Manjeet Kumari* | Shalini Jain
© 2019 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
The present article highlights the impact of variable fluid property for MHD viscous fluid flow containing gyrotactic microorganisms over a permeable stretching sheet. It also considers the variable Prandtl number, variable Schmidt number of mass, variable Schmidt number of gyrotactic microorganisms. It also examined the first and second order velocity slip, temperature jump, concentration slip, microorganism slip, porosity medium, non-liner radiation and non-linear chemical reaction. By using suitable transformation, the governing PDEs corresponding to the momentum, energy, mass and microorganism equations are converted into non-linear coupled ODEs and solved numerically by using R-K 4th order with shooting technique. The effects of physical parameters on the velocity, heat, mass and microorganism are analyzed with the help of graphs and tables.
first and second order velocity slip, temperature jump, concentration slip, microorganism slip, porosity medium
The term bio-convection refers to a macroscopic convection motion of fluid which is caused by the density gradient created by collective swimming of motile microorganisms. These self-propelled motile microorganisms increase the density of the base fluid by swimming in a particular direction, thus causing bio-convection. The concept of nano-fluid bio-convection was introduced by several researchers [1-21]. One of the nice researches on fluid flow over stretching sheet is carried out by Khan and Pop [22]. Nadeem et al. [23] have investigated the radiative slip flow on MHD nanofluid over a stretching sheet. All these researchers considered the flow by taking nanofluids and concluded that motion due to self-propelled microorganisms result in enhancement in mixing and thus preventing nanoparticle cluster.
Heat transfer is a key process throughout a number of residential, industrial, research, engineering and commercial facilities. At all these locations, heat must efficiently and effectively be added, removed or transferred from one process to another to keep the status quo. Also, heat transfer takes place due to many causes. That may be viscous dissipation (frictional heat), non-uniform heat source/sink (internal heat generation/absorption), etc. Further, this model was extensively used by many researchers [24-33] to construct the energy equation and they also discussed the flow and heat transfer behavior of various kinds of Newtonian and non-Newtonian fluids.
The major objective of the present work is to analyse the impact of variable fluid property for MHD viscous fluid flow containing gyrotactic microorganisms over a permeable stretching sheet through porosity medium, non-liner radiation and non-linear chemical reaction. Sheikholeslami et al. [34-35] has studied nanofluid transportation through a permeable media with magnetic force and radiation. Sajjadi et al. [36-37] have proposed the effects of 3-D MHD natural convection on fluid flow and heat transfer. Application of full-spectrum k-distribution method to combined non-gray radiation and forced convection flow in a duct with an expansion was studied by Atashhafrooz et al. [38]. Megahed [47] investigated second order slip velocity and thermal slip effects on MHD viscous Casson fluid flow and heat tranfer. Alama [48] studied the effects of variable fluid properties on unsteady convective boundary flow. Micropolor fluid flow along a non-linear stretching sheet has been studied by Rahmanet al. [49]. Rahman and Eltayeb [50] investigated convective slip flow of rarefied fluids over a wedge with thermal jump and variable transport properties.
Figure 1. Physical model of the problem
We have considered the variable Prandtl number, variable Schmidt number of mass, variable Schmidt number of gyrotactic microorganisms in temperature, mass and microorganisms profiles and first and second order velocity slip, temperature jump, concentration slip, microorganism slip in boundary conditions. R-K 4th method is applied to solve the transformed boundary layer equations of the flow, heat, mass and microorganisms profiles. Further, the influence of pertinent parameters such as magnetic field parameter, thermal Grashof number, mass Grashof number, variable Prandtl number, variable Schmidt number of mass, variable Schmidt number of gyrotactic microorganisms, velocity slip, temperature jump, concentration slip, microorganism slip, temperature difference parameter, on velocity, temperature and concentration fields along with friction factor and Nusselt number are examined and shown in graphs and tables. Finally, a comparison of the current work with the earlier results Nadeem and Hussain [39], Gorla and Sidawi [40], Goyal and Bhargava [41], Andersson et al. [42], Prasad et al. [43], Mukhopadhyay et al. [44], Palani et al. [45] and Gorla et al. [46] is also made for the purpose of validating the results.
MHD viscous fluid containing gyrotactic microorganisms flow over a permeable stretching surface in the presence of a magnetic field. A uniform magnetic field is applied at perpendicular to the fluid flow. Joule heating, viscous dissipation and Hall effect are neglected. Hence the Lorentz force depends only on magnetic field. Surface is stretching ${{u}_{w}}=Bx$ along the x axis. B is the positive constant.
The continuity, momentum, energy, mass and microorganism equations are given by Waqas et al. [1-2] and Naseem et al. [3]
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\,\,$ (1)
$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{1}{{{\rho }_{\infty }}}\frac{\partial }{\partial y}\left( \mu (T)\frac{\partial u}{\partial y} \right)-\left( \frac{\sigma B_{0}^{2}}{{{\rho }_{\infty }}}+\frac{\mu (T)}{{{\rho }_{\infty }}{{k}_{p}}} \right)u$ (2)
$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{1}{{{\rho }_{\infty }}{{C}_{p}}}\frac{\partial }{\partial y}\left( k(T)\frac{\partial T}{\partial y} \right)-\frac{1}{{{\rho }_{\infty }}{{C}_{p}}}\frac{\partial {{q}_{r}}}{\partial y}$ (3)
$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={{D}_{m}}\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}}-{{k}_{n}}{{(C-{{C}_{\infty }})}^{n}}$ (4)
$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}+\frac{{{b}_{c}}{{w}_{c}}}{({{C}_{w}}-{{C}_{\infty }})}\frac{\partial }{\partial y}\left( N\frac{\partial C}{\partial y} \right)={{D}_{n}}\frac{{{\partial }^{2}}N}{\partial {{y}^{2}}}$ (5)
where, $u(x, y)$ and $v(x, y)$ are the horizontal and vertical fluid velocity components. $\mathrm{T}$ and $T_{\infty}$ are temperature and ambient fluid temperature. $\mu (T)={{\mu }_{\infty }}(1+a(T-{{T}_{w}}))$: temperature-dependent viscosity, ${{\mu }_{\infty }}$: ambient viscosity,$k(T)={{k}_{\infty }}(1+b(T-{{T}_{\infty }}))$: temperature dependent thermal conductivity,${{k}_{\infty }}$: ambient thermal conductivity,$\varepsilon =b({{T}_{w}}-{{T}_{\infty }})$: thermal conductivity variation parameter, $\delta =a({{T}_{w}}-{{T}_{\infty }})$: viscosity variation parameter,
Boundary Conditions [46-47]
$\begin{align} & u={{u}_{w}}+{{L}_{1}}\frac{\partial u}{\partial y}+{{L}_{2}}\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}},\,\,v=-{{v}_{w}},\,\,\,T={{T}_{w}}+{{L}_{3}}\frac{\partial T}{\partial y},\,\, \\ & C={{C}_{w}}+\,{{L}_{4}}\frac{\partial C}{\partial y}\,,\,N={{N}_{w}}+\,{{L}_{5}}\frac{\partial N}{\partial y}\,\,at\,y=0 \\\end{align}$
$u\to 0,\,\,T\to {{T}_{\infty }}\,,\,\,C\to {{C}_{\infty }},\,N\to {{N}_{\infty }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,at\,y\to \infty $ (6)
${{v}_{w}}$ suction/injection velocity. Following Rosseland approximation ${{q}_{r}}$, the radiation heat flux is given
${{q}_{r}}=-\left( \frac{4\sigma }{3{{k}^{*}}} \right)\frac{\partial {{T}^{4}}}{\partial y}=-\left( \frac{16\sigma }{3{{k}^{*}}} \right){{T}^{3}}\frac{\partial T}{\partial y}$.
Solution
We now introduce the following relations for u,v as
$u=Bxf'(\eta ),\,\,v=-\sqrt{B{{v}_{\infty }}}f(\eta ),\,\eta =y\sqrt{\frac{B}{{{v}_{\infty }}}}$,$\theta (\eta )=\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}}$,$\phi =\frac{C-{{C}_{\infty }}}{{{C}_{w}}-{{C}_{\infty }}}$ and $N(\eta )=\frac{N-{{N}_{\infty }}}{{{N}_{w}}-{{N}_{\infty }}}$ (7)
Eqns. (2-6) and using Eq. (7) thus reduces to the following non-dimensional form
$\begin{align} & \left( f'\,'\,'\left( 1+\delta (1-\theta ) \right)-\delta \theta 'f''\, \right)-f{{'}^{2}}+f''f \\ & -\left[ M\,+{{K}_{p}}\left( 1+\delta (1-\theta ) \right) \right]f'=0 \\\end{align}$ (8)
$\begin{align} & \theta '\,'\left( 1+\frac{4}{3}R{{\left( ({{\theta }_{w}}-1)\theta +1 \right)}^{3}}+\varepsilon \theta \right)+\varepsilon \theta {{'}^{2}}+ 4R({{\theta }_{w}}-1)\theta {{'}^{2}}{{\left( ({{\theta }_{w}}-1)\theta +1 \right)}^{2}}+{{\Pr }_{\infty }}\left( f\theta ' \right)=0 \\\end{align}$ (9)
$\phi ''-S{{c}_{\infty }}\,\left( {{K}_{n}}{{\phi }^{n}}-f\phi ' \right)=0$ (10)
$\omega ''+S{{n}_{\infty }}\,\left( f\omega '-Pe\left( \omega '\phi '+\phi ''(\omega +\sigma ) \right) \right)=0$ (11)
Variable Prandtl number, Variable Schmidt number of mass and Variable Schmidt number of gyrotactic microorganisms
Prandtl number is a function of viscosity, thermal conductivity and specific heat. Viscosity and thermal conductivity are the function of temperature, the Prandtl number also varies. Therefore, the Prandtl number related to the variable viscosity and variable thermal conductivity is defined as
${{\Pr }_{v}}=\frac{\mu (T){{C}_{p}}}{k(T)}=\frac{{{C}_{p}}{{\mu }_{\infty }}(1+\delta (1-\theta ))}{{{k}_{\infty }}(1+\varepsilon \theta )}=\frac{{{\Pr }_{\infty }}(1+\delta (1-\theta ))}{(1+\varepsilon \theta )}$
At the surface $\eta $=0 of the porous stretching, this can be written as
${{\Pr }_{w}}=\frac{{{\Pr }_{\infty }}(1+\delta )}{(1+\varepsilon )}$
It can be seen that for $\delta $=0,$\varepsilon $=0 the variable Prandtl number ${{\Pr }_{w}}$ is equal to the ambient Prandtl number ${{\Pr }_{\infty }}$. It is mention that $\eta \to \infty $, i.e. outside the boundary-layer,$\theta (\eta )$ becomes zero, therefore,${{\Pr }_{\infty }}$=${{\Pr }_{v}}$ regardless of the values of $\delta$ and $\varepsilon$. Schmidt numberis the ratio of viscous diffusivity to mass diffusivity. If the viscosity of the fluid varies with the temperature, then the Schmidt number varies too. The assumption of constant Schmidt number inside the boundary layer may produce unrealistic results [40-42]. Therefore, the Schmidt number related to the variable viscosity is defined as
$S{{c}_{v}}=\frac{\mu (T)}{{{D}_{m}}{{\rho }_{\infty }}}=\frac{{{\mu }_{\infty }}(1+\delta (1-\theta ))}{{{D}_{m}}{{\rho }_{\infty }}}=S{{c}_{\infty }}(1+\delta (1-\theta ))$
$S{{c}_{N}}=\frac{\mu (T)}{{{D}_{N}}{{\rho }_{\infty }}}=\frac{{{\mu }_{\infty }}(1+\delta (1-\theta ))}{{{D}_{N}}{{\rho }_{\infty }}}=S{{c}_{N\infty }}(1+\delta (1-\theta ))$
At the surface $\eta $= 0 of the porous stretching, this can be written as
$S{{c}_{v}}=\frac{\mu (T)}{{{D}_{m}}{{\rho }_{\infty }}}=\frac{{{\mu }_{\infty }}(1+\delta )}{{{D}_{m}}{{\rho }_{\infty }}}=S{{c}_{\infty }}(1+\delta )$
$S{{c}_{N}}=\frac{\mu (T)}{{{D}_{N}}{{\rho }_{\infty }}}=\frac{{{\mu }_{\infty }}(1+\delta )}{{{D}_{N}}{{\rho }_{\infty }}}=S{{c}_{N\infty }}(1+\delta )$
It can be seen that for $\delta $= 0,$\varepsilon $ = 0 the variable Prandtl number $S{{c}_{w}},S{{c}_{wN}}$ is equal to the ambient Schmidt number of mass and gyrotactic microorganisms $S{{c}_{\infty }},S{{c}_{\infty N}}$. It is mention that $\eta \to \infty $, i.e. outside the boundary-layer,$\theta (\eta )$ becomes zero, therefore,$S{{c}_{\infty }},S{{c}_{\infty N}}$=$S{{c}_{v}},S{{c}_{vN}}$ regardless of the values of $\delta $ and $\varepsilon $.
The non-dimensional temperature Eqns. (8-11) can be expressed as
$\begin{align} & \theta '\,'\left( 1+\frac{4}{3}R{{\left( ({{\theta }_{w}}-1)\theta +1 \right)}^{3}}+\varepsilon \theta \right)+\varepsilon \theta {{'}^{2}} \\ & +4R({{\theta }_{w}}-1)\theta {{'}^{2}}{{\left( ({{\theta }_{w}}-1)\theta +1 \right)}^{2}} \\ & +\frac{{{\Pr }_{v}}(1+\varepsilon \theta )}{(1+\delta (1-\theta ))}\left( f\theta ' \right)=0 \\\end{align}$ (12)
$\phi ''-\,\frac{S{{c}_{v}}}{(1+\delta (1-\theta ))}\left( {{K}_{n}}{{\phi }^{n}}-f\phi ' \right)=0$ (13)
$\begin{align} & \omega ''+\frac{S{{n}_{v}}}{(1+\delta (1-\theta ))} \\ & \left( f\omega '-Pe\left( \omega '\phi '+\phi ''(\omega +\sigma ) \right) \right)=0 \\\end{align}$ (14)
Boundary conditions equations (6) reduces as:
$\begin{align} & \eta =0\,:\,\,\,\,\,\,\,f'(\eta )=1+Sli{{p}_{1}}f'\,'(\eta )+Sli{{p}_{2}}\,f'\,'\,'(\eta ),\,\,\,\, \\ & f(\eta )=S,\,\,\theta (\eta )=1+Sli{{p}_{T}}\theta '(\eta ),\,\, \\ & \phi (\eta )=1+Sli{{p}_{C}}\phi '(\eta ),\,\,\,\,\,\,\omega (\eta )=1+Sli{{p}_{C}}\omega '(\eta ) \\ & \eta \to \infty :\,\,\,\,\,\,\,\,\,\,\,f'(\eta )\to 0,\,\,\theta (\eta )\to 0,\phi (\eta )\to 0,\,\, \\ & \omega (\eta )\to 0 \\\end{align}$. (15)
where, $Sli{{p}_{1}}={{L}_{1}}\sqrt{\frac{B\,}{{{\nu }_{\infty }}}}$; $Sli{{p}_{2}}={{L}_{2}}\frac{B}{{{\nu }_{\infty }}}$$Sli{{p}_{T}}={{L}_{3}}\sqrt{\frac{B}{{{\nu }_{\infty }}}}$, $Sli{{p}_{C}}={{L}_{4}}\sqrt{\frac{B}{{{\nu }_{\infty }}}}$, $Sli{{p}_{N}}={{L}_{5}}\sqrt{\frac{B}{{{\nu }_{\infty }}}}$: first and second order slip velocity parameter, slip temperature parameter, slip concentration parameter and slip microorganism parameter respectively,$R=\frac{4\sigma T_{\infty }^{3}}{{{k}_{\infty }}k*}$; Radiation parameter, k* ; thermal radiation parameter, $M=\frac{\sigma B_{0}^{2}}{{{\rho }_{\infty }}B}$: Magnetic field parameter, ${{K}_{n}}=\frac{{{k}_{n}}}{b}{{({{C}_{w}}-{{C}_{\infty }})}^{n-1}}$: chemical reaction parameter, ${{\theta }_{w}}=\frac{{{T}_{w}}}{{{T}_{\infty }}}$: temperature difference parameter, ${{k}_{\infty }}$: thermal conductivity ${{K}_{p}}=\frac{{{\nu }_{\infty }}}{{{k}_{p}}b}$: porosity parameter,$Pe=\frac{{{b}_{c}}{{w}_{c}}}{{{\nu }_{\infty }}}$: bioconvection Péclet number,$\sigma =\frac{{{N}_{\infty }}}{{{N}_{w}}-{{N}_{\infty }}}$:bioconvection constant,$S{{n}_{\infty }}=\frac{{{\nu }_{\infty }}}{{{D}_{n}}}$: ambient Schmidt number of microorganism,$S{{c}_{\infty }}=\frac{{{\nu }_{\infty }}}{{{D}_{B}}}$: ambient Schmidt number of mass${{\Pr }_{\infty }}=\frac{{{\mu }_{\infty }}}{{{k}_{\infty }}}{{C}_{p}}$ ambient Prandtl number.
In this article, the influence of various physical parameters like magnetic field parameter, first and second order slip velocity parameter, slip temperature parameter, slip concentration parameter and slip microorganism parameter radiation parameter, chemical reaction parameter, temperature difference parameter, porosity parameter, bioconvection Péclet number, bioconvection constant, ambient Schmidt number of microorganism, ambient Schmidt number of mass, ambient Prandtl number is examined. These values are kept as common in entire study except the varied values as shown in respective figures and tables. Figures 2-5 the f, $\theta $, $\phi $and$\omega $ profiles are plotted for the (M) and other parameters are kept fixed. The f of fluid increase as (M) increases and the $\theta $, $\phi $and $\omega$ flux decrease as the value of (M) increase. This is due to the fact that the magnetic field introduces a retarding body force known as Lorentz force. As the Lorentz force is a resistive force which opposes the fluid motion, so heat is produced and as a result, the thermal boundary layer thickness and concentration (volume fraction) boundary layer thickness become thicker for stronger magnetic field. Physically, the drag force increases with an increase in the magnetic field and as a result depreciation occurs in the velocity field.
Figures 6-9 depicts the $f, \theta$ and $\phi$ profiles are plotted for the $\left(k_{p}\right)$ and other parameters are kept fixed. The $f$ of fluid decreases as $\left(k_{p}\right)$ increases and the $\theta, \phi$ and $\omega$ flux increase as the value of $\left(k_{p}\right)$ increases. Figures $10-17$ show the impact of $\left(\text {slip}_{1} \& slip_{2} \text { ) parameter on } f, \theta, \phi \text { and } \omega \text { profiles. It is }\right.$ observed that for increasing values of $\left(s l i p_{1} \& \operatorname{slip}_{2}\right)$ parameter, the f decrease whereas $\theta, \phi$ and $\omega$ profiles increase. Figure 18 shows the impact of $\left(s \operatorname{lip}_{T}\right)$ parameter on $\theta$ profile. It is observed that for increasing values of $\left(s \operatorname{lip}_{T}\right),$ the $\theta$ profile decrease. Figures 19-20 show the impact of $\left(s l i p_{c}\right)$ parameter on $\phi$ and $\omega$ profiles. It is observed that for increasing values of $\left(s l i p_{c}\right)$, the $\phi$ and $\omega$ profiles decrease. Physically, when slip occurs, the slipping fluid shows a decrease in the surface skin-friction between the fluid and the surface, because not all the pulling force of the surface can be transmitted to the fluid. So, increasing the value of the velocity slip parameter will decrease the flow velocity in the region of the boundary layer. Figure 21 shows the impact of (slip2) parameter on$\theta $profile. It is observed that for increasing values of ($sli{{p}_{2}}$), the $\theta $profile decrease. Figures 22-23 show that the heat of the fluid enhances with the increase of (R), (${{\theta }_{w}}$). Generally, an increment in (R) causes a decrease in absorption coefficient, which results rise in the divergence of radiative heat flux. Hence, the rate of radiative heat transfers of the fluid shoots up, so that the heat flux increases. Figure 24 shows the mass profile is plotted for the (n) other parameters are kept fixed. Concentration of fluid increases as (n) increase. Figure 25 depicts that the temperature profile for the several values of the (Pr) and other parameters are kept fixed. Heat of fluid decreases as (Pr) increases. Prandtl number is used to increase the rate of cooling in conducting flow. At high Prandtl number the fluid is very viscous and the viscous dissipation produces heat due to drag between the fluid particles and this extra heat causes an increase of the initial fluid temperature. Figure 26 shows the $\omega $ profile is plotted for the ($S{{n}_{v}}$) other parameters are kept fixed. The $\omega $ profile of fluid decreases as ($S{{n}_{v}}$) increase. Figure 27 depicts that the concentration of the fluid suppresses with the rise the value of (${{K}_{n}}$). Physically, chemical reaction increases the rate of interfacial mass transfer. Chemical reaction suppresses the local concentration and increases its mass gradient and its flux. It is due to the fact that ($S{{c}_{v}}$) is the ratio of velocity to mass diffusivities which means when ($S{{c}_{v}}$) increases, mass diffusivity decreases and there is a reduction in mass. Figures 28-30 show the $\omega $ profile against the similarity variable η for various values of following parameters such as Pé, $\sigma $ and$sli{{p}_{N}}$. We are noted that the $\omega $ profile decreases Pé, $\sigma $ and$sli{{p}_{N}}$ increases. Figures 31-32 show the impact of ($S{{c}_{v}}$) parameter on $\phi $ and$\omega $profiles. It is observed that for increasing values of ($S{{c}_{v}}$) parameter parameter, the $\phi $and$\omega $profiles. Table 1 and 2 shows the comparison of the present results with the existed results of Nadeem and Hussain [39], Gorla and Sidawi [40], Goyal and Bhargava [41], Andersson et al. [42], Prasad et al. [43], Mukhopadhyay et al. [44], Palani et al. [45] and Gorla et al. [46]. Under some special conditions, present results have an excellent agreement with the existed results. This shows the validity of the present results along with the accuracy of the numerical technique used in this study. Table 3-4 shows that the variation of skin friction coefficient, local Nusselt number, local Sherwood number and local gyrotactic microorganisms Sherwood number for various physical parameters.
Table 1. Comparison of $-\theta '(0)$ for different values Pr in the absence of the parameters S= M=R=Kp=We=$sli{{p}_{1}}$=$sli{{p}_{2}}$=$sli{{p}_{T}}$=$\varepsilon $=$\delta $=0.0; ${{\theta }_{w}}$=1
|
Pr |
HAM method Nadeem and Hussain [39] |
Gorla and Sidawi [41] |
FEM method Goyal andBhargava [42] |
RKF45 method Gorla et al. [47] |
R-K 4th order present study |
|
0.2 |
0.169 |
0.1691 |
0.1691 |
0.170259788 |
0.172348764 |
|
0.7 |
0.454 |
0.5349 |
0.4539 |
0.454447258 |
0.453917857 |
|
2 |
0.911 |
0.9114 |
0.9113 |
0.911352755 |
0.911361492 |
|
7 |
— |
1.8905 |
1.8954 |
1.895400395 |
1.895412536 |
|
20 |
— |
3.3539 |
3.3539 |
3.353901838 |
3.353933867 |
Figure 2. Impact of M on velocity profile
Figure 3. Impact of M on temperature profile
Figure 4. Impact of M on mass profile
Figure 5. Impact of M on microorganism profile
Figure 6. Impact of ${{k}_{p}}$ on velocity profile
Figure 7. Impact of ${{k}_{p}}$ on temperature profile
Figure 8. Impact of ${{k}_{p}}$ on mass profile
Figure 9. Impact of ${{k}_{p}}$ on microorganism profile
Figure 10. Impact of$sli{{p}_{1}}$ on velocity profile
Figure 11. Impact of $sli{{p}_{1}}$on temperature profile
Figure 12. Impact of $sli{{p}_{1}}$ on mass profile
Figure 13. Impact of $sli{{p}_{1}}$ on microorganism profile
Figure 14. Impact of $sli{{p}_{2}}$on velocity profile
Figure 15. Impact of $sli{{p}_{2}}$ on temperature profile
Figure 16. Impact of $sli{{p}_{2}}$ on mass
Figure 17. Impact of $sli{{p}_{2}}$ on microorganism profile
Figure 18. Impact of $sli{{p}_{T}}$on temperature profile
Figure 19. Impact of $sli{{p}_{C}}$on mass profile
Figure 20. Impact of $sli{{p}_{C}}$ on microorganism profile
Figure 21. Impact of $\varepsilon $ temperature profile
Figure 22. Impact of ${{\theta }_{w}}$on temperature profile
Figure 23. Impact of R on temperature profile
Figure 24. Impact of n on mass profile
Figure 25. Impact of Prv on temperature profile
Figure 26. Impact of $S{{n}_{V}}$on microorganism profile
Figure 27. Impact of Kn on mass profile
Figure 28. Impact of Pe on microorganism profile
Figure 29. Impact of s on microorganism profile
Figure 30. Impact of $sli{{p}_{N}}$ on concentration profile
Figure 31. Impact of $S{{c}_{v}}$ on concentration profile
Table 2. Comparison of$-f''(0)$ for different values M in the absence of the parameters S= R=Kp=$We$=$sli{{p}_{1}}$ =$sli{{p}_{2}}$ =$sli{{p}_{T}}$ =$\varepsilon $ =$\delta $=0.0; ${{\theta }_{w}}$=1, Pr=0.72
|
M |
Andersson et al. [42] |
Prasad et al. [43] |
Mukhopadhyay et al. [44] |
Palani et al. [45] |
Present study |
|
0.0 |
1.000000 |
1.000174 |
1.000173 |
1.00000 |
1.000000000 |
|
0.5 |
1.224900 |
1.224753 |
1.224753 |
1.224745 |
1.224744871 |
|
1 |
1.414000 |
1.414449 |
1.414450 |
1.414214 |
1.414213562 |
|
1.5 |
1.581000 |
1.581139 |
1.581140 |
1.581139 |
1.581138830 |
|
2 |
1.732000 |
1.732203 |
1.732203 |
1.732051 |
1.732050808 |
Table 3. Variation of local Nusselt number, local Sherwood number and local gyrotactic microorganisms Sherwood number for various physical parameters
|
e |
R |
qw |
n |
Prv |
Sn |
Kn |
Pe |
$\sigma$ |
$s l i p_{N}$ |
$-\theta^{\prime}(0)$ |
$-\phi^{\prime}(0)$ |
$-\omega^{\prime}(0)$ |
|
0.2 |
|
|
|
|
|
|
|
|
|
0.675182220 |
|
|
|
0.4 |
|
|
|
|
|
|
|
|
|
0.791666927 |
|
|
|
0.6 |
|
|
|
|
|
|
|
|
|
0.907057220 |
|
|
|
|
0.0 |
|
|
|
|
|
|
|
|
0.672452398 |
|
|
|
|
0.5 |
|
|
|
|
|
|
|
|
0.857743356 |
|
|
|
|
1.0 |
|
|
|
|
|
|
|
|
0.907057220 |
|
|
|
|
|
1 |
|
|
|
|
|
|
|
0.804888907 |
|
|
|
|
|
1.2 |
|
|
|
|
|
|
|
0.853297614 |
|
|
|
|
|
1.3 |
|
|
|
|
|
|
|
0.873799158 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
1.321873369 |
|
|
|
|
|
2 |
|
|
|
|
|
|
|
1.165526785 |
|
|
|
|
|
3 |
|
|
|
|
|
|
|
1.093296359 |
|
|
|
|
|
|
0.72 |
|
|
|
|
|
0.715699806 |
|
|
|
|
|
|
|
0.85 |
|
|
|
|
|
0.808371394 |
|
|
|
|
|
|
|
1.0 |
|
|
|
|
|
0.907057220 |
|
|
|
|
|
|
|
|
1.5 |
|
|
|
|
|
|
1.148806504 |
|
|
|
|
|
|
2 |
|
|
|
|
|
|
1.428303592 |
|
|
|
|
|
|
2.5 |
|
|
|
|
|
|
1.680515788 |
|
|
|
|
|
|
|
0.0 |
|
|
|
|
0.945201672 |
|
|
|
|
|
|
|
|
0.5 |
|
|
|
|
1.165526785 |
|
|
|
|
|
|
|
|
1.0 |
|
|
|
|
1.332019354 |
|
|
|
|
|
|
|
|
|
0.1 |
|
|
|
|
1.189603361 |
|
|
|
|
|
|
|
|
0.2 |
|
|
|
|
1.428303592 |
|
|
|
|
|
|
|
|
0.3 |
|
|
|
|
1.661265580 |
|
|
|
|
|
|
|
|
|
0.0 |
|
|
|
1.262732944 |
|
|
|
|
|
|
|
|
|
0.5 |
|
|
|
1.428303592 |
|
|
|
|
|
|
|
|
|
1.0 |
|
|
|
1.593874604 |
|
|
|
|
|
|
|
|
|
|
0.0 |
|
|
1.634725755 |
|
|
|
|
|
|
|
|
|
|
0.1 |
|
|
1.428303592 |
|
|
|
|
|
|
|
|
|
|
0.2 |
|
|
1.268167888 |
Table 4. Variation of skin friction coefficient, local Nusselt number, local Sherwood number and local gyrotactic microorganisms Sherwood number for various physical parameters
|
M |
Kp |
$sli{{p}_{1}}$ |
$sli{{p}_{2}}$ |
$sli{{p}_{T}}$ |
$sli{{p}_{C}}$ |
$f''(0)$ |
$-\theta '(0)$ |
$-\phi '(0)$ |
-$-\omega '(0)$ |
|
0 |
|
|
|
|
|
-1.265354 |
1.004250 |
1.191662 |
1.464307 |
|
1 |
|
|
|
|
|
-1.641575 |
0.907057 |
1.165526 |
1.428303 |
|
2 |
|
|
|
|
|
-1.977595 |
0.852901 |
1.149273 |
1.405030 |
|
|
0 |
|
|
|
|
-1.458759 |
0.949793 |
1.177437 |
1.444907 |
|
|
0.5 |
|
|
|
|
-1.641575 |
0.907057 |
1.165526 |
1.428303 |
|
|
1 |
|
|
|
|
-1.813989 |
0.875559 |
1.156204 |
1.415042 |
|
|
|
0.0 |
|
|
|
-2.066579 |
0.988152 |
1.218744 |
1.509977 |
|
|
|
0.1 |
|
|
|
-1.641575 |
0.907057 |
1.165526 |
1.428303 |
|
|
|
0.2 |
|
|
|
-1.374705 |
0.850296 |
1.128698 |
1.369648 |
|
|
|
|
0.0 |
|
|
-1.413180 |
0.858801 |
1.134191 |
1.378525 |
|
|
|
|
0.03 |
|
|
-1.539607 |
0.885967 |
1.151798 |
1.406668 |
|
|
|
|
0.05 |
|
|
-1.641575 |
0.907057 |
1.165526 |
1.428303 |
|
|
|
|
|
0.0 |
|
|
0.912196 |
|
|
|
|
|
|
|
0.1 |
|
|
0.907057 |
|
|
|
|
|
|
|
0.2 |
|
|
0.901785 |
|
|
|
|
|
|
|
|
0.0 |
|
|
1.351283 |
|
|
|
|
|
|
|
0.1 |
|
|
1.165526 |
|
|
|
|
|
|
|
0.2 |
|
|
1.028143 |
|
Figure 32. Impact of $S{{c}_{v}}$ on microorganisms profile
We have discussed variable fluid property for MHD viscous fluid flow containing gyrotactic microorganisms over a permeable stretching sheet and also considered the variable Prandtl number, variable Schmidt number of mass, variable Schmidt number of gyrotactic microorganisms, first and second order velocity slip, temperature jump, concentration slip, microorganism slip, porosity medium, non-liner radiation and non-linear chemical reaction. By using suitable transformation, the governing PDEs corresponding to the momentum, energy, mass and microorganism equations are converted into non-linear coupled ODEs and numerically by using R-K 4th order with shooting technique. The effects of physical parameters on the velocity, heat, mass and microorganism are analyzed with the help of graphs and tables.
The following important results can be drawn from this study:
T temperature
$T_{\infty}$ ambient fluid temperature
$\mu (T) $ temperature-dependent viscosity
${{\mu }_{\infty }}$ ambient viscosity
$k(T) $ temperature dependent thermal conductivity
${{k}_{\infty }}$ ambient thermal conductivity
$\varepsilon$ thermal conductivity variation parameter
$\delta $ viscosity variation parameter
R Radiation parameter
$k*$ thermal radiation parameter
M Magnetic field parameter
${{K}_{n}}$ chemical reaction parameter
${{\theta }_{w}}$ temperature difference parameter
${{k}_{\infty }}$ thermal conductivity
${{K}_{p}}$ porosity parameter
$Pe$ bioconvection Péclet number
$\sigma $ bioconvection constant
$S{{n}_{\infty }}$ ambient Schmidt number of microorganism
$S{{c}_{\infty }}$ ambient Schmidt number of mass
${{\Pr }_{\infty }}$ ambient Prandtl number
$Sli{{p}_{1}}$ first order slip velocity parameter
$Sli{{p}_{2}}$ second order slip velocity parameter
$Sli{{p}_{T}}$ slip temperature parameter
$Sli{{p}_{C}}$ slip concentration parameter
$Sli{{p}_{N}}$ slip microorganism parameter respectively
u(x, y) the horizontal fluid velocity component
v(x, y) vertical fluid velocity component
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