Nomenclature
$a\,$ strength of the squeezed flow $({{\rm{s}}}^{-1});$
$b\,$ index of squeezed flow;
${C}_{p}\,$ specific heat at constant pressure $({{\rm{Jkg}}}^{-1}\,{{\rm{K}}}^{-1});$
${C}_{f}\,$ skin friction coefficient;
$Ec\,$ Eckert number;
${f}_{0}\,$ permeable velocity parameter;
$f^{\prime} \,$ dimensionless velocity;
$h\left(t\right)\,$ height $(m);$
$k\,$ thermal conductivity $({{\rm{Wm}}}^{-1}\,{{\rm{K}}}^{-1});$
${k}^{* }\,$ average absorption coefficient;
$M\,$ magnetic parameter;
$n\,$ power law index;
$p\,$ pressure $({\rm{kg}}\,{{\rm{m}}}^{-1}\,{{\rm{s}}}^{-2});$
$Pr\,$ Prandtl number;
${Q}_{0}\,$ heat source coefficient $({\rm{W}}\,{{\rm{m}}}^{-2}\,{{\rm{K}}}^{-1});$
$Q\,$ heat source parameter;
${q}_{0}\,$ heat flux $({\rm{W}}\,{{\rm{m}}}^{-2});$
${q}_{r}\,$ radiative heat flux $({\rm{W}}\,{{\rm{m}}}^{-2});$
$q\left(x\right)\,$ heat flux $({\rm{W}}\,{{\rm{m}}}^{-2});$
$R{e}_{x}\,\,$ local Reynolds number;
$Rd\,\,$ radiation parameter;
$s\,$ arbitrary constant
$T\,$ fluid temperature $(K);$
${T}_{\infty }\,$ temperature far from the surface $(K);$
$t\,\,$ time $(s);$
$u,\,v\,$ velocity components $({{\rm{ms}}}^{-1});$
$U\,$ free stream velocity $({\rm{m}}\,{{\rm{s}}}^{-1});$
${v}_{0}\left(t\right)\,$ reference velocity of sensor surface $({{\rm{ms}}}^{-1});$
$We\,\,$ Weissenberg number;
Greek symbols
$\eta \,$ independent dimensionless coordinate;
$\theta \,$ dimensionless temperature;
${\rm{\Gamma }}\,$ time constant;
$\sigma \,$ electrical conductivity $\left({{\rm{A}}}^{2}\,{{\rm{s}}}^{3}\,{{\rm{m}}}^{-3}\,{{\rm{kg}}}^{-1}\right);$
${\sigma }^{* }\,$ Stefan-Boltzman constant
$\rho \,$ density $({\rm{kg}}\,{{\rm{m}}}^{-3});$
$\nu \,$ kinematic viscosity $({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1});$
${\tau }_{w}\,$ shear stress ($pascal$);
$\alpha \,\,\,$ thermal diffusivity $({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1});$
$\psi \,$ stream function;
Introduction
A sensor surface holds its significance as one can retrieve its applications in biological and chemical engineering grounds. One such sensitive surface is a micro cantilever which can analyze even a tiny amount of mass. Hence it turns out to be of utmost importance in sensing various compounds simultaneously. It serves a purpose in industrial, military and bio-clinical disciplines. An emphasis of the magnetic field on the flow makes the flow problem more relatable to a practical scenario as it avails itself in a nuclear reactor, installation of an electric furnace and many more applications. Electrolytes are the ones that turn out to be magnetohydrodynamic (MHD) fluid. Because of this, there has been an increased interest in studying this effect. Rashidi et al [1] discussed thermodynamic second law within a rotating porous disk. The simulation turns out to be useful in enhancing heat transmission in sustainable energy systems. Hayat et al [2] investigated MHD 3D flow with convective conditions at the boundaries. The solutions are attained by HAM and their results uphold the theory that the Biot number maximizes the transfer of heat. Prasannakumara et al [3] examined Sisko nanofluid flowing over a stretched sheet. The temperature profiles are more influenced by non-linear radiation than that of linear. Lund et al [4] laid their effect in analyzing dual solutions for Williamson fluid flowing with slippage; the presence of the buoyancy force has contributed to finding dual solutions. Tassaddiq et al [5] investigated Casson fluid flow with Newtonian heating by making use of the fractional model along Mittag–Leffler memory. Kumar et al [6], by considering the vertical plate that is impulsively initiated, studied the impact of thermal radiation and declared that velocity and temperature escalate by maximizing the radiation supply.
The fluids responsible for practical implications do not manifest the property of the Newtonian fluid; hence there was origination of the concept of the fluid that is not Newtonian. Carreau fluid is one such fluid. Ellahi et al [7] emphasized such fluid flow in a rectangle shaped duct. Akbar and Nadeem [8] created a model for flow of the blood utilizing Carreau fluid with stenosis that is axially not symmetric but symmetric radially whereas Khan et al [9] outlined a numerical study of the same fluid by squeezing the plate over the sensor surface by varying thermal conductivity. Kumar et al [10] investigated the radiative transport of heat by suspending fluid particles within the Carreau fluid. Ali et al [11] incorporated the designed microbes in the flow and studied the interactions between them and Carreau fluid biologically with magnetic effects. The Taylor swimming sheet model is used to serve this purpose. The Homann chemically active stagnation point Carreau fluid flow was examined by Khan et al [12]. The implementation of Fick’s and Fourier law in their modified form makes the flow study more interesting. Heat transport impacts on electro-MHD Carreau fluid allowed amid micro but parallel plates were scrutinized by Bhatti et al [13].
Sensors are involved in most computing devices. A micro cantilever is the sensing element used in the functioning as it has the capacity to diagnose many sicknesses. Their high sensitivity property and low cost aids their use in sophisticated applications. To this end, many of researchers have shown their concern in contributing to this field. Dijkstra et al [14] gave their insights on the thermal flow sensor in miniaturized devices with integrated-planar sensor arrangement on semicircular microchannels suspended by silicon-nitride. Akgül and Pakdemirli [15] analyzed Lie group theory for fluid that is not Newtonian, allowed over a pervious surface. Haq et al [16] utilized functionalized metal-nano-sized particles over this surface and studied the squeezed flow. In their research they discussed that $Cu$-nano-sized particles promote a better transport of heat. A compressed flow of a time-reliant tangent hyperbolic fluid on the sensor surface was investigated by Kumar et al [17]. Hussain et al [18] analyzed magneto-pseudoplastic fluid flowing across a sensor surface. An assimilation of the energy equation in their model is obtained by Fourier’s law. Waini et al [19] explored hybrid nano-liquid squeezed flow over a sensor surface. Their results elucidate that the rate at which the heat is transferred is higher for a hybrid fluid than for a regular fluid.
Daniel and Daniel [20] explored buoyancy, the radiation effect on MHD flow using HAM. It is noted that the hydrodynamic and thermal boundary layer escalates with excessive exposure to radiation. Sheikholeslami and Rokni [21] numerically simulated the consequence of Coulomb force on heat transport of a nanofluid in a thermally radiated porous enclosure. They deduced that nano-sized particles that have a platelet shape have a high Nusselt number. Gireesha et al [22] made use of the Cattaneo–Christov model to study dusty Casson fluid with melting heat transport. Comparison of their work with previous one highlights the precision of their method. Muhammad et al [23] revealed the implication of radiation in the Powell-Eyring 3D nanomaterial flow. To understand the transport of heat and mass flux, non-Fourier’s, non-Fick’s hypothesis is applied.
Hayat and Qasim [24] together attained the results for Joule heating influence on Maxwell fluid involving the thermophoresis effect. Sánchez et al [25] analyzed non-Newtonian fluid flow along a slit microchannel; Joule heating modifies the fluid features, which in turn alters the electric potential and further on the flow field. Babu and Narayana [26] elucidated the MHD mixed convective influence on a Jeffrey fluid with power law heat flux. The Deborah number has major influence on thermal and momentum boundary layer. Qayyum et al [27] investigated the comparison of five nano-sized particles in a spinning disk provided with slip. Their study revealed that for a higher volume fraction of particles, the velocity along the axis decays and as the thermal conductivity of $Ag-{H}_{2}O$ is more, the temperature is also high. Nazeer et al [28] analytically described the radiative, Joule heating impact on electro-osmotic liquid flow.
Sheikh et al [29] compared two different fractional derivatives for Casson fluid flowing with heat production. The velocity profile attained from these methods at unit time is noted to be identical whereas the deviations grow higher with the time greater than unity. Khan and Sardar [30] explored heat generated as well as absorbed in 3D Carreau fluid flow. They carried out the analysis by keeping the shear rate infinite. Tlili [31] independently worked on microgravity environment by allowing Jeffrey fluid to flow upon a sheet being stretched. Hafeez and Khan [32] looked into Cattaneo-Christov theory in a spinning disk. The disk undergoes rotation as well as stretching and the influence of heat production is also noted. Further investigations have been carried out by many authors [33–35] in their field of study.
A sensor surface is a geometrical means over which the magnetized Carreau fluid is allowed to flow. A squeezed 2D flow is reliant on time. A sensor surface is exposed to radiation and its impact is scrutinized. Thermal performance of the flow is explored under persuasion of an internal heat source and Joule heating. All these effects are pronounced to form an irresistible flow demeanor. Manifestation of the graphs is crucial in checking the consequence of the flow. Drag force and the Nusselt number are computed together with velocity and thermal profiles. The streamlines are drawn in order to understand the outline of the flow.
Mathematical illustration
Consider a 2D, time-reliant, squeezed Carreau fluid flow driven by a penetrable sensor surface. The height $h(t)$ of the closed compressed channel is considered much higher than the thickness of the boundary layer as in figure 1. Furthermore, the magnetic field ${B}_{0}$ is operated at a normal channel direction. The upper plate is squeezed whereas the lower plate is left to be immovable. Joule heating, radiation and internal heat source impacts are also considered to observe the thermal performance of the flow.
Figure 1. Schematic representation of the flow model. |
The mathematical representations of continuity, momentum and heat are shown as [9]:9 ) in equation (7 ), we get4 ) and (5 ).1 ) is sufficiently satisfied and the remaining equations are transformed as follows:16 ) is given by,
$\begin{eqnarray}\displaystyle \frac{\partial u}{\partial x}+\displaystyle \frac{\partial v}{\partial y}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial u}{\partial t}+u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial y}=-\displaystyle \frac{1}{\rho }\left(\displaystyle \frac{\partial p}{\partial x}\right)\\ \,+\,\nu \left[\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}+\displaystyle \frac{3\left(n-1\right)}{2}{{\rm{\Gamma }}}^{2}{\left(\displaystyle \frac{\partial u}{\partial y}\right)}^{2}\left(\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\right)\right]-\displaystyle \frac{\sigma {B}_{0}^{2}u}{\rho },\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\partial U}{\partial t}+U\displaystyle \frac{\partial U}{\partial x}=-\displaystyle \frac{1}{\rho }\left(\displaystyle \frac{\partial p}{\partial x}\right)-\displaystyle \frac{\sigma {B}_{0}^{2}u}{\rho },\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial T}{\partial t}+u\displaystyle \frac{\partial T}{\partial x}+v\displaystyle \frac{\partial T}{\partial y}=\alpha \displaystyle \frac{{\partial }^{2}T}{\partial {y}^{2}}-\displaystyle \frac{1}{\rho {C}_{p}}\displaystyle \frac{\partial {q}_{r}}{\partial y}\\ \,+\,{Q}_{0}\displaystyle \frac{\left(T-{T}_{\infty }\right)}{\rho {C}_{p}}+\displaystyle \frac{{\sigma }^{2}{B}_{0}^{2}{u}^{2}}{\rho {C}_{p}}.\end{array}\end{eqnarray}$
Here, $u,v$-velocity attributes respectively along $x$ and $y$ directions, $U$ is the free stream velocity, $n$ is the power law index, $\nu $ is the kinematic viscosity, ${\rm{\Gamma }}$ is the time constant, $p$ is the pressure, $\alpha $ is the thermal diffusivity, $T$ represents temperature, $t$ represents time, $\rho $ represents density, $\sigma $ represents electric conductivity, ${C}_{p}$ represents specific heat capacity, ${q}_{r}$ represents radiative heat flux, ${Q}_{0}$ represents internal heat source, ${T}_{\infty }$ represents temperature far from the surface. Further, on removing a pressure attribute from (2) and (3), it takes the form, $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial u}{\partial t}+u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial y}=\displaystyle \frac{\partial U}{\partial t}+U\displaystyle \frac{\partial U}{\partial x}\\ \,+\,\nu \left[\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}+\displaystyle \frac{3\left(n-1\right)}{2}{{\rm{\Gamma }}}^{2}{\left(\displaystyle \frac{\partial u}{\partial y}\right)}^{2}\left(\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\right)\right]\\ \,-\,\displaystyle \frac{\sigma {B}_{0}^{2}u}{\rho },\end{array}\end{eqnarray}$
with tangled peripheral conditions: $\begin{eqnarray*}u\left(x,0,t\right)=0,\,v\left(x,0,t\right)={v}_{0}\left(t\right),\,u\left(x,\infty ,t\right)=U\left(x,t\right),\end{eqnarray*}$
$\begin{eqnarray}-\,k\displaystyle \frac{\partial T\left(x,0,t\right)}{\partial y}=q\left(x\right),\,T\left(x,\infty ,t\right)={T}_{\infty }.\end{eqnarray}$
Here, ${v}_{0}\left(t\right)=\sqrt{a}{v}_{i}$, the sensor surface reference velocity is assumed only if the considered surface is permeable, $q(x)$- the wall heat flux, $k$-the thermal conductivity. As per Rosseland diffusion estimation, the radiative heat flux is deduced to be, $\begin{eqnarray}{q}_{r}=-\displaystyle \frac{4\sigma * }{3k* }\displaystyle \frac{\partial {T}^{4}}{\partial y}.\,\end{eqnarray}$
According to Taylor series expansion ${T}^{4}$ is expressed as $\begin{eqnarray}{T}_{\infty }^{4}+4{T}_{\infty }^{3}\left(T-{T}_{\infty }\right)+6{T}_{\infty }^{2}{\left(T-{T}_{\infty }\right)}^{2}+\cdots \approx {T}^{4}.\,\end{eqnarray}$
On deducting terms of order above first degree, we get $\begin{eqnarray}{T}^{4}\approx 4{T}_{\infty }^{3}T-3{T}_{\infty }^{4}.\,\end{eqnarray}$
Employing ( $\begin{eqnarray}\displaystyle \frac{\partial {q}_{r}}{\partial y}=-\displaystyle \frac{16{\sigma }^{* }}{3{k}^{* }}{T}_{\infty }^{3}\displaystyle \frac{{\partial }^{2}T}{\partial {y}^{2}}.\,\end{eqnarray}$
The below mentioned similarity elements are applied to transform equations ( $\begin{eqnarray}\begin{array}{l}\eta =y\sqrt{\displaystyle \frac{a}{\nu }},\,\psi =x\sqrt{a\nu }f\left(\eta \right),\,a=\displaystyle \frac{1}{s+bt},\\ u=axf^{\prime} \left(\eta \right),\\ v=-\sqrt{a\nu }f\left(\eta \right),\,U=ax,\,\theta \left(\eta \right)=\displaystyle \frac{T-{T}_{\infty }}{{q}_{0}x\sqrt{\displaystyle \frac{\nu }{a}}},\\ q\left(x\right)={q}_{0}x.\end{array}\end{eqnarray}$
Here, $b$ is the index of the squeezed flow, $s$ is an arbitrary constant, $a$ is the strength of squeezed flow, $\psi $ is the stream function, ${q}_{0}$ is the reference heat flux. The height of the channel movements is described by the condition: $h\left(t\right)\,={h}_{0}{e}^{-st}\,for\,b=0\,{\rm{a}}{\rm{n}}{\rm{d}}\,h\left(t\right)=\,\tfrac{{h}_{0}}{{\left(s+bt\right)}^{\displaystyle \tfrac{1}{b}}}for\,b\gt 0,$ where ${h}_{0}$ is a constant. Clearly, equation ( $\begin{eqnarray}\begin{array}{l}f^{\prime\prime\prime} +\displaystyle \frac{3\left(n-1\right)}{2}W{e}^{2}{f}^{{\prime\prime} 2}f^{\prime\prime\prime} +\left(\displaystyle \frac{\eta b}{2}+f\right)f^{\prime\prime} \\ \,-\,{f}^{{\prime} 2}+b\left(f^{\prime} -1\right)+M\left(1-f^{\prime} \right)\,+\,1=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left(1+Rd\right)\theta ^{\prime\prime} +PrQ\theta +PrEcM{f}^{{\prime} 2}\\ \,-\displaystyle \frac{b}{2}\left(\theta -\eta \theta ^{\prime} \right)-f^{\prime} \theta +f\theta ^{\prime} =0,\end{array}\end{eqnarray}$
and the associated boundary conditions are transfused into $\begin{eqnarray}\begin{array}{l}f\left(0\right)=-{f}_{0},\,f^{\prime} \left(0\right)=0,\,\theta ^{\prime} \left(0\right)=-1,\,f^{\prime} \left(\infty \right)\\ \,\to 1,\,\theta \left(\infty \right)\to 0,\end{array}\end{eqnarray}$
where $We=\tfrac{{a}^{3}{x}^{2}{{\rm{\Gamma }}}^{2}}{\nu }$ is the local Weissenberg number, $M=\tfrac{\sigma {B}_{0}^{2}}{\rho a}$ is the Magnetic parameter, $Pr=\tfrac{\nu }{\alpha }$ is the Prandtl number, $Rd\,=\,\tfrac{16\sigma * {T}_{\infty }^{3}}{3kk* }\,\mathrm{is\; the}\,\mathrm{radiation}\,\mathrm{parameter},\,Q\,=\,\tfrac{{Q}_{0}}{a\rho {C}_{p}}$ is the internal heat generation parameter, $Ec=\tfrac{{a}^{2}x}{{C}_{p}{q}_{0}}$- the local Eckert number, and ${f}_{0}=\tfrac{{v}_{i}}{\sqrt{\nu }}$ is the permeable velocity. Now, wall shear stress near the sensor surface is expressed as $\begin{eqnarray}{\tau }_{w}=\displaystyle \frac{\partial u}{\partial y}+{{\rm{\Gamma }}}^{2}\displaystyle \frac{\left(n-1\right)}{2}{\left(\displaystyle \frac{\partial u}{\partial y}\right)}^{3}.\end{eqnarray}$
The elements of drag and the Nusselt number are deduced as $\begin{eqnarray}{C}_{f}=\displaystyle \frac{{\tau }_{w}}{ax\,\sqrt{\left(\displaystyle \frac{a}{\nu }\right)}}.\end{eqnarray}$
The transformed expression for equation ( $\begin{eqnarray}{\sqrt{Re}}_{x}{C}_{f}=\left[f^{\prime\prime} \left(0\right)+\displaystyle \frac{\left(n-1\right)}{2}\,W{e}^{2}{\left(f^{\prime\prime} \left(0\right)\right)}^{3}\right],\end{eqnarray}$
where $R{e}_{x}=x\sqrt{\tfrac{a}{\nu }}$ is the local Reynolds number.Results and discussion
Present scrutiny emphasizes to analyze the thermal transmission and flow of Carreau fluid, upon permeable sensor surface equipped with radiation, Joule heating, internal heat source and applied magnetic field. The numerical solutions are obtained for all flow profiles by varying different parameters. The graphs are plotted using numerical extractions and a detailed discussion has been given subsequently.
Power law index behavior $n$ on velocity is demonstrated in figure 2. As power law index $\left(n\right)$ is closely related to friction factor, it is obvious from figure 2 that with a rise in the power law index there is reduction in velocity at the boundary region. On rising $n$ the fluid exhibits the shear thickening property thus diminishing the velocity. This ascertains that magnification in $n$ causes a reduction in the field which avoids the fluid motion. But an increase in the thermal boundary layer thickness is noted, resulting in boosting the fluid’s temperature in the boundary (figure 3).
Figure 2. Repercussion of power law index $(n)$ on the velocity regime $({f}^{^{\prime} }\left(\eta \right)).$ |
Figure 3. Repercussion of power law index $(n)$ on thermal regime $(\theta \left(\eta \right)).$ |
Figures 4 and 5 depict the variation of permeable velocity $\left({f}_{0}\right)$ on velocity and temperature. A drop in velocity and augmentation in temperature is noticed for suction case. A decrease in velocity is observed due to the increase in temperature results in decrease in wave velocity owing to suction.
Figure 4. Repercussion of permeable velocity $({f}_{0})$ on the velocity regime $(f^{\prime} \left(\eta \right)).$ |
Figure 5. Repercussion of permeable velocity $({f}_{0})$ on thermal regime $(\theta \left(\eta \right)).$ |
Figures 6 and 7 depict the conduct of the squeezed flow index $\left(b\right)$ on $f^{\prime} (\eta )$ and $\theta \left(\eta \right).$ Clearly an increment in $b$ causes depletion in velocity and temperature. Figures 8 and 9 show that fluid velocity swells over a sensor surface which is admitted with the substantial increase in temperature in the boundary as a result of the increase in thermal diffusivity.
Figure 6. Repercussion of squeezed flow parameter $(b)$ on the velocity regime $(f^{\prime} \left(\eta \right)).$ |
Figure 7. Repercussion of squeezed flow parameter $(b)$ on the thermal regime $(\theta \left(\eta \right)).$ |
Figure 8. Repercussion of Magnetic parameter $(M)$ on the velocity regime $(f^{\prime} \left(\eta \right)).$ |
Figure 9. Repercussion of magnetic parameter $(M)$ on the thermal regime $(\theta \left(\eta \right)).$ |
Figures 10 and 11 reflect the Weissenberg number $\left(We\right)$ impact on velocity, temperature profiles. $We$ shows a negative impact on the flow regime for $n\gt 1$ causing a substantial rise in temperature of the fluid at the boundary. This nature of the $We$ is solely characterized by $n,$ On magnification of $n,$ fluid exhibits shear-thickening trait which shows the declination of velocity with a rise in $We.$
Figure 10. Repercussion of Weissenberg number $(We)$ on the velocity regime $(f^{\prime} \left(\eta \right)).$ |
Figure 11. Repercussion of the Weissenberg number $(We)$ on the thermal regime $(\theta \left(\eta \right)).$ |
$Rd$ and $Q$ show similar behavior in the thermal boundary layer (figures 12 and 13 respectively), but the effect of $Q$ has a greater impact in the boundary causing an exponential enhancement in the fluid’s temperature for fixed $Rd.$ When we rise $Q$ there is enrichment of heat production in the interior of the channel. As a consequence, thermal energy enhances and there is a rise in temperature.
Figure 12. Repercussion of the radiation parameter $(Rd)$ on the thermal regime $(\theta \left(\eta \right)).$ |
Figure 13. Repercussion of the heat source parameter $(Q)$ on the thermal regime $(\theta \left(\eta \right)).$ |
A three-dimensional view on the consequence of $\sqrt{R{e}_{x}}{C}_{f}$ against the squeezed flow parameter $b$ for various ranges of $M$ is explained in figure 14. $\sqrt{R{e}_{x}}{C}_{f}$ magnifies for a different value of the squeezed flow parameter $\left(b\right)$ for the increase in magnetic field strength. A variation of $\sqrt{R{e}_{x}}{C}_{f}$ against $b$ and We for varying values of We and M respectively is shown in figures 15 and 16, respectively. With an increased $We$ an increase in the skin friction coefficient is observed for the increase in the squeezed flow parameter.
Figure 14. Repercussion of squeezed flow parameter $(b)$ on the coefficient of skin friction against the magnetic parameter $(M).$ |
Figure 15. Repercussion of Weissenberg number $(We)$ on the coefficient of skin friction against the squeezed flow parameter $(b).$ |
Figure 16. Repercussion of Weissenberg number $(We)$ on coefficient of skin friction against the Magnetic parameter $(M).$ |
Figures 17–19 show the streamlines of the stated flow. Three different sets of streamlines for various ranges of $M\,\,$as such $0.2,\,1.2,\,2$ are drawn to illustrate the flow characteristics which shows that an increase in $M$ drags the streamlines towards the $x$-axis. The rationale for the stretching of streamlines towards the $x$-axis is the increased presence of the applied magnetic field. Table 1 shows the variation of skin friction coefficient for different values of $We$ and $n.$
Figure 17. Streamlines of the flow when $M=0.1.$ |
Figure 18. Streamlines of the flow when $M=1.2.$ |
Figure 19. Streamlines of the flow when $M=2.$ |
Table 1. Variation of skin friction coefficient for various values of $We$ and $n.$ |
| $We$ | $n$ | ${C}_{f}$ |
|---|---|---|
| 0.1 | 1.2 | 1.263 572 881 |
| 0.5 | 1.2 | 1.274 7735 |
| 0.9 | 1.2 | 1.298 086 614 |
| 0.1 | 1.5 | 1.264 299 498 |
| 0.5 | 1.5 | 1.290 753 42 |
| 0.9 | 1.5 | 1.339 953 045 |
| 0.1 | 1.8 | 1.265 022 319 |
| 0.5 | 1.8 | 1.305 301 177 |
| 0.9 | 1.8 | 1.374 548 367 |
Conclusion
This work focused on analyzing the Carreau fluid allowed to flow above a sensor surface when the flow is affected by a magnetic field and liable to radiation. The results are obtained by an effective numerical method and the plots are explained. A summary of them is given as follows:
| • | Velocity is recorded as declining by enhancing $n,\,{f}_{0},\,b,\,We$ and velocity is amplified by rising $M.$ |
| • | It is noted that the parameters like $n,\,{f}_{0},\,M,\,We,\,Rd\,{\rm{a}}{\rm{n}}{\rm{d}}\,Q$ maximize temperature whereas it is minimized by $b.$ |
| • | On escalating $b$ on the flow the drag force is magnified when compared to the magnetic parameter. |
| • | For a different range of $We$ the drag force is augmented when examined against $b.$ |
| • | Streamlines for the flow on various strengths of $M$ enlightens the fluid particles path. |
| • | The benefit of flow driven by a penetrable sensor surface finds its applications in numerous grounds including biomedical, chemical sensing and engineering. |
Acknowledgments
The authors are thankful to the Department of Science and Technology, Government of India under DST-FIST Program (Ref No. SR/FST/MS-I/2018–2023) for supporting the Department of Mathematics, Kuvempu University, Shankaraghatta.
Declaration of conflict of interest
All the authors acknowledge that there is no conflict interest to this article.