*Magnetic Dipole*

The magnetic field features impacted the ferrofluid flow with magnetic dipole effects detected mostly by magnetic scalar potential Φ1, as given in Equation (8):

$$\Phi\_1 = \frac{\gamma}{2\pi} \frac{x}{x^2 + (s+c)^2},\tag{8}$$

Considering *Hx* and *Hs* to be the components of magnetic field, with *γ* as the magnetic field strength at the source, see Equations (9) and (10):

$$H\_{\mathbf{x}} = -\frac{\partial \Phi\_1}{\partial \mathbf{x}} = \frac{\gamma}{2\pi} \frac{\mathbf{x}^2 - (\mathbf{s} + d)^2}{[\mathbf{x}^2 + (\mathbf{s} + d)^2]^2} \tag{9}$$

$$H\_{\rm s} = -\frac{\partial \Phi\_1}{\partial \mathbf{s}} = \frac{\gamma}{2\pi} \frac{2\mathbf{x}(\mathbf{s} + d)}{[\mathbf{x}^2 + (\mathbf{s} + d)^2]^2}. \tag{10}$$

As the strength of a magnetic body is normally approximately equal to the *Hx* and *Hs* gradients, it is therefore given as in (11):

$$H = \sqrt{H\_x^2 + H\_s^2}.\tag{11}$$

Equation (12) displays the approximate linearized relation of the magnetization *M* as function of temperature *T*,

$$M = -K\_1(T\_\infty - T),\tag{12}$$

with *K*<sup>1</sup> identified as the ferromagnetic coefficient. Figure 1 depicts the physical configuration of the heated ferrofluid.

Considering the following transformations, given the stream function as Φ(*x*,*s*), such that

$$u = \frac{1}{r} \frac{\partial \Phi}{\partial s} \qquad \qquad \qquad w = -\frac{1}{r} \frac{\partial \Phi}{\partial x} \tag{13}$$

with <sup>Φ</sup>(*x*,*s*) = *<sup>ν</sup>frRax <sup>f</sup>*(*ζ*) and *Rax* is the Rayleigh number given by *Rax* <sup>=</sup> *<sup>ρ</sup> <sup>f</sup> <sup>β</sup><sup>T</sup> <sup>g</sup>*(*Tw*−*T*∞)*x*3*cos*(*α*) *ν*2 *f* , therefore, the following are given:

$$\begin{aligned} u &= \frac{\nu\_f R a\_x^{\frac{1}{2}}}{\ge} f'(\zeta), \ w = -\frac{\nu\_f R a\_x^{\frac{1}{2}}}{\ge} \left(\zeta f'(\zeta) - f(\zeta)\right), \ \zeta &= \frac{s}{\chi} R a\_x^{\frac{1}{2}}, \\ \zeta (T\_w - T\_\infty)\theta(\zeta) &= (T - T\_\infty), \ (\mathbb{C}\_w - \mathbb{C}\_\infty)\phi(\zeta) = (\mathbb{C} - \mathbb{C}\_\infty), \ (\mathbb{N}\_w - \mathbb{N}\_\infty)\chi(\zeta) = (\mathbb{N} - \mathbb{N}\_\infty). \end{aligned} \tag{14}$$

**Figure 1.** A picture scheme of the problem.

Taking *r* to be approximately the cone local radius, for the thermal boundary layer becoming thin, it will be along the *x* coordinate with *r* = *xsin*(*α*).

By using the above transformations, Equation (1) will be satisfactory, and Equations (2)–(5) will be

$$f''' - P\_1 f' + f f'' - F\_r f'^2 + \frac{2\beta}{(\zeta + a\_1)^4} (1 + \theta) + N\_c \phi + Ra\_b \chi = 0,\tag{15}$$

$$\begin{aligned} \left(1+Rd\right)\theta'' + Prf\theta' + Nb\phi'\theta' + Nt(\theta')^2 + \frac{2Pr\beta\lambda(\theta-\varepsilon)(f-\mathbb{f}\_{\varepsilon}'f')}{(\mathbb{f}+da\_1)^3} \\ + Pr\beta\lambda(\theta-\varepsilon)\left[\frac{2f'}{(\mathbb{f}+a\_1)^4} + \frac{4(\mathbb{f}\_{\varepsilon}'-f)}{(\mathbb{f}+a\_1)^5}\right] + Pr\varepsilon c(f')^2 = 0,\end{aligned} \tag{16}$$

$$
\phi'' + \frac{\text{Nt}}{\text{N}b} \theta'' + \text{Scf}\phi' - \delta \text{Sc}\phi = 0,\tag{17}
$$

$$
\chi^{\prime\prime} - \text{Pe}\left[\phi^{\prime}\chi^{\prime} + \phi^{\prime\prime}\chi + \delta\_n\phi^{\prime\prime}\right] + Lbf\chi^{\prime} = 0. \tag{18}
$$

Moreover, with the new boundary conditions:

$$\begin{aligned} f' = 1, f = \mathbb{S}, \theta &= 1, \, \text{N}b\phi' + \text{N}t\theta = 0, \, \text{ $\chi = 1$ } & \text{at} \, \text{ $\tilde{\chi} = 0$ }, \\\ f' &\to 0, \, \theta \to 0, \, \phi \to 0, \, \text{ $\chi \to 0$ }, \, a \,\text{s} \, \text{ $\tilde{\chi} \to \infty$ } \end{aligned} \tag{19}$$

where *α*<sup>1</sup> is dimensionless distance, *Nc* is the ratio due to buoyancy force, *Rab* is the bioconvection Rayleigh number, *β* is the ferrohydrodynamic interaction parameter, *ε* the Curie temperature, *λ* is the heat dissipation parameter, *S* is the heat generation/absorption parameter (*S* > 0 for suction and *S* < 0 for injection), *Nb* and *Nt* are the Brownian motion and thermophoresis parameters, the Prandtl number is *Pr*, the Eckert number is *Ec*, the radiation parameter is *Rd*, the local inertia parameter is *Fr*, the chemical reaction parameter is *δ*, the porosity parameter is *P*1, the Schmidt number is *Sc*, the Lewis number of bioconvection is *Lb*, the Peclet number *Pe*, *δ<sup>n</sup>* is the bioconvection constant, and quantities are defined by

*<sup>S</sup>* <sup>=</sup> *Wwx <sup>ν</sup><sup>f</sup> Ra* <sup>1</sup> 2 , *Nc* <sup>=</sup> *<sup>g</sup>βC*(*Cw* <sup>−</sup> *<sup>C</sup>*∞)*x*3*cos*(*α*1) *ν*2 *<sup>f</sup> Rax* , *Rab* <sup>=</sup> *<sup>g</sup>βN*(*Nw* <sup>−</sup> *<sup>N</sup>*∞)*x*3*cos*(*α*) *ν*2 *<sup>f</sup> Rax* , *<sup>P</sup>*<sup>1</sup> <sup>=</sup> *<sup>μ</sup><sup>f</sup> k*∗ *o* , *Li* <sup>=</sup> *Cb k*∗ *o* , *<sup>β</sup>* <sup>=</sup> *γλoKρf*(*Tw* <sup>−</sup> *<sup>T</sup>*∞) 2*πμ*<sup>2</sup> *f* , *Pr* <sup>=</sup> *<sup>μ</sup><sup>f</sup> cp k f* , *Ec* <sup>=</sup> *<sup>W</sup>*<sup>2</sup> *w cp*(*Tw* − *T*∞) , *<sup>λ</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>2</sup> *f <sup>ρ</sup>f*(*Tw* <sup>−</sup> *<sup>T</sup>*∞)*Ra* <sup>3</sup> 4 *x* , (20) *<sup>δ</sup>* <sup>=</sup> *<sup>K</sup>*1*x*<sup>2</sup> *<sup>ν</sup><sup>f</sup> Ra* <sup>1</sup> 2 *x* , *Rd* <sup>=</sup> <sup>16</sup>*σ*∗*T*<sup>3</sup> ∞ 3*k*∗*k <sup>f</sup>* , *Sc* <sup>=</sup> *<sup>ν</sup><sup>f</sup> DB* , *Pe* <sup>=</sup> *bWc Dn* , *Lb* <sup>=</sup> *<sup>ν</sup><sup>f</sup> Dn* , <sup>=</sup> *<sup>T</sup>*<sup>∞</sup> *T*<sup>∞</sup> − *Tw* , *<sup>d</sup>* <sup>=</sup> *aRax <sup>x</sup>* , *Nt* <sup>=</sup> *<sup>τ</sup>DT*(*Tw* <sup>−</sup> *<sup>T</sup>*∞) *T*∞*α<sup>f</sup>* , *Nb* <sup>=</sup> *<sup>τ</sup>DB*(*Cw* <sup>−</sup> *<sup>C</sup>*∞) *αf* , *<sup>δ</sup><sup>n</sup>* <sup>=</sup> *<sup>N</sup>*<sup>∞</sup> *Nw* − *N*<sup>∞</sup> .

The local Nusselt, Sherwood, and local Density expressions, as well as the coefficient of skin friction, will be computed by

$$\mathcal{C}\_{f} = \frac{2\tau\_{\text{w}}}{\rho\_{f}\mathcal{W}\_{\text{w}}^{2}}, \text{Nu}\_{\text{x}} = \frac{q\_{\text{h}}\mathbf{x}}{k\_{f}(T\_{\text{w}} - T\_{\text{co}})}, \text{Sh}\_{\text{x}} = \frac{q\_{\text{m}}\mathbf{x}}{D\_{\text{B}}(\mathbb{C}\_{\text{w}} - \mathbb{C}\_{\text{co}})}, \text{Sn}\_{\text{x}} = \frac{q\_{\text{n}}\mathbf{x}}{D\_{\text{n}}(N\_{\text{w}} - N\_{\text{co}})}, \tag{21}$$

$$\tau\_w = \mu\_f u\_s|\_{s=0\prime} \, q\_h = [-k\_f T\_s + q\_r]|\_{s=0\prime} \, q\_m = -D\_B C\_s|\_{s=0\prime} \, q\_n = -D\_n N\_s|\_{s=0\prime} \tag{22}$$

$$Ra\_x^{\frac{1}{4}}\mathbb{C}\_f = 2f^{\prime\prime}(0),\\Nu = -Ra\_x^{\frac{1}{4}}(1+Rd)\theta^{\prime}(0),\\Sh = -Ra\_x^{\frac{1}{4}}\phi^{\prime}(0),\\Sn = -Ra\_x^{\frac{1}{4}}\chi^{\prime}(0). \tag{23}$$
