80.3869200.6659351.90.3258270.6749422.00.3095360.8318991.80.3869200.6659351.90.4158940.0829862.00.3642780.3756431.80.3869200.665935Table 3
Values ofM,β,andω for the skin friction coefficient.
βMω1/2CfRe[12]0.11.31.30.474983−0.8393020.20.638577−0.9370150.30.768523−1.0539340.10.474983−0.8393021.40.480390−0.8294151.50.485796−0.9338321.30.474983−0.8393021.40.471195−1.0264991.50.467776−0.4783021.30.474983−0.38643


## 7. Concluding Remarks

We have discussed the unsteady the flow of Casson fluid by assuming the Riga plate over the oscillating surface. The obtained results which are given below:(1)
The behavior ofβ and modified Hartmann number M increase by positively increasing the values of both parameters for Fη, while ω decreases with increasing the values of ω for Fη(2)
The graph ofβ shows upward behavior for distinct values for both Gη and G′η for velocity portray(3)
The Prandtl number andβ for the temperature profile of θη and θ1η goes downward with increasing parameters(4)
Nusselt number graph for temperature shows an increment



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