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The role of relaxation and retardation phenomenon of Oldroyd-B fluid flow through Stehfest’s and Tzou’s algorithms

The role of relaxation and retardation phenomenon of Oldroyd-B fluid flow through Stehfest’s and... 1IntroductionIn recent years, non-Newtonian fluids have become more important due to their applications in the industrial and engineering fields. Non-Newtonian fluids includes paint, suspension, colloidal solutions, specific oil, exotic lubricants, clay coatings, cosmetic products, and polymer solutions. There is not a single constitutive demonstration that can foresee all the notable highlights of non-Newtonian liquids due to different physical structures of these liquids. We analyzed and worked on the rate type fluid model, called Oldroyd-B fluid. The best subclass of rate type liquid is Maxwell liquid; in any case, this liquid demonstrates as it was depicted in terms of its relaxation time, whereas there is no evidence of its retardation time. Fetecau et al. [1] demonstrated Oldroyd-B fluid flow over a plate. This idea gained attention of many researchers. Vieru et al. [2] inspected the influences of Oldroyd-B fluid due to a constantly accelerating plate. Chang et al. [3] investigated the Walters-B viscoelastic flow at wall suction. They examined the numerical results of convective heat transport of fluid flow at the wall and gained the most important results. Hayat et al. [4] discussed and analyzed the flow of Oldroyd-B fluid in a porous channel. Azeem Khan et al. [5] highlighted the Oldroyd-B nanomaterial fluid flow effects due to stretching sheets. Awan et al. [6,7,8,9] examined the flow of non-Newtonian liquids by varying shear stress in different circumstances. In recent days, similar studies have been carried out in various circumstances but few researchers have developed interest in analyzing the non-Newtonian fluid’s effects on the stretching surface due to various assumptions, see latest attempts [10,11,12,13,1415] and references therein.Boundary layer flow is the most significant application in routine life. The liquids used in technologies and industries do not follow Newton's law of viscosity, for example, greases, shampoo, food, yogurt, ketchup, and polymer melts. These fluids revealed the complicated relationship between the rate of strain and shear stress. The boundary layer flow and heat transfer examination of these liquids on a persistently moving surface have a wide range of applications in building and mechanical forms, for example, fabrication of plastic sheets, polymeric sheets, artificial fibers, plastic froth preparing, the expulsion of polymer sheet from a pass on, warm materials voyaging between a bolster roll, and so on. Soundalgekar [16] pioneered and analyzed the fluid flow at an infinite oscillating plate in the presence of impacts of free convection. Mass transfer of fluid flow at the oscillating vertical plate in the presence of the free convection impacts has been investigated by Soundalgekar and Akolkar [17]. Hocking [18] investigated the waving flow in the oscillating vertical plate. Chang and Lin [19] analyzed the reverse flow at the oscillating channel. Recently, a few investigators have analyzed the flow over oscillatory sheet, see references [20–26].From the last three decades, fractional differential conditions have picked up significance and ubiquity, primarily because of their exhibited applications in various material science and designing fields. Numerous significant phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, material science, likelihood and measurements, electrochemistry of erosion, concoction physical science, and sign preparing are all depicted by fractional differential equations [27,28,29,30]. Consequently, special consideration has been given to discover solutions of fractional differential equations. When all is said in done, it is hard to get an exact answer. The fractional studies on viscoelastic fluid [27,28,29,30] and numerical and analytical studies on viscoelastic fluid [31,32,33,34,35,36,37,38,39,40] can be overviewed.Motivated by the above discussions, the current study aims to deal with the concept for a fractional derivative on Oldroyd-B fluid over a flat vertical plate that moves to employ an oscillating velocity in its plane. We have calculated the temperature and velocity fields and shear stress with the constant wall and slip condition at the boundary using the Laplace transformation technique. We investigated the Laplace inverse through “Stehfest’s and Tzou’s algorithms.” The inspirations of different fractional as well as physical parameters are plotted by using Mathcad software.2Methods2.1Problem formulationFree convection flow of time-dependent Oldroyd-B fluid over a flat vertical and oscillating plate moving with an oscillating velocity is sketched in Figure 1(a).Figure 1(a) Flow analysis of oscillating vertical plate. (b) The velocity for the variants of dimensionless factor Gr at different times.Starting at t = 0 the fluid and plate have an ambient fluid temperature T∞. For the time t1 = 0+, the plate begins to oscillate with the velocity, w=R0H1(t1)cos(ωt1)i,w={R}_{0}{H}_{1}({t}_{1})\cos (\omega {t}_{1})i,where, R0, H1(t1), and i, are unit-step function and oscillating frequency, respectively, and i is the direction of vertical flow. The temperature is variable for the plate which can either be raised or be lowered to Tw. The governing equations of an Oldroyd-B fluid are described through the resulting differential equations [6]:(1)ρ1+λ2∂∂t1∂w∂t1=μ1+λr∂∂t1∂2w∂y12+1+λ2∂∂t1ρβg(T1−T∞),\rho \left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\frac{\partial w}{\partial {t}_{1}}=\mu \left(1+{\lambda }_{\text{r}}\frac{\partial }{\partial {t}_{1}}\right)\frac{{\partial }^{2}w}{\partial {{y}_{1}}^{2}}+\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\rho \beta g({T}_{1}-{T}_{\infty }),(2)1+λ2∂∂t1τ1=μ∂w∂y1,\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right){\tau }_{1}=\mu \frac{\partial w}{\partial {y}_{1}},(3)ρCp∂T1∂t1=k1+16σ˜T∞3kk˜∂2T1∂y12,\rho {C}_{\text{p}}\frac{\partial {T}_{1}}{\partial {t}_{1}}=k\left(1+\frac{16\tilde{\sigma }{T}_{\infty }}{3k\tilde{k}}\right)\frac{{\partial }^{2}{T}_{1}}{\partial {{y}_{1}}^{2}},where w is the velocity in the x-direction and T1 is the temperature field. The other parameters μ,ρ,β,g,k\mu ,\hspace{.25em}\rho ,\hspace{.25em}\beta ,\hspace{.25em}g,\hspace{.25em}k, and Cp denote viscosity, density, heat transmission factor, acceleration due to gravity, heat conduction, and heat capability at a particular pressure, respectively. Now, we define the Heaviside unit step function H1(t1)=12(1+sign(t1)){H}_{1}({t}_{1})=\frac{1}{2}(1+\text{sign}({t}_{1})); boundary conditions are as follows:(4)w(y1,0)=0,T1(y1,0)=T∞,w({y}_{1},0)=0,{T}_{1}({y}_{1},0)={T}_{\infty },(5)w(0,t1)−b1∂w(y1,t1)∂t1=RoH1(t1)cos(ωt1),w(0,{t}_{1})-{b}_{1}\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}}={R}_{\text{o}}{H}_{1}({t}_{1})\text{cos}(\omega {t}_{1}),(6)T1(0,t1)=Tw,∂w(y1,t1)∂t1t1=0,{T}_{1}(0,{t}_{1})={T}_{\text{w}},{\left.\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}},\right|}_{{t}_{1}=0},(7)w(∞,y1)=0,T1(∞,t1)=T∞.w(\infty ,{y}_{1})=0,{T}_{1}(\infty ,{t}_{1})={T}_{\infty }.The non-dimensional parameters are as follows:(8)w⁎=wWo,y1⁎=y1Woν,t1⁎=Wo2t1ν,b1⁎=b1Woν,ω⁎=ωWo2,τ1∗=τ1μWo2,θ1=T1−T∞Tz−T∞,λ⁎=λ2Wo2ν,Pr=μCpk,Preff=Pr1+Nr,Nr=16σ˜T∞33kk˜,Gr=βg(Tz−T∞)w03,λr⁎=λrwo2ν.\left.\begin{array}{c}{w}^{\ast }=\frac{w}{{W}_{\text{o}}},{y}_{1}^{\ast }=\frac{{y}_{1}{W}_{\text{o}}}{\nu },{t}_{1}^{\ast }=\frac{{W}_{\text{o}}^{2}{t}_{1}}{\nu },{b}_{1}^{\ast }=\frac{{b}_{1}{W}_{o}}{\nu },\\ {\omega }^{\ast }=\frac{\omega }{{W}_{\text{o}}^{2}},{\tau }_{1}^{\ast }=\frac{{\tau }_{1}}{\mu {W}_{\text{o}}^{2}},{\theta }_{1}=\frac{{T}_{1}-{T}_{\infty }}{{T}_{z}-{T}_{\infty }},{\lambda }^{\ast }=\frac{{\lambda }_{2}{W}_{\text{o}}^{2}}{\nu },\\ \Pr =\frac{\mu {C}_{\text{p}}}{k},{\Pr }_{\text{eff}}=\frac{\Pr }{1+\text{Nr}},\text{Nr}=\frac{16\tilde{\sigma }{T}_{\infty }^{3}}{3k\tilde{k}},\\ \text{Gr}=\frac{\beta g({T}_{z}-{T}_{\infty })}{{w}_{0}^{3}},\hspace{0.25em}{\lambda }_{\text{r}}^{\ast }=\frac{{\lambda }_{\text{r}}{w}_{\text{o}}^{2}}{\nu }.\end{array}\right\}In Eqs. (1)–(7), we obtain it by dropping star notation:(9)1+λ2∂∂t1∂w∂t1=1+λr∂∂t1∂2w∂y12+1+λ2∂∂t1Grθ1,\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\frac{\partial w}{\partial {t}_{1}}=\left(1+{\lambda }_{\text{r}}\frac{\partial }{\partial {t}_{1}}\right)\frac{{\partial }^{2}w}{\partial {{y}_{1}}^{2}}+\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\text{Gr}{\theta }_{1},(10)1+λ2∂∂t1τ1=∂w∂y1,\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right){\tau }_{1}=\frac{\partial w}{\partial {y}_{1}},(11)Pr∂θ1∂t1=(1+Nr)∂2θ1∂y12,\Pr \frac{\partial {\theta }_{1}}{\partial {t}_{1}}=(1+\text{Nr})\frac{{\partial }^{2}{\theta }_{1}}{\partial {y}_{1}^{2}},where b1 is the slip factor, Gr, Nr, Pr are the parameters, and the boundary conditions become(12)w(y1,0)=0,θ1(y1,0)=0,w({y}_{1},0)=0,{\theta }_{1}({y}_{1},0)=0,(13)w(0,t1)−b1∂w(0,t1)∂y1=RoH1(t1)cos(ωt),w(0,{t}_{1})-{b}_{1}\frac{\partial w(0,{t}_{1})}{\partial {y}_{1}}={R}_{\text{o}}{H}_{1}({t}_{1})\cos (\omega t),(14)θ1(0,t1)=1,∂w(y1,t1)∂t1t1=0=0,{\theta }_{1}(0,{t}_{1})=1,{\left.\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}}\right|}_{{t}_{1}=0}=0,(15)w(y1,t1)=0,θ1(y1,t1)=0,asy1→∞.w({y}_{1},{t}_{1})=0,\hspace{.25em}{\theta }_{1}({y}_{1},{t}_{1})=0,\hspace{.5em}\text{as}\hspace{.5em}{y}_{1}\to \infty .In this work, we use Caputo-Fabrizio derivative (CFD) of order α1∈(0,1).{\alpha }_{1}\in (0,1).(16)(1+λ2Dt1α1)∂w(y1,t1)∂t1=(1+λrDt1α1)∂2w(y1,t1)∂y12+Gr(1+λ2Dt1α1)θ1(y1,t1),\begin{array}{c}(1+{\lambda }_{2}{D}_{{t}_{1}}^{{\alpha }_{1}})\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}}\\ \hspace{1em}=(1+{\lambda }_{\text{r}}{D}_{{t}_{1}}^{{\alpha }_{1}})\frac{{\partial }^{2}w({y}_{1},{t}_{1})}{\partial {{y}_{1}}^{2}}+\text{Gr}(1+{\lambda }_{2}{D}_{{t}_{1}}^{{\alpha }_{1}}){\theta }_{1}({y}_{1},{t}_{1}),\end{array}(17)(1+λ2Dt1α1)τ1(y1,t1)=∂w(y1,t1)∂y1,(1+{\lambda }_{2}{D}_{{t}_{1}}^{{\alpha }_{1}}){\tau }_{1}({y}_{1},{t}_{1})=\frac{\partial w({y}_{1},{t}_{1})}{\partial {y}_{1}},(18)PreffDt1α1θ1(y1,t1)=∂2θ1(y1,t1)∂y12,{\Pr }_{\text{eff}}{D}_{{t}_{1}}^{{\alpha }_{1}}{\theta }_{1}({y}_{1},{t}_{1})=\frac{{\partial }^{2}{\theta }_{1}({y}_{1},{t}_{1})}{\partial {y}_{1}^{2}},where CFD is defined as:(19)Dt1α1w(y1,t1)=11−α1∫0t1e−α1(t1−τ1)1−α1ẇ(y1,t1)dτ1;0<α1<1,\begin{array}{c}{D}_{{t}_{1}}^{{\alpha }_{1}}w({y}_{1},{t}_{1})\\ \hspace{1em}=\frac{1}{1-{\alpha }_{1}}\underset{0}{\overset{{t}_{1}}{\int }}{\text{e}}^{\left(\frac{-{\alpha }_{1}({t}_{1}-{\tau }_{1})}{1-{\alpha }_{1}}\right)}\dot{w}({y}_{1},{t}_{1})\text{d}{\tau }_{1};\hspace{.5em}0\lt {\alpha }_{1}\lt 1,\end{array}(20)ℒ{Dt1α1w(y1,t1)}=q1ℒ{w(y1,t1)}−w(y1,0)(1−α1)q1+α1.{\mathcal L} \{{D}_{{t}_{1}}^{{\alpha }_{1}}w({y}_{1},{t}_{1})\}=\frac{{q}_{1} {\mathcal L} \{w({y}_{1},{t}_{1})\}-w({y}_{1},0)}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}.2.2Solution to the problemWe use the transformation, namely Laplace transform that is defined in the following procedure.2.2.1Temperature fieldUsing the Laplace transformation to Eqs. (18), (14)1, (15)2, and using the initial condition (12)2 with Eq. (20), we obtain(21)γ1Preffq1θ¯1(y1,q1)q1+γ1α1=∂2θ¯1(y1,q1)∂y12,whereγ1=11−α1.\begin{array}{c}\frac{{\gamma }_{1}{\Pr }_{\text{eff}}{q}_{1}{\bar{\theta }}_{1}({y}_{1},{q}_{1})}{{q}_{1}+{\gamma }_{1}{\alpha }_{1}}=\frac{{\partial }^{2}{\bar{\theta }}_{1}({y}_{1},{q}_{1})}{\partial {y}_{1}^{2}},\\ \hspace{1em}\text{where}\hspace{.5em}{\gamma }_{1}=\frac{1}{1-{\alpha }_{1}}.\end{array}(22)θ¯1(0,q1)=1q1,θ1¯(y1,q1)→0;asy1→∞.{\overline{\theta }}_{1}(0,{q}_{1})=\frac{1}{{q}_{1}},\overline{{\theta }_{1}}({y}_{1},{q}_{1})\to 0;\hspace{.5em}\text{as}\hspace{.5em}{y}_{1}\to \infty .The solution of Eq. (21) subjected to conditions Eq. (22) is(23)θ1¯(y1,q1)=1q1exp−Preffq1+α1γ1y1=θ¯1(y1,q1;Preffγ1,α1γ1).\overline{{\theta }_{1}}({y}_{1},{q}_{1})=\frac{1}{{q}_{1}}\exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1}={\overline{\theta }}_{1}({y}_{1},{q}_{1};\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1}).Eq. (23) can be written as(24)θ1¯(y1,q1;a,b)=1q1exp−aq1q1+by1,\overline{{\theta }_{1}}({y}_{1},{q}_{1};\hspace{0.25em}a,b)=\frac{1}{{q}_{1}}\exp \left(-\frac{\sqrt{a{q}_{1}}}{\sqrt{{q}_{1}+b}}\right){y}_{1},using the following formula:ψ1(y,q;a,b,c)=ℒ−1{y,q;a,b,c}=expct−aqc+b−1−2aπ∫0∞sin(yx)x(ac+(c+b)x2)exp−btx2(a+x2)dx.\begin{array}{c}{\psi }_{1}(y,q;\hspace{0.25em}a,b,c)={ {\mathcal L} }^{-1}\{y,q;\hspace{0.25em}a,b,c\}\\ \hspace{1em}=\exp \left(ct-\frac{\sqrt{aq}}{\sqrt{c+b}}\right)-1-\frac{2a}{\pi }\underset{0}{\overset{\infty }{\int }}\frac{\sin (yx)}{x(ac+(c+b){x}^{2})}\exp \left(-\frac{bt{x}^{2}}{(a+{x}^{2})}\right)\text{d}x.\end{array}We obtain the inverse Laplace transform of Eq. (24) as(25)θ1(y1,t1;Preffγ1,α1γ1)=1−2Preffγ1π∫0∞sin(y1x)x(Preffγ1+x2)exp−α1γ1tx2Preffγ1+x2dx,0<α1<1\begin{array}{c}{\theta }_{1}({y}_{1},{t}_{1};\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})=1-\frac{2{\Pr }_{\text{eff}}{\gamma }_{1}}{\pi }\underset{0}{\overset{\infty }{\int }}\frac{\sin ({y}_{1}x)}{x({\Pr }_{\text{eff}}{\gamma }_{1}+{x}^{2})}\\ \hspace{1em}\exp \left(-\frac{{\alpha }_{1}{\gamma }_{1}t{x}^{2}}{{\Pr }_{\text{eff}}{\gamma }_{1}+{x}^{2}}\right)\text{d}x,\hspace{.5em}0\lt {\alpha }_{1}\lt 1\end{array}and(26)ℒ−1{θ¯1(y,q;Preffγ1,α1γ1)}=θ1(y,t;Preffγ1,α1γ1).{ {\mathcal L} }^{-1}\{{\overline{\theta }}_{1}(y,q;\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})\}={\theta }_{1}(y,t;\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1}).For ordinary case, put α1=1{\alpha }_{1}=1.The expression for temperature equivalent to the ordinary case is found based on the property of the CFD, namely,(27)θ1(y1,t1)=limα1→0θ¯1(y1,t1;Preffγ1,α1γ1)=limγ1→∞θ¯1(y1,t1;Preffγ1,α1γ1)=1−2π∫0∞sin(yx)xexp−t1x2Preffdx,\begin{array}{c}{\theta }_{1}({y}_{1},{t}_{1})=\mathop{\mathrm{lim}}\limits_{\hspace{.25em}{\alpha }_{1}\to 0}{\overline{\theta }}_{1}({y}_{1},{t}_{1};\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})=\mathop{\mathrm{lim}}\limits_{{\gamma }_{1}\to \infty }{\overline{\theta }}_{1}({y}_{1},{t}_{1};{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})=1-\frac{2}{\pi }\underset{0}{\overset{\infty }{\int }}\frac{\sin (yx)}{x}\exp \left(-\frac{{t}_{1}{x}^{2}}{{\Pr }_{\text{eff}}}\right)\text{d}x,\end{array}using the formula(28)∫0∞sin(y1x)xexp(−ax2)dx=π2erfb2a,\underset{0}{\overset{\infty }{\int }}\frac{\sin ({y}_{1}x)}{x}\exp (-a{x}^{2})\text{d}x=\frac{\pi }{2}\text{erf}\left(\frac{b}{2\sqrt{a}}\right),we obtain(29)θ1(y1,t1)=1−erfy1Pr2t1=erfcy1Pr2t1,{\theta }_{1}({y}_{1},{t}_{1})=1-\text{erf}\left(\frac{{y}_{1}\sqrt{\Pr }}{2\sqrt{{t}_{1}}}\right)=\text{erfc}\left(\frac{{y}_{1}\sqrt{\Pr }}{2\sqrt{{t}_{1}}}\right),where erf(x) is the error and erfc(x) is the error function of complementary.2.2.2Velocity distributionApplying the Laplace transform to Eqs. (16), (13), (15)1, and using the initial condition (12)1, (14)2 using Eq. (20), we obtain the following transformed problem:(30)1+λ2q1(1−α1)q1+α1qw¯(y1,q1)=1+λrq1(1−α1)q1+α1∂2w¯(y1,q1)∂y12+Gr1+λ2q1(1−α1)q1+α1q1×exp−Preffq1+α1γ1y1,\begin{array}{l}\left(1+\frac{{\lambda }_{2}{q}_{1}}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}\right)q\bar{w}({y}_{1},{q}_{1})\\ \hspace{1em}=\left(1+\frac{{\lambda }_{\text{r}}{q}_{1}}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}\right)\frac{{\partial }^{2}\bar{w}({y}_{1},{q}_{1})}{\partial {{y}_{1}}^{2}}\\ \hspace{2em}+\frac{\text{Gr}\left(1+\frac{{\lambda }_{2}{q}_{1}}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}\right)}{{q}_{1}}\\ \hspace{2em}\times \exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1},\end{array}with the boundary conditions(31)w¯(0,q1)−b1∂w¯(0,q1)∂y1=q1q12+ω2,w¯(y1,q1)→0,asy1→∞.\begin{array}{c}\bar{w}(0,{q}_{1})-{b}_{1}\frac{\partial \bar{w}(0,{q}_{1})}{\partial {y}_{1}}=\frac{{q}_{1}}{{q}_{1}^{2}+{\omega }^{2}},\hspace{.5em}\bar{w}({y}_{1},{q}_{1})\to 0,\hspace{.5em}\text{as}\hspace{.5em}{y}_{1}\to \infty .\end{array}The solution of Eq. (30), along with the boundary conditions in Eq. (31), is(32)w¯(y1,q1)=(α1+(1−α1)q1+λrq1)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12×q1q12+ω2×exp−y1q1(α1+(1−α1)q1+λ2q12)(α1+(1−α1)q1+λrq1)+(α1+(1−α1)q1+λrq1)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12exp−y1q1(α1+(1−α1)q1+λ2q12)(α1+(1−α1)q1+λrq1)×Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)+bPreffγ1q1(q1+α1γ1)Gr(q1(1−α1)+α1+λ2q1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)−Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)exp−Preffγ1q1q1+α1γ1y1(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13).\overline{w}({y}_{1},{q}_{1})=\frac{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\times \frac{{q}_{1}}{{q}_{1}^{2}+{\omega }^{2}}\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}\right)+\left(\frac{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\right)\exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}\right)\hspace{5em}\times \left[\begin{array}{c}\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}\\ +\frac{b\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}({q}_{1}+{\alpha }_{1}{\gamma }_{1})}Gr({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}\end{array}\right]-\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})\exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1}}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}.2.3Shear stressThe shear stress is attained by using the following relation:τ¯1(y1,q1)=q1(1−α1)+α1q1(1−α1)+α1+λ2q1∂w¯(y1,q1)∂y1,{\bar{\tau }}_{1}({y}_{1},{q}_{1})=\frac{{q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}}{{q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1}}\left(\frac{\partial \overline{w}({y}_{1},{q}_{1})}{\partial {y}_{1}}\right),we obtain(33)τ¯1(y1,q1)=q1(1−α1)+α1(q1(1−α1)+α1+λ2q1)×q1q12+ω2×(α1q1+(1−α1)q12+λ2q12)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12×exp−y1q1(α1+(1−α1)q1+λ2q12)α1+(1−α1)q1+λrq1−(α1q1+(1−α1)q12+λ2q12)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12×exp−y1q1(α1+(1−α1)q1+λ2q12)α1+(1−α1)q1+λrq1+Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)+(α1q+(1−α1)q2+λ2q2)(α1+(1−α1)q+λrq)+b(1−α1)q2+qα1+λ2q2×bPreffγ1q1(q1+α1γ1)Gr(q1(1−α1)+α1+λ2q1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13){\overline{\tau }}_{1}({y}_{1},{q}_{1})=\frac{{q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}}{({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})}\times \frac{{q}_{1}}{{q}_{1}^{2}+{\omega }^{2}}\times \left(\frac{\sqrt{({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\right)\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{{q}_{1}}^{2})}}{\sqrt{{\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1}}}\right)-\left(\frac{\sqrt{({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\right)\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{{\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1}}}\right)+\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}+\left(\frac{\sqrt{({\alpha }_{1}q+(1-{\alpha }_{1}){q}^{2}+{\lambda }_{2}{q}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1})q+{\lambda }_{\text{r}}q)}+b\sqrt{(1-{\alpha }_{1}){q}^{2}+q{\alpha }_{1}+{\lambda }_{2}{q}^{2}}}\right)\times \frac{b\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{\rm{G}}{\rm{r}}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}×exp−y1q1(α1+(1−α1)q1+λ2q12)α1+(1−α1)q1+λrq1+Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)×Preffγ1q1q1+α1γ1×exp−Preffγ1q1q1+α1γ1y1.\hspace{1em}\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{{\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1}}}\right)+\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}\times \left(\frac{\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right)\times \exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1}.3Results and discussionWe analyzed the time-dependent Oldroyd-B fluid flow at the oscillatory vertical plate under the slip effects. The results of the dimensionless system are obtained through an analytical technique such as Laplace transform, its inversion, and semi-analytical solution of the shear stress, velocity, and temperature. The numerical scheme of Stehfest’s and Tzou’s algorithms is used to achieve shear stress and velocity results. The effects of different fractional as well as physical parameters are plotted by using Mathcad software. The results of the graphs have clearly described that the velocity declines with the upsurge in the Prandtl number and the Grashof number. Velocity increases with the time magnitude. The results are displayed in Figures 1(b) and 2, respectively. Figure 3 depicts the changes in fractional parameter at several values of the time. The influence of relaxation time on fluid velocity is represented in Figure 4; it is pointed out that the fluid velocity increases as we increase the time. The velocity behavior for deviation of slip parameter is shown in Figure 5 with different estimations of time. It is pointed out from Figures 6 and 7 that for different values of the fractional parameter and Prandtl number, the behavior is the same for the temperature field at different time estimations. Also, shear stress increases with the Grashof number but declines with the growth of retardation. These results are shown in Figures 8–10. Equivalence relation is described in Figure 11 as well as in Table 1, to check for the validation of the results by using the numerical inversion of Laplace transforms, namely, Stehfest’s and Tzou’s algorithms for velocity and shear stress. Table 1 shows the values of two different algorithms, namely, Stehfest’s and Tzou’s, to find the inverse Laplace for the velocity field and shear stress solution. Numerically, it is clear that the values taken from these two different algorithms are approximated to each other.Figure 2The velocity profile with different dimensionless Prandtl numbers at different times.Figure 3The velocity for the variants of a fractional factor for different times.Figure 4The velocity profile for the variants of relaxation time at different times.Figure 5The velocity profile for the variation of the slip parameter at different times.Figure 6The fluid temperature for the variants of the fractional factor.Figure 7The fluid temperature for the variation of the Prandtl number.Figure 8The shear stress for the variation of the Grashof number.Figure 9The shear stress for various estimations of retardation time.Figure 10The shear stress profile for variation of slip parameter.Figure 11The validation of obtained results for velocity and stress function, with α1=0.2{\alpha }_{1}=0.2and t1=0.5{t}_{1}=0.5.Table 1Comparison of velocity and shear stress with two different “numerical algorithms, i.e., Stehfest’s and Tzou’s” at α1=0.2{\alpha }_{1}=0.2and t1=0.5{t}_{1}=0.5y1Velocity (Stehfest’s)Velocity (Tzou’s)y1Shear stress (Stehfest’s)Shear stress (Tzou’s)00.9470.93500.6020.5990.20.8520.8530.20.5050.5050.40.7430.7430.40.4130.4130.60.6290.6290.60.3290.330.80.520.520.80.2580.2581.00.4210.4211.00.1990.1991.20.3350.3351.20.1510.1511.40.2630.2641.40.1150.1151.60.2060.2061.60.0870.0871.80.160.161.80.0670.0672.00.1260.1262.00.0520.0524ConclusionThe analytical and semi-analytical solutions for fractional Oldroyd-B fluid with wall slip conditions are found by the latest and technical approach of the fractional derivatives such as the Caputo and the Fabrizio. Numerical inversion procedures named “Stehfest’s and Tzou’s” have been used for finding the inverse Laplace transformation for the non-dimensional problem. The following points are concluded from the present work.Velocity is increased by decreasing the estimations of relaxation time λ as well as slip parameter.The magnitude of velocity is increased for significant estimations of the time.Temperature is increasing for a large estimation of the fractional factor.Shear stress and Grashof number have the same behavior of enhancing.Shear stress is decreased as the retardation time increased.Our solutions achieved by the use of inversion algorithms, i.e., Stehfest’s and Tzou’s, are equivalent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Engineering de Gruyter

The role of relaxation and retardation phenomenon of Oldroyd-B fluid flow through Stehfest’s and Tzou’s algorithms

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de Gruyter
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© 2022 Aziz Ullah Awan et al., published by De Gruyter
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2192-8029
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2192-8029
DOI
10.1515/nleng-2022-0006
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Abstract

1IntroductionIn recent years, non-Newtonian fluids have become more important due to their applications in the industrial and engineering fields. Non-Newtonian fluids includes paint, suspension, colloidal solutions, specific oil, exotic lubricants, clay coatings, cosmetic products, and polymer solutions. There is not a single constitutive demonstration that can foresee all the notable highlights of non-Newtonian liquids due to different physical structures of these liquids. We analyzed and worked on the rate type fluid model, called Oldroyd-B fluid. The best subclass of rate type liquid is Maxwell liquid; in any case, this liquid demonstrates as it was depicted in terms of its relaxation time, whereas there is no evidence of its retardation time. Fetecau et al. [1] demonstrated Oldroyd-B fluid flow over a plate. This idea gained attention of many researchers. Vieru et al. [2] inspected the influences of Oldroyd-B fluid due to a constantly accelerating plate. Chang et al. [3] investigated the Walters-B viscoelastic flow at wall suction. They examined the numerical results of convective heat transport of fluid flow at the wall and gained the most important results. Hayat et al. [4] discussed and analyzed the flow of Oldroyd-B fluid in a porous channel. Azeem Khan et al. [5] highlighted the Oldroyd-B nanomaterial fluid flow effects due to stretching sheets. Awan et al. [6,7,8,9] examined the flow of non-Newtonian liquids by varying shear stress in different circumstances. In recent days, similar studies have been carried out in various circumstances but few researchers have developed interest in analyzing the non-Newtonian fluid’s effects on the stretching surface due to various assumptions, see latest attempts [10,11,12,13,1415] and references therein.Boundary layer flow is the most significant application in routine life. The liquids used in technologies and industries do not follow Newton's law of viscosity, for example, greases, shampoo, food, yogurt, ketchup, and polymer melts. These fluids revealed the complicated relationship between the rate of strain and shear stress. The boundary layer flow and heat transfer examination of these liquids on a persistently moving surface have a wide range of applications in building and mechanical forms, for example, fabrication of plastic sheets, polymeric sheets, artificial fibers, plastic froth preparing, the expulsion of polymer sheet from a pass on, warm materials voyaging between a bolster roll, and so on. Soundalgekar [16] pioneered and analyzed the fluid flow at an infinite oscillating plate in the presence of impacts of free convection. Mass transfer of fluid flow at the oscillating vertical plate in the presence of the free convection impacts has been investigated by Soundalgekar and Akolkar [17]. Hocking [18] investigated the waving flow in the oscillating vertical plate. Chang and Lin [19] analyzed the reverse flow at the oscillating channel. Recently, a few investigators have analyzed the flow over oscillatory sheet, see references [20–26].From the last three decades, fractional differential conditions have picked up significance and ubiquity, primarily because of their exhibited applications in various material science and designing fields. Numerous significant phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, material science, likelihood and measurements, electrochemistry of erosion, concoction physical science, and sign preparing are all depicted by fractional differential equations [27,28,29,30]. Consequently, special consideration has been given to discover solutions of fractional differential equations. When all is said in done, it is hard to get an exact answer. The fractional studies on viscoelastic fluid [27,28,29,30] and numerical and analytical studies on viscoelastic fluid [31,32,33,34,35,36,37,38,39,40] can be overviewed.Motivated by the above discussions, the current study aims to deal with the concept for a fractional derivative on Oldroyd-B fluid over a flat vertical plate that moves to employ an oscillating velocity in its plane. We have calculated the temperature and velocity fields and shear stress with the constant wall and slip condition at the boundary using the Laplace transformation technique. We investigated the Laplace inverse through “Stehfest’s and Tzou’s algorithms.” The inspirations of different fractional as well as physical parameters are plotted by using Mathcad software.2Methods2.1Problem formulationFree convection flow of time-dependent Oldroyd-B fluid over a flat vertical and oscillating plate moving with an oscillating velocity is sketched in Figure 1(a).Figure 1(a) Flow analysis of oscillating vertical plate. (b) The velocity for the variants of dimensionless factor Gr at different times.Starting at t = 0 the fluid and plate have an ambient fluid temperature T∞. For the time t1 = 0+, the plate begins to oscillate with the velocity, w=R0H1(t1)cos(ωt1)i,w={R}_{0}{H}_{1}({t}_{1})\cos (\omega {t}_{1})i,where, R0, H1(t1), and i, are unit-step function and oscillating frequency, respectively, and i is the direction of vertical flow. The temperature is variable for the plate which can either be raised or be lowered to Tw. The governing equations of an Oldroyd-B fluid are described through the resulting differential equations [6]:(1)ρ1+λ2∂∂t1∂w∂t1=μ1+λr∂∂t1∂2w∂y12+1+λ2∂∂t1ρβg(T1−T∞),\rho \left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\frac{\partial w}{\partial {t}_{1}}=\mu \left(1+{\lambda }_{\text{r}}\frac{\partial }{\partial {t}_{1}}\right)\frac{{\partial }^{2}w}{\partial {{y}_{1}}^{2}}+\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\rho \beta g({T}_{1}-{T}_{\infty }),(2)1+λ2∂∂t1τ1=μ∂w∂y1,\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right){\tau }_{1}=\mu \frac{\partial w}{\partial {y}_{1}},(3)ρCp∂T1∂t1=k1+16σ˜T∞3kk˜∂2T1∂y12,\rho {C}_{\text{p}}\frac{\partial {T}_{1}}{\partial {t}_{1}}=k\left(1+\frac{16\tilde{\sigma }{T}_{\infty }}{3k\tilde{k}}\right)\frac{{\partial }^{2}{T}_{1}}{\partial {{y}_{1}}^{2}},where w is the velocity in the x-direction and T1 is the temperature field. The other parameters μ,ρ,β,g,k\mu ,\hspace{.25em}\rho ,\hspace{.25em}\beta ,\hspace{.25em}g,\hspace{.25em}k, and Cp denote viscosity, density, heat transmission factor, acceleration due to gravity, heat conduction, and heat capability at a particular pressure, respectively. Now, we define the Heaviside unit step function H1(t1)=12(1+sign(t1)){H}_{1}({t}_{1})=\frac{1}{2}(1+\text{sign}({t}_{1})); boundary conditions are as follows:(4)w(y1,0)=0,T1(y1,0)=T∞,w({y}_{1},0)=0,{T}_{1}({y}_{1},0)={T}_{\infty },(5)w(0,t1)−b1∂w(y1,t1)∂t1=RoH1(t1)cos(ωt1),w(0,{t}_{1})-{b}_{1}\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}}={R}_{\text{o}}{H}_{1}({t}_{1})\text{cos}(\omega {t}_{1}),(6)T1(0,t1)=Tw,∂w(y1,t1)∂t1t1=0,{T}_{1}(0,{t}_{1})={T}_{\text{w}},{\left.\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}},\right|}_{{t}_{1}=0},(7)w(∞,y1)=0,T1(∞,t1)=T∞.w(\infty ,{y}_{1})=0,{T}_{1}(\infty ,{t}_{1})={T}_{\infty }.The non-dimensional parameters are as follows:(8)w⁎=wWo,y1⁎=y1Woν,t1⁎=Wo2t1ν,b1⁎=b1Woν,ω⁎=ωWo2,τ1∗=τ1μWo2,θ1=T1−T∞Tz−T∞,λ⁎=λ2Wo2ν,Pr=μCpk,Preff=Pr1+Nr,Nr=16σ˜T∞33kk˜,Gr=βg(Tz−T∞)w03,λr⁎=λrwo2ν.\left.\begin{array}{c}{w}^{\ast }=\frac{w}{{W}_{\text{o}}},{y}_{1}^{\ast }=\frac{{y}_{1}{W}_{\text{o}}}{\nu },{t}_{1}^{\ast }=\frac{{W}_{\text{o}}^{2}{t}_{1}}{\nu },{b}_{1}^{\ast }=\frac{{b}_{1}{W}_{o}}{\nu },\\ {\omega }^{\ast }=\frac{\omega }{{W}_{\text{o}}^{2}},{\tau }_{1}^{\ast }=\frac{{\tau }_{1}}{\mu {W}_{\text{o}}^{2}},{\theta }_{1}=\frac{{T}_{1}-{T}_{\infty }}{{T}_{z}-{T}_{\infty }},{\lambda }^{\ast }=\frac{{\lambda }_{2}{W}_{\text{o}}^{2}}{\nu },\\ \Pr =\frac{\mu {C}_{\text{p}}}{k},{\Pr }_{\text{eff}}=\frac{\Pr }{1+\text{Nr}},\text{Nr}=\frac{16\tilde{\sigma }{T}_{\infty }^{3}}{3k\tilde{k}},\\ \text{Gr}=\frac{\beta g({T}_{z}-{T}_{\infty })}{{w}_{0}^{3}},\hspace{0.25em}{\lambda }_{\text{r}}^{\ast }=\frac{{\lambda }_{\text{r}}{w}_{\text{o}}^{2}}{\nu }.\end{array}\right\}In Eqs. (1)–(7), we obtain it by dropping star notation:(9)1+λ2∂∂t1∂w∂t1=1+λr∂∂t1∂2w∂y12+1+λ2∂∂t1Grθ1,\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\frac{\partial w}{\partial {t}_{1}}=\left(1+{\lambda }_{\text{r}}\frac{\partial }{\partial {t}_{1}}\right)\frac{{\partial }^{2}w}{\partial {{y}_{1}}^{2}}+\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right)\text{Gr}{\theta }_{1},(10)1+λ2∂∂t1τ1=∂w∂y1,\left(1+{\lambda }_{2}\frac{\partial }{\partial {t}_{1}}\right){\tau }_{1}=\frac{\partial w}{\partial {y}_{1}},(11)Pr∂θ1∂t1=(1+Nr)∂2θ1∂y12,\Pr \frac{\partial {\theta }_{1}}{\partial {t}_{1}}=(1+\text{Nr})\frac{{\partial }^{2}{\theta }_{1}}{\partial {y}_{1}^{2}},where b1 is the slip factor, Gr, Nr, Pr are the parameters, and the boundary conditions become(12)w(y1,0)=0,θ1(y1,0)=0,w({y}_{1},0)=0,{\theta }_{1}({y}_{1},0)=0,(13)w(0,t1)−b1∂w(0,t1)∂y1=RoH1(t1)cos(ωt),w(0,{t}_{1})-{b}_{1}\frac{\partial w(0,{t}_{1})}{\partial {y}_{1}}={R}_{\text{o}}{H}_{1}({t}_{1})\cos (\omega t),(14)θ1(0,t1)=1,∂w(y1,t1)∂t1t1=0=0,{\theta }_{1}(0,{t}_{1})=1,{\left.\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}}\right|}_{{t}_{1}=0}=0,(15)w(y1,t1)=0,θ1(y1,t1)=0,asy1→∞.w({y}_{1},{t}_{1})=0,\hspace{.25em}{\theta }_{1}({y}_{1},{t}_{1})=0,\hspace{.5em}\text{as}\hspace{.5em}{y}_{1}\to \infty .In this work, we use Caputo-Fabrizio derivative (CFD) of order α1∈(0,1).{\alpha }_{1}\in (0,1).(16)(1+λ2Dt1α1)∂w(y1,t1)∂t1=(1+λrDt1α1)∂2w(y1,t1)∂y12+Gr(1+λ2Dt1α1)θ1(y1,t1),\begin{array}{c}(1+{\lambda }_{2}{D}_{{t}_{1}}^{{\alpha }_{1}})\frac{\partial w({y}_{1},{t}_{1})}{\partial {t}_{1}}\\ \hspace{1em}=(1+{\lambda }_{\text{r}}{D}_{{t}_{1}}^{{\alpha }_{1}})\frac{{\partial }^{2}w({y}_{1},{t}_{1})}{\partial {{y}_{1}}^{2}}+\text{Gr}(1+{\lambda }_{2}{D}_{{t}_{1}}^{{\alpha }_{1}}){\theta }_{1}({y}_{1},{t}_{1}),\end{array}(17)(1+λ2Dt1α1)τ1(y1,t1)=∂w(y1,t1)∂y1,(1+{\lambda }_{2}{D}_{{t}_{1}}^{{\alpha }_{1}}){\tau }_{1}({y}_{1},{t}_{1})=\frac{\partial w({y}_{1},{t}_{1})}{\partial {y}_{1}},(18)PreffDt1α1θ1(y1,t1)=∂2θ1(y1,t1)∂y12,{\Pr }_{\text{eff}}{D}_{{t}_{1}}^{{\alpha }_{1}}{\theta }_{1}({y}_{1},{t}_{1})=\frac{{\partial }^{2}{\theta }_{1}({y}_{1},{t}_{1})}{\partial {y}_{1}^{2}},where CFD is defined as:(19)Dt1α1w(y1,t1)=11−α1∫0t1e−α1(t1−τ1)1−α1ẇ(y1,t1)dτ1;0<α1<1,\begin{array}{c}{D}_{{t}_{1}}^{{\alpha }_{1}}w({y}_{1},{t}_{1})\\ \hspace{1em}=\frac{1}{1-{\alpha }_{1}}\underset{0}{\overset{{t}_{1}}{\int }}{\text{e}}^{\left(\frac{-{\alpha }_{1}({t}_{1}-{\tau }_{1})}{1-{\alpha }_{1}}\right)}\dot{w}({y}_{1},{t}_{1})\text{d}{\tau }_{1};\hspace{.5em}0\lt {\alpha }_{1}\lt 1,\end{array}(20)ℒ{Dt1α1w(y1,t1)}=q1ℒ{w(y1,t1)}−w(y1,0)(1−α1)q1+α1.{\mathcal L} \{{D}_{{t}_{1}}^{{\alpha }_{1}}w({y}_{1},{t}_{1})\}=\frac{{q}_{1} {\mathcal L} \{w({y}_{1},{t}_{1})\}-w({y}_{1},0)}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}.2.2Solution to the problemWe use the transformation, namely Laplace transform that is defined in the following procedure.2.2.1Temperature fieldUsing the Laplace transformation to Eqs. (18), (14)1, (15)2, and using the initial condition (12)2 with Eq. (20), we obtain(21)γ1Preffq1θ¯1(y1,q1)q1+γ1α1=∂2θ¯1(y1,q1)∂y12,whereγ1=11−α1.\begin{array}{c}\frac{{\gamma }_{1}{\Pr }_{\text{eff}}{q}_{1}{\bar{\theta }}_{1}({y}_{1},{q}_{1})}{{q}_{1}+{\gamma }_{1}{\alpha }_{1}}=\frac{{\partial }^{2}{\bar{\theta }}_{1}({y}_{1},{q}_{1})}{\partial {y}_{1}^{2}},\\ \hspace{1em}\text{where}\hspace{.5em}{\gamma }_{1}=\frac{1}{1-{\alpha }_{1}}.\end{array}(22)θ¯1(0,q1)=1q1,θ1¯(y1,q1)→0;asy1→∞.{\overline{\theta }}_{1}(0,{q}_{1})=\frac{1}{{q}_{1}},\overline{{\theta }_{1}}({y}_{1},{q}_{1})\to 0;\hspace{.5em}\text{as}\hspace{.5em}{y}_{1}\to \infty .The solution of Eq. (21) subjected to conditions Eq. (22) is(23)θ1¯(y1,q1)=1q1exp−Preffq1+α1γ1y1=θ¯1(y1,q1;Preffγ1,α1γ1).\overline{{\theta }_{1}}({y}_{1},{q}_{1})=\frac{1}{{q}_{1}}\exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1}={\overline{\theta }}_{1}({y}_{1},{q}_{1};\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1}).Eq. (23) can be written as(24)θ1¯(y1,q1;a,b)=1q1exp−aq1q1+by1,\overline{{\theta }_{1}}({y}_{1},{q}_{1};\hspace{0.25em}a,b)=\frac{1}{{q}_{1}}\exp \left(-\frac{\sqrt{a{q}_{1}}}{\sqrt{{q}_{1}+b}}\right){y}_{1},using the following formula:ψ1(y,q;a,b,c)=ℒ−1{y,q;a,b,c}=expct−aqc+b−1−2aπ∫0∞sin(yx)x(ac+(c+b)x2)exp−btx2(a+x2)dx.\begin{array}{c}{\psi }_{1}(y,q;\hspace{0.25em}a,b,c)={ {\mathcal L} }^{-1}\{y,q;\hspace{0.25em}a,b,c\}\\ \hspace{1em}=\exp \left(ct-\frac{\sqrt{aq}}{\sqrt{c+b}}\right)-1-\frac{2a}{\pi }\underset{0}{\overset{\infty }{\int }}\frac{\sin (yx)}{x(ac+(c+b){x}^{2})}\exp \left(-\frac{bt{x}^{2}}{(a+{x}^{2})}\right)\text{d}x.\end{array}We obtain the inverse Laplace transform of Eq. (24) as(25)θ1(y1,t1;Preffγ1,α1γ1)=1−2Preffγ1π∫0∞sin(y1x)x(Preffγ1+x2)exp−α1γ1tx2Preffγ1+x2dx,0<α1<1\begin{array}{c}{\theta }_{1}({y}_{1},{t}_{1};\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})=1-\frac{2{\Pr }_{\text{eff}}{\gamma }_{1}}{\pi }\underset{0}{\overset{\infty }{\int }}\frac{\sin ({y}_{1}x)}{x({\Pr }_{\text{eff}}{\gamma }_{1}+{x}^{2})}\\ \hspace{1em}\exp \left(-\frac{{\alpha }_{1}{\gamma }_{1}t{x}^{2}}{{\Pr }_{\text{eff}}{\gamma }_{1}+{x}^{2}}\right)\text{d}x,\hspace{.5em}0\lt {\alpha }_{1}\lt 1\end{array}and(26)ℒ−1{θ¯1(y,q;Preffγ1,α1γ1)}=θ1(y,t;Preffγ1,α1γ1).{ {\mathcal L} }^{-1}\{{\overline{\theta }}_{1}(y,q;\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})\}={\theta }_{1}(y,t;\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1}).For ordinary case, put α1=1{\alpha }_{1}=1.The expression for temperature equivalent to the ordinary case is found based on the property of the CFD, namely,(27)θ1(y1,t1)=limα1→0θ¯1(y1,t1;Preffγ1,α1γ1)=limγ1→∞θ¯1(y1,t1;Preffγ1,α1γ1)=1−2π∫0∞sin(yx)xexp−t1x2Preffdx,\begin{array}{c}{\theta }_{1}({y}_{1},{t}_{1})=\mathop{\mathrm{lim}}\limits_{\hspace{.25em}{\alpha }_{1}\to 0}{\overline{\theta }}_{1}({y}_{1},{t}_{1};\hspace{0.25em}{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})=\mathop{\mathrm{lim}}\limits_{{\gamma }_{1}\to \infty }{\overline{\theta }}_{1}({y}_{1},{t}_{1};{\Pr }_{\text{eff}}{\gamma }_{1},{\alpha }_{1}{\gamma }_{1})=1-\frac{2}{\pi }\underset{0}{\overset{\infty }{\int }}\frac{\sin (yx)}{x}\exp \left(-\frac{{t}_{1}{x}^{2}}{{\Pr }_{\text{eff}}}\right)\text{d}x,\end{array}using the formula(28)∫0∞sin(y1x)xexp(−ax2)dx=π2erfb2a,\underset{0}{\overset{\infty }{\int }}\frac{\sin ({y}_{1}x)}{x}\exp (-a{x}^{2})\text{d}x=\frac{\pi }{2}\text{erf}\left(\frac{b}{2\sqrt{a}}\right),we obtain(29)θ1(y1,t1)=1−erfy1Pr2t1=erfcy1Pr2t1,{\theta }_{1}({y}_{1},{t}_{1})=1-\text{erf}\left(\frac{{y}_{1}\sqrt{\Pr }}{2\sqrt{{t}_{1}}}\right)=\text{erfc}\left(\frac{{y}_{1}\sqrt{\Pr }}{2\sqrt{{t}_{1}}}\right),where erf(x) is the error and erfc(x) is the error function of complementary.2.2.2Velocity distributionApplying the Laplace transform to Eqs. (16), (13), (15)1, and using the initial condition (12)1, (14)2 using Eq. (20), we obtain the following transformed problem:(30)1+λ2q1(1−α1)q1+α1qw¯(y1,q1)=1+λrq1(1−α1)q1+α1∂2w¯(y1,q1)∂y12+Gr1+λ2q1(1−α1)q1+α1q1×exp−Preffq1+α1γ1y1,\begin{array}{l}\left(1+\frac{{\lambda }_{2}{q}_{1}}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}\right)q\bar{w}({y}_{1},{q}_{1})\\ \hspace{1em}=\left(1+\frac{{\lambda }_{\text{r}}{q}_{1}}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}\right)\frac{{\partial }^{2}\bar{w}({y}_{1},{q}_{1})}{\partial {{y}_{1}}^{2}}\\ \hspace{2em}+\frac{\text{Gr}\left(1+\frac{{\lambda }_{2}{q}_{1}}{(1-{\alpha }_{1}){q}_{1}+{\alpha }_{1}}\right)}{{q}_{1}}\\ \hspace{2em}\times \exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1},\end{array}with the boundary conditions(31)w¯(0,q1)−b1∂w¯(0,q1)∂y1=q1q12+ω2,w¯(y1,q1)→0,asy1→∞.\begin{array}{c}\bar{w}(0,{q}_{1})-{b}_{1}\frac{\partial \bar{w}(0,{q}_{1})}{\partial {y}_{1}}=\frac{{q}_{1}}{{q}_{1}^{2}+{\omega }^{2}},\hspace{.5em}\bar{w}({y}_{1},{q}_{1})\to 0,\hspace{.5em}\text{as}\hspace{.5em}{y}_{1}\to \infty .\end{array}The solution of Eq. (30), along with the boundary conditions in Eq. (31), is(32)w¯(y1,q1)=(α1+(1−α1)q1+λrq1)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12×q1q12+ω2×exp−y1q1(α1+(1−α1)q1+λ2q12)(α1+(1−α1)q1+λrq1)+(α1+(1−α1)q1+λrq1)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12exp−y1q1(α1+(1−α1)q1+λ2q12)(α1+(1−α1)q1+λrq1)×Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)+bPreffγ1q1(q1+α1γ1)Gr(q1(1−α1)+α1+λ2q1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)−Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)exp−Preffγ1q1q1+α1γ1y1(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13).\overline{w}({y}_{1},{q}_{1})=\frac{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\times \frac{{q}_{1}}{{q}_{1}^{2}+{\omega }^{2}}\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}\right)+\left(\frac{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\right)\exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}}\right)\hspace{5em}\times \left[\begin{array}{c}\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}\\ +\frac{b\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}({q}_{1}+{\alpha }_{1}{\gamma }_{1})}Gr({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}\end{array}\right]-\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})\exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1}}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}.2.3Shear stressThe shear stress is attained by using the following relation:τ¯1(y1,q1)=q1(1−α1)+α1q1(1−α1)+α1+λ2q1∂w¯(y1,q1)∂y1,{\bar{\tau }}_{1}({y}_{1},{q}_{1})=\frac{{q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}}{{q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1}}\left(\frac{\partial \overline{w}({y}_{1},{q}_{1})}{\partial {y}_{1}}\right),we obtain(33)τ¯1(y1,q1)=q1(1−α1)+α1(q1(1−α1)+α1+λ2q1)×q1q12+ω2×(α1q1+(1−α1)q12+λ2q12)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12×exp−y1q1(α1+(1−α1)q1+λ2q12)α1+(1−α1)q1+λrq1−(α1q1+(1−α1)q12+λ2q12)(α1+(1−α1)q1+λrq1)+b(1−α1)q12+q1α1+λ2q12×exp−y1q1(α1+(1−α1)q1+λ2q12)α1+(1−α1)q1+λrq1+Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)+(α1q+(1−α1)q2+λ2q2)(α1+(1−α1)q+λrq)+b(1−α1)q2+qα1+λ2q2×bPreffγ1q1(q1+α1γ1)Gr(q1(1−α1)+α1+λ2q1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13){\overline{\tau }}_{1}({y}_{1},{q}_{1})=\frac{{q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}}{({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})}\times \frac{{q}_{1}}{{q}_{1}^{2}+{\omega }^{2}}\times \left(\frac{\sqrt{({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\right)\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{{q}_{1}}^{2})}}{\sqrt{{\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1}}}\right)-\left(\frac{\sqrt{({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1})}+b\sqrt{(1-{\alpha }_{1}){q}_{1}^{2}+{q}_{1}{\alpha }_{1}+{\lambda }_{2}{q}_{1}^{2}}}\right)\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{{\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1}}}\right)+\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}+\left(\frac{\sqrt{({\alpha }_{1}q+(1-{\alpha }_{1}){q}^{2}+{\lambda }_{2}{q}^{2})}}{\sqrt{({\alpha }_{1}+(1-{\alpha }_{1})q+{\lambda }_{\text{r}}q)}+b\sqrt{(1-{\alpha }_{1}){q}^{2}+q{\alpha }_{1}+{\lambda }_{2}{q}^{2}}}\right)\times \frac{b\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{\rm{G}}{\rm{r}}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}×exp−y1q1(α1+(1−α1)q1+λ2q12)α1+(1−α1)q1+λrq1+Gr(q1(1−α1)+α1+λ2q1)(q1+α1γ1)(Preffγ1q12)(α1q1+(1−α1)q12+λrq12)−(q12+α1q1γ1)(q13(1−α1)+α1q12+λ2q13)×Preffγ1q1q1+α1γ1×exp−Preffγ1q1q1+α1γ1y1.\hspace{1em}\times \exp \left(-{y}_{1}\frac{\sqrt{{q}_{1}({\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{2}{q}_{1}^{2})}}{\sqrt{{\alpha }_{1}+(1-{\alpha }_{1}){q}_{1}+{\lambda }_{\text{r}}{q}_{1}}}\right)+\frac{\text{Gr}({q}_{1}(1-{\alpha }_{1})+{\alpha }_{1}+{\lambda }_{2}{q}_{1})({q}_{1}+{\alpha }_{1}{\gamma }_{1})}{({\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}^{2})({\alpha }_{1}{q}_{1}+(1-{\alpha }_{1}){q}_{1}^{2}+{\lambda }_{\text{r}}{q}_{1}^{2})-({q}_{1}^{2}+{\alpha }_{1}{q}_{1}{\gamma }_{1})({q}_{1}^{3}(1-{\alpha }_{1})+{\alpha }_{1}{q}_{1}^{2}+{\lambda }_{2}{q}_{1}^{3})}\times \left(\frac{\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right)\times \exp \left(-\frac{\sqrt{{\Pr }_{\text{eff}}{\gamma }_{1}{q}_{1}}}{\sqrt{{q}_{1}+{\alpha }_{1}{\gamma }_{1}}}\right){y}_{1}.3Results and discussionWe analyzed the time-dependent Oldroyd-B fluid flow at the oscillatory vertical plate under the slip effects. The results of the dimensionless system are obtained through an analytical technique such as Laplace transform, its inversion, and semi-analytical solution of the shear stress, velocity, and temperature. The numerical scheme of Stehfest’s and Tzou’s algorithms is used to achieve shear stress and velocity results. The effects of different fractional as well as physical parameters are plotted by using Mathcad software. The results of the graphs have clearly described that the velocity declines with the upsurge in the Prandtl number and the Grashof number. Velocity increases with the time magnitude. The results are displayed in Figures 1(b) and 2, respectively. Figure 3 depicts the changes in fractional parameter at several values of the time. The influence of relaxation time on fluid velocity is represented in Figure 4; it is pointed out that the fluid velocity increases as we increase the time. The velocity behavior for deviation of slip parameter is shown in Figure 5 with different estimations of time. It is pointed out from Figures 6 and 7 that for different values of the fractional parameter and Prandtl number, the behavior is the same for the temperature field at different time estimations. Also, shear stress increases with the Grashof number but declines with the growth of retardation. These results are shown in Figures 8–10. Equivalence relation is described in Figure 11 as well as in Table 1, to check for the validation of the results by using the numerical inversion of Laplace transforms, namely, Stehfest’s and Tzou’s algorithms for velocity and shear stress. Table 1 shows the values of two different algorithms, namely, Stehfest’s and Tzou’s, to find the inverse Laplace for the velocity field and shear stress solution. Numerically, it is clear that the values taken from these two different algorithms are approximated to each other.Figure 2The velocity profile with different dimensionless Prandtl numbers at different times.Figure 3The velocity for the variants of a fractional factor for different times.Figure 4The velocity profile for the variants of relaxation time at different times.Figure 5The velocity profile for the variation of the slip parameter at different times.Figure 6The fluid temperature for the variants of the fractional factor.Figure 7The fluid temperature for the variation of the Prandtl number.Figure 8The shear stress for the variation of the Grashof number.Figure 9The shear stress for various estimations of retardation time.Figure 10The shear stress profile for variation of slip parameter.Figure 11The validation of obtained results for velocity and stress function, with α1=0.2{\alpha }_{1}=0.2and t1=0.5{t}_{1}=0.5.Table 1Comparison of velocity and shear stress with two different “numerical algorithms, i.e., Stehfest’s and Tzou’s” at α1=0.2{\alpha }_{1}=0.2and t1=0.5{t}_{1}=0.5y1Velocity (Stehfest’s)Velocity (Tzou’s)y1Shear stress (Stehfest’s)Shear stress (Tzou’s)00.9470.93500.6020.5990.20.8520.8530.20.5050.5050.40.7430.7430.40.4130.4130.60.6290.6290.60.3290.330.80.520.520.80.2580.2581.00.4210.4211.00.1990.1991.20.3350.3351.20.1510.1511.40.2630.2641.40.1150.1151.60.2060.2061.60.0870.0871.80.160.161.80.0670.0672.00.1260.1262.00.0520.0524ConclusionThe analytical and semi-analytical solutions for fractional Oldroyd-B fluid with wall slip conditions are found by the latest and technical approach of the fractional derivatives such as the Caputo and the Fabrizio. Numerical inversion procedures named “Stehfest’s and Tzou’s” have been used for finding the inverse Laplace transformation for the non-dimensional problem. The following points are concluded from the present work.Velocity is increased by decreasing the estimations of relaxation time λ as well as slip parameter.The magnitude of velocity is increased for significant estimations of the time.Temperature is increasing for a large estimation of the fractional factor.Shear stress and Grashof number have the same behavior of enhancing.Shear stress is decreased as the retardation time increased.Our solutions achieved by the use of inversion algorithms, i.e., Stehfest’s and Tzou’s, are equivalent.

Journal

Nonlinear Engineering – de Gruyter

Published: Jan 1, 2022

Keywords: viscoelastic fluid; slip condition; fractional model; Laplace transform; non-singular kernel; semi-analytical solution

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