Stagnation Point Flow Investigation of Micropolar Viscoelastic Fluid with Modified Fourier and Ficks Law Scientific Reports

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Here we consider a constant flow, 2D incompressible, stagnation point of a second degree micropolar fluid on a stretch surface with modified heat and mass flow. To examine aspects of mass heat transport, the effects of thermal radiation and activation energy are considered. Furthermore, thermal and stratification boundary conditions are implemented on the sheet surface. The physical model is displayed in Fig.1. The sheet is stretching with a speed of \(u=cx\) (here, c is the positive constant). The magnetic effect of force \({B}_{0}\) is used along the y direction. The temperature on the sheet is maintained \({T}_{w}\) and away from the sheet is \({T}_{\infty }\).

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Figure 1
Figure 1

The 2nd degree fluid model, the Cauchy stress tensor is defined as51,

$$T=-PI+\mu {A}_{1}+{\alpha }_{1}{A}_{2}+{\alpha }_{2}{A}_{1}^{2 },$$

(1)

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Where \(I\) is the identity tensor, \(\mu\) is the dynamic viscosity, \(P\) is the pressure, \({\alpha }_{i}(i=\text{1,2})\) the material constants, \({A}_{1}\) AND \({A}_{2}\) are two RivlinEricksen tensors

$${A}_{1}=\left(degree\, v\right)+{\left(degree\, v\right)}^{T},$$

(2)

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$${A}_{2}=\frac{d{A}_{1}}{dt}+{A}_{1}\left(grad\, v\right)+{\left(grad\ , v\right)}^{T}{A}_{1}.$$

(3)

Here, \(V\) is the speed and \(\frac{d}{dt}\) is the material time derivative. Keep in mind that when the fluid is in equilibrium and at rest locally, the Clausius-Duhem inequality is satisfied and the Helmboltz free energy is minimal.

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$$\mu \ge 0, {\alpha }_{1}\ge 0, {\alpha }_{1}+{\alpha }_{2}=0.$$

(4)

It should be noted that the constitutive equation for a second degree fluid simplifies to an equation for a viscous fluid when \({\alpha }_{1}={\alpha }_{2}=0\).

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The equations of momentum, continuity and heat with some effects are as follows52.53.

$$\frac{\u partial}{\x partial}+\frac{\v partial}{\y partial}=0,$$

(5)

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$$u\frac{\u partial}{\x partial}+v\frac{\u partial}{\y partial}={U}_{\infty }\frac{d{U}_{\infty } }{dx}+\left(\frac{\mu +k}{\rho }\right)\frac{{\partial }^{2}u}{\partial {z}^{2}}+\frac {k}{\rho }\frac{\partial N}{\partial z}+\frac{1}{\rho }\sigma {B}_{0}^{2}\left({U}_{ \infty }-u\right)-\frac{\nu }{{k}_{1}}u+\frac{{\alpha }_{1}}{{c}_{p}\rho }\left[\frac{\partial }{\partial x}\left(u\frac{{\partial }^{2}u}{\partial {y}^{2}}\right)+\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial {v}^{2}}+v\frac{{\partial }^{3}u}{\partial {y}^{3}}\right],$$

(6)

$$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}=\frac{\gamma }{j\rho }\frac{{\partial }^{ 2}N}{\partial {y}^{2}}-\frac{k}{j\rho }\frac{\partial v}{\partial z}-\frac{2k}{j\rho }N ,$$

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(7)

$$u\frac{\T partial}{\x partial}+v\frac{\T partial}{\y partial}+{\lambda }_{E}\left[\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\frac{\partial T}{\partial x}+\left(v\frac{\partial v}{\partial y}+u\frac{\partial v}{\partial y}\right)\frac{\partial T}{\partial y}+2uv\frac{{\partial }^{2}T}{\partial x\partial y}+{u}^{2}\frac{{\partial }^{2}T}{\partial {x}^{2}}+{v}^{2}\frac{\partial v}{\partial x}\right]=\alpha \frac{{\partial }^{2}T}{\partial {y}^{2}}+\frac{{\alpha }_{1}}{\rho {c}_{p} }\Left[\frac{\partial u}{\partial y}\left[\frac{\partial }{\partial y}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\right]\right]+\left(\frac{\mu +k}{\rho {c}_{p}}\right){\left(\frac{\u partial}{\y partial}\right)}^ {2}-\frac{1}{\rho {c}_{p}}\frac{\partial {q}_{r}}{\partial y},$$

(8)

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$$u\frac{\partial do}{\partial x}+v\frac{\partial do}{\partial y}+{\lambda }_{c}\left[\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\frac{\partial C}{\partial x}+\left(u\frac{\partial v}{\partial y}+v\frac{\partial v}{\partial y}\right)\frac{\partial C}{\partial y}+2uv\frac{{\partial }^{2}C}{\partial x\partial y}+{u}^{2}\frac{{\partial }^{2}C}{\partial {x}^{2}}+{v}^{2}\frac{{\partial }^{2}C}{\partial {y}^{2}}\right]={D}_{B}\frac{{\partial }^{2}C}{\partial {y}^{2}}-{{K}_{r}}^{2}{\left( \frac{T}{{T}_{\infty }}\right)}^{m}\text{exp}\left(\left(\frac{-{E}_{a}}{\kappa T }\right)\right)\left(C-{C}_{\infty }\right)-\frac{\partial }{\partial y}\left({V}_{T}C\right). $$

(9)

In any case, the following boundary criteria apply:

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$$u={U}_{w}=cx, v=0, T={T}_{0}+{\in }_{1}x, C={C}_{w}={C }_{0}+{\in }_{2}x, N=-n\frac{\partial u}{\partial y} \text{ at }y\to 0,$$

$$u\to {U}_{\infty } , T\to {T}_{\infty }={T}_{0}+{\in }_{3}x, C\to {C} _{\infty }={C}_{0}+{\in }_{4}x, N\to 0 \, {\text{at}} \, y\to \infty .$$

(10)

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The similarity transformations are as follows,

$$u=cx{f}^{^{\prime}}, v=-\sqrt{c\nu }f , N=cxg, \theta =\frac{T-{T}_{\infty }} {{T}_{w}-{T}_{\infty }}, \phi =\frac{C-{C}_{\infty }}{{C}_{w}-{C}_{ \infty }}, \eta =\sqrt{\frac{c}{\nu }}y.$$

(11)

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Here, \(\left(u,v\right)\)are the velocity components corresponding to \(\left(x,y\right). \text{The symbols }{U}_{e}\), \(\mu\), \(K\), \(\rho\), \(N\), \({c}_{p}\), \({\alpha }_{1}\), \(\sigma\), \({B}_{0}\), \(G\), \({\lambda}_{1}\), \(T\), \({T}_{\infty }\), \({\lambda}_{2}\), \({\lambda}_{3}\), \(C\), \({C}_{\infty }\), \({\lambda}_{4}\), \(\nu\), \({k}_{1}\), \(\range\), \(J\), \({\lambda}_{E}\), \(\alpha\), \({Q}_{0}\), \({q}_{r}\), \({\lambda}_{c}\), \({D}_{B}\), \({k}_{r}\), \(M\), \({It’s at}\), \({U}_{m}\), \({T}_{0}\), \({\epsilon}_{1}\), \({\epsilon}_{2}\), \({\epsilon}_{3}\), \({\epsilon}_{4}\), \({C}_{w}\), \({C}_{0}\) en the free flow velocity, dynamic viscosity, vortex viscosity, fluid density, heat capacity, second degree fluid coefficient, electrical conductivity, magnetic field strength, gravity force are represented , linear expansion coefficient, temperature, free flow temperature, linear mass expansion, linear mass expansion coefficient, fluid concentration, free flow concentration, nonlinear mass expansion coefficient, kinematic viscosity, permeability of porous medium, spin gradient, micro-inertia density, heat flux relaxation, thermal conductivity, heat sink /source4 coefficient, thermal radiation coefficient, mass flux relaxation, Brownian motion coefficient, velocity coefficient of reaction, adjusted rate constant, activation energy coefficient, wall velocity along x-axis, concentration, reference temperature, positive constants, wall fluid concentration, reference concentration and rotation parameter respectively.

The flow expressions in the non-dimensional form become,

$$\left(1+K\right){f}^{\prime\prime\prime}+K{g}^{{{\prime}}}-{{f}^{{{\prime}} }}^{2}+f{f}^{\prime\prime}-{M}^{2}\left(\in -{f}^{{{\prime}}}\right)+{\ in }^{2}+\beta \left[2{f}^{{{\prime}}}{f}^{\prime\prime\prime}-{{f}^{\prime\prime}}^{2}-f{f}^{\left(iv\right)}\right]-\varepsilon {f}^{{{\prime}}}=0,$$

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(12)

$$\left(1+\frac{K}{2}\right){g}^{\prime\prime}+f{g}^{\prime\prime}-{f}^{{{\prime }}}gK\left(2g+{f}^{\prime\prime}\right)=0,$$

(13)

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$${P}{r}f{\theta }^{{{\prime}}}-{P}{r}\left(\theta +S\right){f}^{{{\prime}} }-{{P}{r}\delta }_{e}\left[f{f}^{{{\prime}}}{\theta }^{{{\prime}}}+{\theta }^{\prime\prime }{f}^{2}+\left(\theta +S\right)\left({{f}^{{{\prime}}}}^{2}-f{f}^{\prime\prime}\right)\right]+{P}{r}\beta {f}^{\prime\prime}\left[{f}^{{{\prime}}}{f}^{\prime\prime}-f{f}^{\prime\prime}\right]+\left(1+K\right){P}{r}{{f}^{{{\prime}}}}^{2}+{\left[1+R\left({\left(1+\left({\theta }_{w}-1\right)\theta \right)}^{3}\right){\theta }^{{{\prime}}}\right]}^{{{\prime}}}=0,$$

(14)

$${\phi }^{\prime\prime}-{Sc}\left(\phi +{S}^{*}\right){f}^{\prime}+{Sc}{f}{\ phi }^{{{\prime}}}-{Sc}{\delta }_{c}\left[f{f}^{{{\prime}}}{\phi }^{{{\prime}}}+{\phi }^{\prime\prime }{f}^{2}+\left(\phi +{S}^{*}\right)\left({{f}^{{{\prime}}}}^{2}-f{f}^{\prime\prime}\right)\right]+{Sc}{\epsilon }_{5}{\left(1+{\epsilon }_{6}\right)}^{m}{e}^{-\frac{E}{\left(1 +{\epsilon }_{6}\theta \right)}}\phi -{Sc}\tau \left({\theta }^{{{\prime}}}{\phi }^{{{\prime }}}+{\theta }^{\prime\prime}\phi \right)=0.$$

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(15)

The comparable boundary conditions are as follows:

$$f=0, {f}^{^{\prime}}=0, \theta =1-S , \phi =1-{S}^{*}, g=-n{f}^{\ prime\prime}\left(0\right)\text{ as }\age \to 0,$$

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$${f}^{{{\prime}}}\to 1 , \theta \to 0, \phi \to 0, g\to 0\text{ at }\eta \to \infty ,$$

(16)

Where, \(K=\frac{k}{\mu }\) (parameter of micropolar fluid), \(M=\frac{{\sigma }_{0}{B}_{0}^{2}}{\rho c}\) (Magnetic effect parameter), \(\varepsilon =\frac{a}{c}\) (Speed ​​ratio parameter), \(\beta =\frac{{\alpha }_{1}c}{\nu }\) (Second degree fluid parameter), \(\in =\frac{\nu }{{k}_{0}c}\) (medium porous parameter), \({L}_{e}=\frac{k}{\rho {c}_{p}{D}_{B}}\) (Lewis number), \({S}_{c}=\frac{\nu }{{D}_{B}}\) (similar number), \({\delta }_{c}={\lambda }_{c}c\) (Parameter of concentration relaxation time), \({\in }_{5}=\frac{{k}_{r}^{2}}{c}\) (Chemical reaction parameter), \({\in }_{6}=\frac{{T}_{w}-{T}_{\infty }}{{T}_{\infty }}\) (Temperature difference parameter), \(E=\frac{{E}_{a}}{\kappa {T}_{\infty }}\) (Activation energy parameter), \(S=\frac{{\in }_{3}}{{\in }_{1}}\) (thermal stratification parameter), \({S}^{*}=\frac{{\in }_{4}}{{\in }_{2}}\) (concentration stratification parameter), \({P}{r}=\frac{\rho {c}_{p}\nu }{k}\) (Prandtl number), \(R=\frac{16{\sigma }^{*}{T}_{\infty }^{3}}{3k{k}^{*}}\) (Radiation parameter), \({\theta }_{w}=\frac{{T}_{w}}{{T}_{\infty }}\) (Temperature ratio parameter), \({\delta }_{e}={\lambda }_{E}c\) (thermal relaxation time parameter).

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Skin friction

From an engineering point of view, skin friction is a very important physical quantity, which is stated by,

$${C}_{fx}= \frac{{\tau }_{w}}{\frac{1}{2}\rho {{u}_{w}}^{2}},$$

(17)

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$${\tau }_{w}={\left[\frac{{\alpha }_{1}}{\rho }\left(u\frac{{\partial }^{2}u}{\partial {y}^{2}}-2\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}+u\frac{{\partial }^{2}u}{\partial x\partial y}\right)+ \left(\frac{\mu +k}{\rho }\right)\frac{\partial u}{\partial y}+k{N}_{1}\right]}_{|y=0}.$$

(18)

In dimensionless form,

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$${{C}_{fx}}_{|\age =0}=\left[\beta \left(3{f}^{{{\prime}}}\left(0\right){f}^{\prime\prime}\left(0\right)-{ff}^{\prime\prime{^{\prime}}}\left(0\right)\right)+(1+K)f{^{\prime}}{^{\prime}}(0)\right]{\left(\frac{{R}_{ex}}{2}\right)}^{-\frac{1}{2}},$$

(19)

Where \({R}_{ex}=\frac{{xu}_{w}}{\nu }\) is the Reynolds number.

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