Unsteady Viscous Incompressible Bingham Fluid Flow Through a Parallel Plate

inventions

Article Unsteady Viscous Incompressible Bingham Fluid Flow through a Parallel Plate

Muhammad Minarul Islam 1, Md. Tusher Mollah 1 , Sheela Khatun 1, Mohammad Ferdows 2,* and Md. Mahmud Alam 3

1 Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj 8100, Bangladesh 2 Department of Applied Mathematics, Dhaka University, Dhaka 100, Bangladesh 3 Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh * Correspondence: [email protected]

Received: 15 June 2019; Accepted: 14 August 2019; Published: 27 August 2019

Abstract: Numerical investigation for unsteady, viscous, incompressible Bingham fluid flow through parallel plates is studied. The upper plate drifts with a constant uniform velocity and the lower plate is stationary. Both plates are studied at different fixed temperatures. To obtain the dimensionless equations, the governing equations for this study have been transformed by usual transformations. The obtained dimensionless equations are solved numerically using the explicit finite difference method (FDM). The studio developer Fortran (SDF) 6.6a and MATLAB R2015a are both used for numerical simulations. The stability criteria have been established and the system is converged for Prandtl number Pr 0.08 with ∆Y = 0.05 and ∆τ = 0.0001 as constants. As a key outcome, ≥ the steady-state solutions have been occurred for the dimensionless time τ = 4.00 The influence of parameters on the flow phenomena and on shear stress, including Nusselt number, are explained graphically. Finally, qualitative and quantitative comparison are shown.

Keywords: Bingham fluid; explicit finite difference method; stability analysis

1. Introduction A Bingham fluid is a non-Newtonian viscoplastic fluid that possesses a yield strength that must be outstripped before the fluid will flow. In many geological and industry materials, Bingham fluids are used as a general mathematical basis of mud flow in drilling engineering, including in the handling of slurries, granular suspensions, etc. Bingham fluid is named after Eugene C. Bingham, who declared its mathematical explanation. In this regard, an exploration of the laws for plastic flow has been studied by Bingham [1]. Darby and Melson [2] formulated an empirical expression to guess the friction loss factor for the drift of a Bingham fluid. Bird et al. [3] analyzed the rheology and flow phenomena of viscoplastic materials. Convective heat transfer for Bingham plastic inside a circular pipe and the numerical approach for hydro-dynamically emerging flow and the simultaneously emerging flow were studied by Min et al. [4]. Liu and Mei [5] considered the slow spread of a Bingham fluid sheet on an inclined plane. For Bingham fluids, the Couette–Poiseuille flow between two porous plates, taking slip conditions into consideration, was investigated by Chen and Zhu [6]. Sreekala and Kesavareddy [7] mentioned the Hall impacts on magneto-hydro dynamics (MHD) Bingham plastic flow over a porous medium, including uniform suction and injection. For Bingham fluids, the MHD flow for an unsteady case considering Hall currents was described by Parvin et al. [8]. Rees and Bassom [9] considered Bingham fluids over a porous medium following a rapid modification of surface heat flux.

Inventions 2019, 4, 51; doi:10.3390/inventions4030051 www.mdpi.com/journal/inventions Inventions 2019, 4, 51 2 of 11 Inventions 2019, 4, x 2 of 11

Mollah etet al.al. [ 10[10]] numerically numerically studied studied the the ion-slip ion-slip and Halland impactsHall impacts for MHD for BinghamMHD Bingham fluid flowing fluid flowingthrough through parallel plates,parallel including plates, including the considerations the considerations of unsteady of unsteady cases and cases suction. and suction Gupta .Gupta et al. [11 et] al.considered [11] considered the MHD the mixed MHD convective mixed convective stagnation pointstagnation flow andpoint the flow heat transferand the ofheat an incompressibletransfer of an incompressiblenano fluid over annano inclined fluid over stretching an inclined sheet. Mollahstretching et al. sheet. [12] investigatedMollah et al. Bingham [12] investigated fluid flow Bingham through fluidan oscillatory flow through porous an plateoscillatory with an porous ion-slip plate and with hall an current. ion-slip and hall current. Along with these studies, ourour aimaim is to numerically investigateinvestigate unsteadyunsteady BinghamBingham fluidfluid flow flow between parallel plates. The The current current study is conc concernederned with the generalized Ohm’s law considering the ion-slip and Hall Hall current current is is absent. absent. Additional Additionally,ly, the Couette Couette flow flow is is considered considered where where the the viscous viscous dissipation, pressurepressure gradient, gradient, and and rheology rheology of of Bingham Bingham fluids fluids are are involved. involved. The The explicit explicit FDM FDM has been has beenused toused demonstrate to demonstrate the dimensionless the dimensionless non-linear, non-line coupled,ar, coupled, partial di partialfferential differential equations. equations. The achieved The achievedresults are results shown are graphically. shown graphically.

2. Mathematical Mathematical Formulation Formulation The physical configuration configuration of the considered problem that corresponds with the boundary conditions is shown in FigureFigure1 .1. AA laminar,laminar, incompressible,incompressible, non-Newtoniannon-Newtonian BinghamBingham fluidfluid isis considered toto bebe flowingflowing among among two two infinite infinite plates plates situated situated atthe at they = 𝑦=ℎh planesplanes and lengtheningand lengthening from ± fromx = 0 𝑥=0 to . 𝑡𝑜. ∞ ∞

Figure 1. Physical configuration. conﬁguration.

The upper plate has a uniform velocity, U𝑈0 andand thethe lowerlower plateplate isis stationary.stationary. Two constant temperatures, T 𝑇2and andT 𝑇1are are considered considered respectively respectively for for both both upper upper and and lower lower plates, plates, where whereT2 >𝑇T>1. A constant pressure gradient dP is driven on the ﬂuid along X-direction and the body force are neglected. 𝑇. A constant pressure gradientdx is driven on the fluid along 𝑋-direction and the body force are Within the above considerations, the model is governed by the following equations: neglected. ContinuityWithin equation: the above considerations, the model is governed by the following equations: Continuity equation: ∂u ∂v + = 0 (1) 𝜕𝑢 ∂x𝜕𝑣 ∂y + =0 (1) 𝜕𝑥 𝜕𝑦 Momentum equation: Momentum equation: ! ∂u ∂u ∂u 1 dp 1 ∂ ∂u 𝜕𝑢 𝜕𝑢+ u 𝜕𝑢+ v =1 𝑑𝑝 1 +𝜕 𝜕𝑢µ (2) +𝑢∂t +𝑣∂x =−∂y −ρ+dx ρ𝜇∂y ∂y (1) 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜌 𝑑𝑥 𝜌 𝜕𝑦 𝜕𝑦 Energy equation: Energy equation: 2 ! !2 ∂T ∂T ∂T κ ∂ T µ ∂u 𝜕𝑇 𝜕𝑇+ u 𝜕𝑇+ v κ= 𝜕𝑇 𝜇+ 𝜕𝑢 (3) 2 +𝑢∂t +𝑣∂x =∂y cpρ+∂y ρcp∂ y (2) 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝑐𝜌 𝜕𝑦 𝜌𝑐 𝜕𝑦 where the apparent viscosityviscosity of the BinghamBingham ﬂuids:fluids: 𝜏 𝜇=Κ + τ0 µ = K+ (3)(4) ∂u ∂y Where Κ represents the plastic viscosity of the Bingham fluid and 𝜏 represents the yield stress. The corresponding initial and boundary conditions can be given as follows: Inventions 2019, 4, 51 3 of 11

where K represents the plastic viscosity of the Bingham ﬂuid and τ0 represents the yield stress.

The corresponding initial and boundary conditions can be given as follows: t 0, u = 0, T = T everywhere ≤ 1

u = 0, T = T1 at x = 0 t > 0, u = 0, T = T at y = h (5) 1 − u = U0, T = T1 at y = h

For the numerical solution, the above Equations (1)–(5) are required to transfer dimensionless form. The dimensionless quantities (6) that have been used which are considered as follows:

x y u v tU T T µ p X = , Y = , U = , V = , τ = 0 , θ = 1 , µ = ,P = (6) − 2 h h U0 U0 h T2 T1 K ρU − 0 The non-dimensional variables (6) are used in Equations (1)–(5) to obtain the dimensionless governing Equations (7)–(11): ∂U ∂V + = 0 (7) ∂X ∂Y ! ∂U ∂U ∂V dP 1 ∂ ∂U + U + V = + µ (8) ∂τ ∂X ∂Y −dX Re ∂Y ∂Y ! !2 ∂θ ∂θ ∂θ 1 ∂2θ ∂U + U + V = + Ecµ (9) ∂τ ∂X ∂Y Pr ∂Y2 ∂Y where τ µ = 1 + D (10) ∂U ∂Y The non-dimensional parameters are as follows:

τ0h ρU0h ρcpU0h Bingham number, τD = ; Reynolds number, Re = ; Prandtl number, Pr = ; and KU0 K k U0K Eckert number, Ec = . ρcph(T T ) 2− 1 The subjected dimensionless conditions can be noted as follows: τ 0, U = 0, θ = 0 everywhere ≤ U = 0, θ = 0 at X = 0 τ > 0, U = 0, θ = 0 at Y = 1 (11) − U = 1, θ = 1 at Y = 1

3. Shear Stress and Nusselt Number The shear stresses at both the stationary and moving plates are studied from the velocity profile. The local shear stress in X-direction for stationary plate is τ = µ ∂U and for the moving plate L1 ∂Y Y= 1 − is τ = µ ∂U . Moreover, the rate of heat transfer, or the Nusselt number, for the stationary and L2 ∂Y Y=1 moving plates are studied from the temperature fields. The Nusselt number in the X-direction for ∂T ∂T ( ∂Y )Y= 1 ( ∂Y )Y=1 the stationary plate is Nu = − and for the moving plate is Nu = , where Tm is the L1 Tm L2 (Tm 1) − R 1 − − 1 UθdY dimensionless mean fluid temperature and is given by Tm = R−1 . 1 UdY − 4. Numerical Technique A set of finite difference approaches are required to solve dimensionless coupled non-linear partial differential Equations (7)–(10). The dimensionless coupled non-linear partial differential Inventions 2019, 4, 51 4 of 11

Equations (7)–(10) have been solved by using explicit FDM, subject to the boundary conditions (11). There are number of experiments to solve this kind of unsteady time-dependent problem using FDM where the pressure gradient is constant. The stability and convergence criteria of the explicit FDM are establishedInventions 2019 for, 4, x ﬁnding the restriction of the values of various parameters to get converged solutions4 of 11 (Mollah et al. [10,12] and Akter et al. [13]). Here, thethe regionregion insideinside thethe boundary boundary layer layer is is distributed distributed into into grid grid spaces spaces of of lines lines parallel parallel to to the theX and 𝑋 andY axes; 𝑌 whereaxes; where the X axis the is 𝑋 towards axis is thetowards plate andthe Yplateaxis isand normal 𝑌axis to is the normalX axis. Itto isthe determined 𝑋 axis. It that is determined that the height of the plate is 𝑋 = 40, i.e.,𝑋extends from 0 to 40 and 𝑌 = the height of the plate is Xmax = (40), i.e., X extends from 0 to 40 and Ymax = (2) which corresponds to Y2 which, i.e., correspondsY limits from to 0 to𝑌 2 (length). → ∞, i.e., 𝑌 There limits are fromm = 040 toand 2 (length).n = 40 grid There spacing are lines𝑚 in= the40 X𝑎𝑛𝑑and 𝑛= → ∞ Y40directions, grid spacing respectively, lines in the as shown𝑋 and 𝑌 in directions,Figure2. It isrespectively, shown that as∆ Xshownand ∆ inY Figureare constant 2. It is mesh shown or gridthat space∆𝑋 and lines ∆𝑌 towardsare constantX and meshY directions, or grid space respectively, lines towards and is 𝑋 noted and 𝑌 as directions, follows, ∆ Xrespectively,= 1.0 (0 x and40 is) ≤ ≤ and,noted with as follows, the smaller ∆𝑋 time-step = 1.0 0≤𝑥≤40 of ∆τ =0.0001. and, with the smaller time-step of ∆𝜏 = 0.0001.

FigureFigure 2. Finite 2. Finite difference diﬀerence grid grid space. space.

U𝑈0 andand θ 𝜃0 denote denote the the values values ofof U 𝑈and andθ 𝜃at at the the last last step step of of time, time, respectively. respectively. ByBy applyingapplying the explicit FDM, the setset ofof ﬁnitefinite didifferenceﬀerence equationsequations forfor thethe modelmodel areare expressedexpressed asas follows:follows:

𝑈, −𝑈, 𝑉, −𝑉, Ui,j Ui +1,j Vi,j Vi=0,j 1 (12) ∆𝑋− − + ∆𝑌 − − = 0 (12) ∆X ∆Y 𝑈, −𝑈, 𝑈, −𝑈, 𝑈, −𝑈, 𝑑𝑃 +𝑈, +𝑉, =− ∆𝜏 Ui0,j Ui,j ∆𝑋Ui,j Ui 1,j Ui∆𝑌,j Ui,j 1 𝑑𝑋 − − − − − dP (13) + Ui,j X + Vi,j Y = 𝜇∆̅ τ −𝜇̅ 𝑈∆ −𝑈 ∆ 𝑈 −−2𝑈dX +𝑈 (13) 1 , 1 ,µi,j µi,j ,1 Ui,j,Ui,j 1 ,Ui,j+1 2Ui,j,+Ui,j 1 , + + − − − +𝜇− +̅ µ − − Re ∆Y ∆Y , i,j 2 𝑅 ∆𝑌 ∆𝑌 (∆∆𝑌Y) θ θ 𝜃 −𝜃 i0,j i𝜃,j −𝜃 θi,j θi 1,j 𝜃 −𝜃θi,j θi,j 1 𝜃 θi,j+−2𝜃1 2θi,j+θ+𝜃i,j 1 , , − , ,− − , ,− − 1 1, − , − , + Ui,j + Vi,j = 2 +𝑈∆,τ ∆+𝑉X , ∆Y = Pr (∆Y) ∆𝜏 ∆𝑋 ∆𝑌 " 𝑃 2# ∆𝑌 (14) Ui,j Ui,j 1 (14) + Ec µ − − i𝑈,j , −𝑈∆,Y +𝐸 𝜇̅ , ∆𝑌

𝜏 𝜇̅, =1+ (15) , , ∆ Further, the finite difference subjected conditions may be described as follows: 𝑈 =0, 𝜃 = 0 𝑎𝑡 𝐿 =−1 , , (16) 𝑈, =1, 𝜃, = 1 𝑎𝑡 𝐿 = 1 Inventions 2019, 4, 51 5 of 11

τD µi,j = 1 + (15) Ui,j Ui,j 1 − − ∆Y Further, the ﬁnite diﬀerence subjected conditions may be described as follows:

U = 0, θ = 0 at L = 1 i,L i,L − (16) Ui,L = 1, θi,L = 1 at L = 1

5. Stability and Convergence Analysis The stability and convergence criteria of the explicit FDM are established for finding the restriction of the values of various parameters to get converged solutions. Excluding the stability criteria and convergence analysis of the acquired finite difference method, the analysis will remain incomplete unless an explicit process is being used. For the constant grid spaces, the stability analyses of the design have been performed as follows. Equations (12) and (15) will be disregarded, because ∆τ does not exist. At a time arbitrarily called t = 0, the Fourier expansion for U and θ at are all eiαXeiβY apart from a constant, where i = √ 1. − The expressions for U and θ at time t = τ can be defined as follows:

U : ψ(τ)eiαXeiβY (17) θ : φ(τ)eiαXeiβY

After the time-step, these terms have taken the form as follows:

iαX iβY U : ψ0(τ)e e iαX iβY (18) θ : φ0(τ)e e

Substituting Equations (15) and (17) into Equations (13) and (14), concerning the coeﬃcients U and V as constants over any one time-step, the obtained equations can be described as follows: ( ) iα∆X ( ) iβ∆Y ψ0(τ) ψ(τ) ψ τ 1 e− ψ τ 1 e− − + U − + V − = 0 (19) ∆τ ∆X ∆Y

iα∆X iβ∆Y   ( ) ( ) φ(τ) 1 e− φ(τ) 1 e− ( )( ) φ0 τ φ τ 1 2φ τ cos β∆Y 1  − + U − + V − =  −  (20) ∆τ ∆X ∆Y Pr  (∆Y)2 

The simpliﬁcation of Equations (19) and (20) can be denoted in matrix form, and are expressed as follows: " # " #" # ψ0 A 0 ψ = , that is η0 = Tη (21) φ0 0 B φ where, " # " # " # ψ0 A 0 ψ U∆τ iα∆X V∆τ iβ∆Y η0 = , T = , η = , A = 1 ∆X 1 e− ∆Y 1 e− , φ0 0 B φ − − − − U∆τ iα∆X V∆τ iβ∆Y 2∆τ B = 1 ∆X 1 e− ∆Y 1 e− + 2 (cos β∆Y 1). − − − − Pr(∆Y) − The Eigen values of the above matrix T have been computed to acquire the stability condition and ﬁnally the stability conditions of the problem can be expressed in the following Equation (22):

U∆τ V ∆τ 2∆τ + 1 (22) | | 2 ∆X − ∆Y Pr(∆Y) ≤

Using ∆Y = 0.05, ∆τ = 0.0001 and the initial conditions, the above equation gives Pr 0.08. ≥ Inventions 2019, 4, 51 6 of 11

6. Results and Discussion In order to assess the physical characteristics of this developed mathematical model, the steady-state numerical solution have been simulated for the dimensionless primary velocity U and temperature fields θ within the boundary layer. The influences of Bingham number τD, Reynolds number Re, Prandtl number Pr and Eckert number Ec on velocity and temperature fields, shear stress, and Nusselt number, where Re = 2.0, Pr = 0.08, Ec = 0.10 and τD = 0.001 and following the procedures described by Akter et al. [13]. The total results were discussed as the following five parts:

1. Examine the mesh sensitivity 2. Examine the sensitivity of MATLAB and FORTRAN coding output. 3. Finding the steady state solution 4.E ﬀect of parameters Inventions5. Comparison 2019, 4, x with the published results 6 of 11

6.1. Validation Validation

6.1.1. Mesh Mesh Sensitivity: Sensitivity: To obtainobtain thethe appropriateappropriate grid grid space space for form 𝑚and 𝑎𝑛𝑑n the 𝑛 the computations computations have have been been carried carried out for out three for threedifferent different grids, grids,(m = 20,𝑚 20) =; (20,m = 20 ; 30,𝑚=30,30 30) and ( m 𝑎𝑛𝑑= 40, 𝑚=40,𝑛=40n = 40), which, which are shown are shown in Figure in Figure3. It is 3.seen It is from seen this from figure this thatfigure the that graph the for graph(m = for40, 𝑚n = 40 = )40,is more 𝑛 =40 closedis more than closed others. than Also others. it has Also been it hasverified been to verified enlarge to the enlarge quoted the point quoted of the point Figure of 3thea,b. Figure The enlargement 3a,b. The enlargement sub-figures ofsub-figures Figure3a,b of Figureshow that 3a,b the show dash that and the doted dash lines and aredot closeded lines to are each closed other. to Thus, each (other.m = 40, Thus,n = 40𝑚=40,𝑛=40) can be chosen can as bethe chosen appropriate as the mesh.appropriate For further mesh. consideration, For further consideration, the results of the velocity results and of velocity temperature and temperature fields, shear fields,stress, shear and Nusselt stress, and number Nusselt have number been simulated have been at simulated(m = 40, n at= 40𝑚=40,𝑛=40). .

(a) Velocity Profile (b) Temperature Profile Figure 3. Effects of grid space lines on (a) velocity field at moving plate, (b) temperature field at Figure 3. Effects of grid space lines on (a) velocity field at moving plate, (b) temperature field at moving moving plate, where 𝑅 =2.0,𝑃 =0.08,𝐸 = 0.10 and 𝜏 = 0.001. plate, where Re = 2.0, Pr = 0.08, Ec = 0.10 and τD = 0.001.

6.1.2. Sensitivity Sensitivity of of MATLAB MATLAB R2015a and SDF 6.6a The accuracy is determined for for the simulated re resultssults by by MATLAB R2015a and SDF 6.6a tools and explicit explicit finite finite difference difference is is used used as as a asoluti solutionon technique. technique. Using Using MATLAB MATLAB and andFortran Fortran code, code, the Figurethe Figure 4a,b4 a,bare areshown shown for for velocity velocity field field and, and, Figu Figurere 4c,d4c,d are are shown shown for for the the shear shear stress respectively. The impact of velocity field field and shear stress at moving plate have been illustrated for several values of Reynolds Reynolds number number 𝑅(Re.) The. The identical identical results results are areobtained obtained for bo forth boththe tools the (see tools Figure (see Figure 4a–d).4 Thus,a–d). theThus, sensitivity the sensitivity of coding of coding achieved achieved accuracy. accuracy.

Inventions 2019, 4, x 6 of 11

6.1. Validation

6.1.1. Mesh Sensitivity: To obtain the appropriate grid space for 𝑚 𝑎𝑛𝑑 𝑛 the computations have been carried out for three different grids, 𝑚 = 20, 20 ; 𝑚=30,30 𝑎𝑛𝑑 𝑚=40,𝑛=40, which are shown in Figure 3. It is seen from this figure that the graph for 𝑚 = 40, 𝑛 =40 is more closed than others. Also it has been verified to enlarge the quoted point of the Figure 3a,b. The enlargement sub-figures of Figure 3a,b show that the dash and doted lines are closed to each other. Thus, 𝑚=40,𝑛=40 can be chosen as the appropriate mesh. For further consideration, the results of velocity and temperature fields, shear stress, and Nusselt number have been simulated at 𝑚=40,𝑛=40.

(a) Velocity Profile (b) Temperature Profile Figure 3. Effects of grid space lines on (a) velocity field at moving plate, (b) temperature field at

moving plate, where 𝑅 =2.0,𝑃 =0.08,𝐸 = 0.10 and 𝜏 = 0.001.

6.1.2. Sensitivity of MATLAB R2015a and SDF 6.6a The accuracy is determined for the simulated results by MATLAB R2015a and SDF 6.6a tools and explicit finite difference is used as a solution technique. Using MATLAB and Fortran code, the Figure 4a,b are shown for velocity field and, Figure 4c,d are shown for the shear stress respectively. The impact of velocity field and shear stress at moving plate have been illustrated for several values

of Reynolds number 𝑅. The identical results are obtained for both the tools (see Figure 4a–d). Thus, Inventionsthe 2019sensitivity, 4, 51 of coding achieved accuracy. 7 of 11

Inventions 2019, 4, x 7 of 11

Figure 4. Effects of Reynolds number𝑅 on (a) velocity profile at moving plate (MATLAB R2015a); (b) velocity profile at moving plate (SDF 6.6a); (c) shear stress at moving plate (MATLAB R2015a); (d) Figure 4. Effects of Reynolds number (Re) on (a) velocity profile at moving plate (MATLAB R2015a); Figure 4. Effects of Reynolds number𝑅 on (a) velocity profile at moving plate (MATLAB R2015a); (shearb) velocity stress profileat moving at moving plate (SDF plate 6.6a), (SDF where 6.6a); 𝑃 (c=) shear0.08, 𝐸 stress = 0.10 at and moving 𝜏 =0.001 plate (MATLAB. R2015a); (b) velocity profile at moving plate (SDF 6.6a); (c) shear stress at moving plate (MATLAB R2015a); (d) (d) shear stress at moving plate (SDF 6.6a), where Pr = 0.08, Ec = 0.10 and τD = 0.001. 6.2. Steady-Stateshear stress Solution: at moving plate (SDF 6.6a), where 𝑃 = 0.08, 𝐸 = 0.10 and 𝜏 =0.001. 6.2. Steady-State Solution: To6.2. acquireSteady-State the Solution:steady-state solution of this developed mathematical model, the computations for velocityTo acquireTo andacquire the temperature steady-statethe steady-state have solution solution been of continu thisof this developed eddeveloped for different mathematical mathematical dimensionless model,model, the times, computations such as 𝜏 for = velocity 0.20, for0.70, andvelocity 1.20, temperature 3.00,and temperature4.00, have 5.00, been6.00, have continuedetc. been It iscontinu shown for died ffin forerent Figure different dimensionless 5a,b dimensionless that the times, computations times, such such as τ for =as 0.20, 𝑈𝜏 𝑎𝑛𝑑 = 0.70, 𝜃 1.20,have 3.00, 0.20,shown 4.00,0.70, little 1.20, 5.00, change 3.00, 6.00, 4.00, etc. after 5.00, It is𝜏= 6.00, shown3.00 etc. in andIt Figureis shownalso5 showna,b in Figure that negligible the 5a,b computations that changethe computations forafterU 𝜏=and for4.00. θ 𝑈have Thus,𝑎𝑛𝑑 shown 𝜃 the littlesolutions changehave ofshown all after variables littleτ = change3.00 for and after𝜏= also 4.00𝜏= shown 3.00 are andtaken negligible also essentially shown change negligible as after the stτchangeeady-state= 4.00. after Thus, 𝜏=solutions. the4.00. solutions Thus, Figure the of 5a,b all variablesshowsolutions that for theτ of =both all4.00 variables velocity are taken for filed essentially𝜏= and4.00 aretemperature as taken the steady-stateessentially field asattain solutions.the ststeady-stateeady-state Figure solutions.5 a,bmonotonically. show Figure that the5a,b It both also velocityshouldshow be filed thatnoted and the temperaturethat both the velocity temperature field filed attain and profiles temperature steady-state are reachedfield monotonically. attain steady steady-state state It also position monotonically. should than be noted the It alsovelocity that the should be noted that the temperature profiles are reached steady state position than the velocity temperatureprofiles. profiles are reached steady state position than the velocity profiles. profiles.

(a) Velocity Profile (b) Temperature Profile Figure 5(.a Illustration) Velocity Profileof time variation𝜏 on (a) velocity field (b and) Temperature (b) temperature Profile field at moving Figureplate, 5. Illustration where 𝑅 =2.0, of time 𝑃 =variation 0.08, 𝐸 =𝜏 0.10 on and(a) velocity𝜏 =0.001 field. and (b) temperature field at moving Figure 5. Illustration of time variation (τ) on (a) velocity ﬁeld and (b) temperature ﬁeld at moving plate, where 𝑅 =2.0, 𝑃 = 0.08, 𝐸 = 0.10 and 𝜏 =0.001. plate,6.3. Effects where of RParameters:e = 2.0, Pr = 0.08, Ec = 0.10 and τD = 0.001. 6.3. EffectsTo of Parameters:attain the clear conception of physical properties of the present study, the impact of parameters, namely Reynolds number 𝑅 and Bingham number 𝜏 on velocity and temperature, To attain the clear conception of physical properties of the present study, the impact of shear stress and Nusselt number at Prandtl number 𝑃 =0.08 and Eckert number 𝐸 = 0.10 parameters, namely Reynolds number 𝑅 and Bingham number 𝜏 on velocity and temperature, are illustrated in Figures 6 and 7. For brevity, the impact of Prandtl number 𝑃 and Eckert number 𝑃 =0.08 𝐸 = 0.10 shear𝐸 stress are not and shown. Nusselt number at Prandtl number and Eckert number are illustratedThe influence in Figures of 6Reynolds and 7. For number brevity, 𝑅 onthe velocity impact field,of Prandtl temperature number field, 𝑃 shear and Eckert stress andnumber 𝐸 areNusselt not shown. number at moving plate are shown in Figure 6a–d. It is seen from Figure 6a,b, the velocity Theand temperatureinfluence of distributions Reynolds decreasenumber wi 𝑅th theon increasevelocity of field, Reynolds temperature number 𝑅 field,. Also itshear is seen stress from and Nusselt number at moving plate are shown in Figure 6a–d. It is seen from Figure 6a,b, the velocity

and temperature distributions decrease with the increase of Reynolds number 𝑅. Also it is seen from Inventions 2019, 4, x 8 of 11

Figure 6b that the temperature distribution has minor decreasing effect. It has been checked to observe the effects by enlargement of the marked point on the curves of Figure 6b. The enlarge sub- figure of Figure 6b shows clearly the minor decreasing effects. It is observed from Figure 6c,d that both the shear stress and the Nusselt number at moving plate reduce with the increment of Reynolds number 𝑅. The effects of Bingham number 𝜏 on velocity field, temperature field, shear stress and Nusselt number at moving plate are shown in Figure 7a–d. It is observed from Figures 7a,b that both the Inventionsvelocity 2019and, 4 ,temperature 51 distribution at moving plate enhance with the increment of Bingham8 of 11 number 𝜏. To check clearly the effects of velocity and temperature field, it has been enlarged the marked point of the curved of Figures 7a,b. The enlarged figures show clearly the enhancement of 6.3. Eﬀects of Parameters: velocity and temperature fields. ToIt is attain observed the clear from conception Figure 7c,d of physicalthat both properties the profiles of of the the present shear study, stress theand impact the Nusselt of parameters, number namelyat moving Reynolds plate enhance number withRe andthe increment Bingham of number BinghamτD onnumber velocity 𝜏. andIt has temperature, been checked shear to observe stress andthe effects Nusselt by number enlargement at Prandtl of the number marked( Ppointr = 0.08 on )thande curves Eckert of numberFigure 7c,d(Ec respectively.= 0.10) are illustrated The enlarge in Figuressub-figures6 and of7. Figure For brevity, 7c,d show the impact the minor of Prandtl decreasing number effectsPr clearly.and Eckert number Ec are not shown.

(a) (b)

FigureFigure 6.6. EEffectffect of of Reynolds Reynolds number number(R e𝑅) on on (a) ( Velocitya) Velocity profile profile (b) ( Temperatureb) Temperature profile profile (c) profile(c) profile for Shearfor Shear stress stress (d) profile (d) forprofile Nusselt for numberNusselt atnumber the moving at the plate; moving where Pplate;r = 0.08, whereEc = 𝑃0.10 =and 0.08,τ 𝐸D == 0.0010.10 and at time 𝜏 =τ 0.001= 4.00 at (Steady time 𝜏 State). = 4.00 (Steady State).

The influence of Reynolds number Re on velocity field, temperature field, shear stress and Nusselt number at moving plate are shown in Figure6a–d. It is seen from Figure6a,b, the velocity and temperature distributions decrease with the increase of Reynolds number Re. Also it is seen from Figure6b that the temperature distribution has minor decreasing e ffect. It has been checked to observe the effects by enlargement of the marked point on the curves of Figure6b. The enlarge sub-figure of Figure6b shows clearly the minor decreasing e ffects. It is observed from Figure6c,d that both the shear stress and the Nusselt number at moving plate reduce with the increment of Reynolds number Re. The effects of Bingham number τD on velocity field, temperature field, shear stress and Nusselt number at moving plate are shown in Figure7a–d. It is observed from Figure7a,b that both the velocity and temperature distribution at moving plate enhance with the increment of Bingham number τD. To check clearly the effects of velocity and temperature field, it has been enlarged the marked Inventions 2019, 4, 51 9 of 11 point of the curved of Figure7a,b. The enlarged figures show clearly the enhancement of velocity and temperature fields. It is observed from Figure7c,d that both the profiles of the shear stress and the Nusselt number at moving plate enhance with the increment of Bingham number τD. It has been checked to observe the effects by enlargement of the marked point on the curves of Figure7c,d respectively. The enlarge sub-figuresInventions 2019 of, 4, Figure x 7c,d show the minor decreasing e ffects clearly. 9 of 11

(a) (b)

FigureFigure 7.7.E Effectffect of of Bingham Bingham number number(τ D𝜏)on on (a )( velocitya) velocity profile profile (b) ( temperatureb) temperature profile profile (c) Profile(c) Profile for

shearfor shear stress stress (d) proﬁles (d) forprofiles Nusselt for number Nusselt at thenumber moving at plate; the wheremovingRe =plate;2.0, P wherer = 0.08 𝑅and =2.0,𝑃Ec = 0.10 =

at0.08 time andτ =𝐸4.00=0.10 (steady at time state). 𝜏 = 4.00 (steady state). 6.4. Comparison 6.4. Comparison Finally, qualitative and quantitative comparison of the study subject to the published results of Finally, qualitative and quantitative comparison of the study subject to the published results of Parvin et al. [8] are discussed in Figures8 and9. Parvinet al. [8] are discussed in Figures 8 and 9. Thus, in Figures8 and9, the obtained results are quantitatively not identical but quite similar qualitatively with the published results of Parvin et al. [8]. The problem has been solved numerically by explicit FDM. Since the present estimation has been veriﬁed by SDF 6.6a and MATLAB R2015a both software. Hence, our model is less ﬂawed concerning published results.

Figure 8. Illustration of time variation 𝜏 for velocity profile. Inventions 2019, 4, x 9 of 11

(a) (b)

Figure 7. Effect of Bingham number 𝜏 on (a) velocity profile (b) temperature profile (c) Profile

for shear stress (d) profiles for Nusselt number at the moving plate; where 𝑅 =2.0,𝑃 =

0.08 and 𝐸 =0.10 at time 𝜏 = 4.00 (steady state).

6.4. Comparison Finally, qualitative and quantitative comparison of the study subject to the published results of Inventions 2019, 4, 51 10 of 11 Parvinet al. [8] are discussed in Figures 8 and 9.

Inventions 2019, 4, x 10 of 11 Figure 8. IllustrationIllustration of of time time variation variation (τ𝜏) for for velocityvelocity proﬁle.profile.

FigureFigure 9. 9. IllustrationIllustration of of time time variation variation (𝜏τ) for for velocityvelocityproﬁle profile by by Parvin Parvin et et al. al. [ 8[8].].

7. ConclusionsThus, in Figures 8 and 9, the obtained results are quantitatively not identical but quite similar qualitativelyIn this study, with thethe unsteadypublished viscous results incompressibleof Parvin et al. Bingham[8]. The problem fluid flow ha throughs been solved a non-conducting numerically parallelby explicit plate FDM. has Since been investigatedthe present estimation numerically has by been explicit verified finite by di SDFfference 6.6a method.and MATLAB The physical R2015a both software. Hence, our model is less flawed concerning published results. properties are illustrated graphically for several values of Reynolds number (Re) and Bingham number (τD). For brevity, the impact of Prandtl number (Pr) and Eckert number (Ec) are not shown. Some of 7. Conclusions the important observations from the graphical illustration of the presented results are listed below. In this study, the unsteady viscous incompressible Bingham fluid flow through a non- 1.conducting The velocity parallel and plate temperature has been investigated fields rise with nume therically increase by ofexplicitRe. finite difference method. The 2. The velocity and temperature fields reduce with the increment of τ . physical properties are illustrated graphically for several values of ReynoldsD number 𝑅 and 3.Bingham The shearnumber stress 𝜏 and. For Nusselt brevity, number the impact reduce of with Prandtl the increment number of𝑃Re .and Eckert number 𝐸 4.are notThe shown. shear Some stress of and the Nusselt important number observations rise with thefrom increment the graphical of τD .illustration of the presented 5.results The are temperature listed below. reaches the steady-state quickly with the comparison of velocity distribution.

1. The velocity and temperature fields rise with the increase of 𝑅. Author Contributions: S.K. has performed the literature review. S.K., M.M.I. and𝜏 M.M.A. have developed the geometrical2. The velocity configuration and temperature and Mathematical fields model.reduce With with the the consultation increment ofof other. authors, S.K. has analyzed the3. mathematicalThe shear stress model and and Nusselt converted number the mathematical reduce with model the into increment explicit finiteof 𝑅 di. fference form with stability analysis.4. The With shear the stress help ofand other Nusselt authors, number S.K. and rise M.T.M. with have the illustratedincrement the of solution𝜏. algorithm and have written two codes by using Math Lab and SDF 6.6a tools. All authors have checked the code and the simulated data. S.K.5. hasThe drawn temperature the graph reaches and written the steady-state manuscript with quickly the consultation with the comparison of other authors. of velocity M.F. and distribution. M.M.A. have discussed the results and commented on the manuscript. M.M.A. and M.M.I. have supervised this research project. Author Contributions: S.K. has performed the literature review. S.K., M.M.I. and M.M.A. have developed the Funding: This research received no external or institutional funding. geometrical configuration and Mathematical model. With the consultation of other authors, S.K. has analyzed Conflicts of Interest: the mathematical modelThe and authors converted declare the no mathematical conflict of interest. model into explicit finite difference form with stability analysis. With the help of other authors, S.K. and M.T.M. have illustrated the solution algorithm and have written two codes by using Math Lab and SDF 6.6a tools. All authors have checked the code and the simulated data. S.K. has drawn the graph and written manuscript with the consultation of other authors. M.F. and M.M.A. have discussed the results and commented on the manuscript. M.M.A. and M.M.I. have supervised this research project.

Funding: This research received no external or institutional funding.

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Bingham, E.C. An investigation of the laws of plastic flow. Bull. Bur. Stand.1916, 13, 309–353. 2. Darby, R.; Melson, J. How to Predict the Friction Factor for Flow of Bingham Plastics. Chem. Eng. 1981, 28, 59–61. 3. Bird, R.B.; Dai, G.; Yarusso, B.J. The Rheology and Flow of Viscoplastic Materials. Rev. Chem. Eng.1983, 1, 1–70. Inventions 2019, 4, 51 11 of 11

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