Two-dimensional dissipative flux transport properties of a hybrid nanofluid with Joule heating and thermal radiation

The two-dimensional magnetic flux of viscous hybrid nanofluid is considered along a curved surface having a radius (R) and as a function of Joule heating and viscous dissipation, as shown in Fig. 1. In addition, the flow rate depends on thermal radiation, surface temperature (({T}_{w})) and heat flow (({q}_{w })) as well as the hybrid nanofluid consists of alumina ((A{l}_{2}{O}_{3})) and copper ((Cu)) nanoparticles associated with the base fluid ethylene glycol (({C}_{2}{H}_{6}{O}_{2})). The ambient temperature is taken as ({T}_{infty})where the stretch and the far-field velocity are respectively taken to be (u=cs)and (uto {u}_{e}left(sright)=as). The flow is directed according to the coordinates (s) and the magnetic flux (({B}_{0})) is considered according to the coordinates (r) which is taken in the normal direction of the tangential vector.

Figure 1

The equations governing the boundary layer and the associated boundary conditions are described in the light of said assumptions as shown below34,35,36.

$$frac{partial }{partial r}left[(r+R)vright]=-Rfrac{partial u}{partial s},$$


$$frac{1}{{rho }_{hnf}}frac{partial p}{partial r}-frac{{u}^{2}}{r+R}=0,$ $


$$frac{1}{{rho }_{hnf}}frac{R}{r+R}frac{partial p}{partial s}={v}_{hnf}left( frac{{partial }^{2}u}{partial {r}^{2}}+frac{1}{r+R}frac{partial u}{partial r}-frac {u}{{left(r+Rright)}^{2}}right)-vfrac{partial u}{partial r}-frac{Ru}{r+R}frac {partial u}{partial s}-frac{uv}{r+R}+frac{{sigma }_{nhf}{{B}_{0}}^{2}}{{ rho }_{hnf}}u,$$


$$frac{{k}_{hnf}}{{left(rho {C}_{p}right)}_{hnf}}left(frac{{partial }^{2} T}{partial {r}^{2}}+frac{1}{r+R}frac{partial T}{partial r}right)-left(vfrac{partial T }{partial r}+frac{Ru}{r+R}frac{partial T}{partial s}right)+frac{{sigma }_{hnf}}{{left( rho {C}_{p}right)}_{hnf}}{{B}_{0}}^{2}{u}^{2}+frac{{mu }_{nf} }{{left(rho {C}_{p}right)}_{hnf}}{left(frac{partial u}{partial r}-frac{u}{r+R }right)}^{2}-frac{1}{r+R}frac{partial {q}_{r}}{partial r}=0.$$


The term ({q}_{r}) in eq. (4) refers to radiative heat flux. Given the Rosseland relation, this term can be defined as37.38.

$${q}_{r}=frac{-4{sigma }^{*}}{3{k}^{*}}frac{partial {T}^{4}}{partial r}$$


In eq. (5), the nonlinear term ({T}^{4}) can be simplified by applying the Taylor series expansion on ({T}_{infty}). Assuming the elimination of higher order terms, we can obtain the following equation.

$${T}^{4}approxeq 4{T}_{infty }^{3}T-3{sigma }^{*}{T}_{infty}^{4}$$


The association of eq. (6) and eq. (5) leads to the following equation.

$${q}_{r}=frac{-16{sigma }^{*}{T}_{infty}^{3}}{3{k}^{*}}frac{ partial T}{partial r}$$


In eq. (4), replace the values ​​of ({q}_{r}) presented in Eq. (7) gives the following equation.

$$frac{{k}_{hnf}}{{left(rho {C}_{p}right)}_{hnf}}left(frac{{partial }^{2} T}{partial {r}^{2}}+frac{1}{r+R}frac{partial T}{partial r}right)-left(vfrac{partial T }{partial r}+frac{Ru}{r+R}frac{partial T}{partial s}right)+frac{1}{r+R}frac{16{sigma }^{*}{T}_{infty }^{3}}{3{k}^{*}}frac{{partial }^{2}T}{partial {r}^{2 }}+frac{{mu }_{hnf}}{{left(rho {C}_{p}right)}_{hnf}}{left(frac{partial u}{ partial r}-frac{u}{r+R}right)}^{2}+frac{{sigma }_{hnf}}{{left(rho {C}_{p} right)}_{hnf}}{{B}_{0}}^{2}{u}^{2}=0.$$


The following equations. (9–11) represents the boundary conditions associated with Eqs. (1–3, 8).

$$left.begin{array}{c}u=cs,v={v}_{w}=0, ,,at,, r=0, uto {u} _{e}left(sright)=as, frac{partial u}{partial r}to 0, ,,as,, rto infty .end{array} right}$$


I. Prescribed Surface Temperature (PST)

$$T={T}_{w} ,,at ,,r=0,,mathrm{and},, Tto {T}_{infty } , ,as,, rto infty$$


II. Prescribed heat flux (PHF)

$$-kfrac{partial T}{partial r}={q}_{w }=D{left(frac{s}{l}right)}^{2},, at,, r=0, ,,and,, Tto {T}_{infty } ,,as,, rto infty$$


The boundary conditions in the two equations above. (10) and (11) refer to binary heating processes, where ({{T}_{w}>T}_{infty}) and (D) is constant.

Table 1 specifies the thermophysical characteristics of the hybrid nanofluid.

Table 1 Mathematical formulation of the thermophysical properties of ({C}_{2}{H}_{6}{O}_{2}-A{l}_{2}{O}_{3}) and (Cu-{Al}_{2}{O}_{3}/{C}_{2}{H}_{6}{O}_{2})39.

Using the following similarity transformations40the governing equations and boundary conditions can be converted to dimensionless form.

$$left.begin{array}{c}u={u}_{e}{f}{^{prime}}left(eta right)=as{f}{^{prime }}left(eta right), v=-frac{R}{r+R}sqrt{frac{{{v}_{f}u}_{e}}{s}}f left(eta right)=-frac{R}{r+R}sqrt{{av}_{f}}fleft(eta right), eta =sqrt{ frac{{u}_{e}}{{sv}_{f}}}r=sqrt{frac{a}{{v}_{f}}}r, p={rho }_{ f}{{u}_{e}}^{2}Pleft(eta right), PST: theta left(eta right)=frac{T-{T}_{ infty }}{{T}_{w}-{T}_{infty }}, PHF: T={T}_{infty }+ frac{D}{k}{left(frac {s}{l}right)}^{2}sqrt{frac{{v}_{f}}{a}}gleft(eta right).end{array}right }.$$


Considering the above transformations, Eq. (1) is satisfied automatically, although Eqs. (2, 3, 8) and Eqs. (9–11) leads to the following equations.

$$frac{{rho }_{f}}{{rho }_{hnf}}frac{partial P}{partial eta }=frac{1}{eta +K}{ f{^{first}}}^{2},$$


$$frac{{rho }_{f}}{{rho }_{hnf}}frac{2K}{eta +K}P=frac{{v}_{hnf}}{{ v}_{f}}left(f{^{prime}}{^{prime}}{^{prime}}-frac{1}{{left(eta +Kright) }^{2}}f{^{prime}}+frac{1}{eta +K}f{^{prime}}{^{prime}}right)-frac{K} {eta +K}{left({f}{^{prime}}right)}^{2}+frac{K}{eta +K}f{f}^{{^{ first}}{^{prime}}}+frac{K}{{left(eta +Kright)}^{2}}f{f}{^{prime}}-{M} ^{2}frac{{sigma }_{hnf}}{{sigma }_{f}}frac{{rho }_{f}}{{rho }_{hnf}}{f }{^{first}},$$


$$frac{1}{Pr}frac{{k}_{hnf}}{{k}_{f}}frac{{left(rho {C}_{p}right)} _{f}}{{left(rho {C}_{p}right)}_{hnf}}(1+frac{4}{3}Rd)left(theta {^{ prime}}{^{prime}}+frac{1}{eta +K}theta {^{prime}}right)+frac{K}{eta +K}f{theta }{^{prime}}+{frac{{mu }_{hnf}}{{mu }_{f}}frac{{left(rho {C}_{p}right )}_{f}}{{left(rho {C}_{p}right)}_{hnf}}E}_{c}{left({f}^{{^{prime }}{^{prime}}}-frac{{f}{^{prime}}}{eta +K}right)}^{2}quad+{{M}^{2}E }_{c}frac{{left(rho {C}_{p}right)}_{f}}{{left(rho {C}_{p}right)}_{ hnf}}frac{{sigma }_{hnf}}{{sigma }_{f}}{left({f}{^{prime}}right)}^{2}=0. $$


In the eq. (13) and (14), the term (P) refers to pressure. This term can be omitted from both equations assuming the differentiation of Eq. (14) with regard to (eta) to arrive at the possible comparison. The resulting equations are given below.

$${f}^{iv}+frac{2}{eta +K}{f}^{{^{prime}}{^{prime}}{^{prime}}}- frac{1}{{left(eta +Kright)}^{2}}{f}^{{^{prime}}{^{prime}}}+frac{1}{{ left(eta +Kright)}^{3}}{f}{^{prime}}+frac{{v}_{f}}{{v}_{hnf}}left[frac{K}{eta +K}left(f{f}^{{^{prime}}{^{prime}}{^{prime}}}-{f}{^{prime}}{f}^{{^{prime}}{^{prime}}}right)+frac{K}{{left(eta +Kright)}^{2}}left(f{f}^{{^{prime}}{^{prime}}}-{{f}{^{prime}}}^{2}right)-frac{K}{{left(eta +Kright)}^{3}}f{f}{^{prime}}-{M}^{2}frac{{sigma }_{hnf}}{{sigma }_{f}}frac{{rho }_{f}}{{rho }_{hnf}}({f}^{{^{prime}}{^{prime}}}+frac{1}{eta +K}{f}{^{prime}})right]=0.$$


$$frac{1}{Pr}frac{{k}_{hnf}}{{k}_{f}}frac{{left(rho {C}_{p}right)} _{f}}{{left(rho {C}_{p}right)}_{nf}}(1+frac{4}{3}Rd)left(theta {^{ prime}}{^{prime}}+frac{1}{eta +K}theta {^{prime}}right)+frac{K}{eta +K}f{theta }{^{prime}}+{frac{{mu }_{hnf}}{{mu }_{f}}frac{{left(rho {C}_{p}right )}_{f}}{{left(rho {C}_{p}right)}_{hnf}}E}_{c}{left({f}^{{^{prime }}{^{prime}}}-frac{{f}{^{prime}}}{eta +K}right)}^{2}+{{M}^{2}E} _{c}frac{{left(rho {C}_{p}right)}_{f}}{{left(rho {C}_{p}right)}_{hnf }}frac{{sigma }_{nf}}{{sigma }_{f}}{left({f}{^{prime}}right)}^{2}=0.$ $


The associated dimensionless conditions are shown below.

$$left.begin{array}{c}fleft(0right)=0, {f}{^{prime}}left(0right)=lambda =frac{c} {a}, theta left(eta right)=1, {f}{^{prime}}left(eta right)=1, {f}^{{^{prime }}{^{prime}}}left(eta right)=0, theta left(eta right)=0,, as,, eta to infty . end {array}right}.$$


In terms of PHF, just (theta) is replaced by (g) in eq. (17) although the whole equation remains unchanged40. However, the boundary conditions are modified which are presented below.

$${g}{^{prime}}left(0right)=-1, gleft(eta right)=0 ,,as,, eta to infty$ $


The parameters appearing in Eqs. (16–19) are defined below

$$lambda =frac{c}{a},K=Rsqrt{frac{a}{{v}_{f}}} , {M}^{2}=frac{{sigma }_{f}{{B}_{0}}^{2}}{a{rho }_{f}}, Pr=frac{{nu }_{f}}{{alpha } _{f}}, {E}_{c}=frac{{u}^{2}}{{C}_{p}Delta T}=frac{{a}^{2}{s }^{2}}{{C}_{p}({T}_{w}-{T}_{infty })}, Rd=frac{4{sigma }^{*}{T }_{infty}^{3}}{{k}_{f}{k}^{*}}$$

The local Nusselt number (({Bad})) and the coefficient of skin friction (({C}_{f})) refers to properties of technical interest which can be illustrated as41.

$${C}_{fr}=frac{{tau }_{w}}{{frac{1}{2}rho }_{f}{{u}_{e}}^{ 2}(s)}, {Nu}_{L}=frac{s{q}_{w}}{{k}_{f}left({T}_{w}-{T}_ {infty }right)}.$$



$${tau }_{w}={mu }_{hnf}{left.left(frac{partial u}{partial r}-frac{u}{r+R} right)right|}_{r=0}, {q}_{w}=-{k}_{hnf}{left.frac{partial T}{partial r}right|}_ {r=0}.$$


In the above system (20), the term ({tau}_{w}) and ({q}_{w}) denotes respectively the shear stress and the heat flux through the surface which are defined above in (21).

Considering the use of Eq. (12) in system (20) leads to the dimensionless structure as shown below.

$$begin{aligned} ~C_{f} {text{~}} & = left( {Re_{s} } right)^{{frac{1}{2}}} {text{ ~}}C_{{fr}} = frac{{mu _{{hnf}} }}{{mu _{f} }}left{ {f^{primeprime}left( 0 right) – frac{lambda }{K}} right}, Nu & = left( {Re_{s} } right)^{{ – 1/2}} {text{ ~}}Nu_{L} {text{~}} = – frac{{k_{{hnf}} }}{{k_{f} }}left( {1 + frac{4}{3} Rd} right)theta ^{prime}left( 0 right). end{aligned}$$


where ({Re}_{s} =frac{a{s}^{2} }{{nu }_{f}}) represent the localized Reynolds number.

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