*2.2. Governing Equations and Boundary Conditions*

By introducing the Boussinesq approximation, the governing equations for the heat transfer and fluid flow can be written as follows:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}$$

$$\varepsilon \left( u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial y} \right) = -\varepsilon \frac{1}{\rho\_{\rm nf}} \frac{\partial p}{\partial \mathbf{x}} + \varepsilon v\_{\rm nf} \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial y^2} \right) - \varepsilon^2 \frac{\nu\_{\rm nf}}{K} u \tag{2}$$

$$\left(u\frac{\partial v}{\partial \mathbf{x}} + v\frac{\partial v}{\partial y}\right) = -\varepsilon \frac{1}{\rho\_{\rm nf}} \frac{\partial p}{\partial y} + \varepsilon v\_{\rm nf} \left(\frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial y^2}\right) - \varepsilon^2 \frac{\nu\_{\rm nf}}{K} v + \varepsilon^2 g\beta (T - T\_0) \tag{3}$$

$$\left(\rho c\_{\rm P}\right)\_{\rm nf} \left(u \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y}\right) = k\_{\rm mnf} \left(\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2}\right) \tag{4}$$

where (*x*, *y*) are the Cartesian coordinates of the geometry, (*u*, *v*) are the velocity components, *T* and *p* are the temperature and pressure, respectively, *ρ*, (*ρc*p), *ν*, *β* and *α* denote the density, heat capacitance, kinematic viscosity, thermal expansion coefficient, and thermal diffusion, respectively, *k* is the thermal conductivity, *ε* is the porosity, and *K* is the medium permeability. The subscripts nf and mnf designate nanofluid and porous medium filled with nanofluid.

The effective thermal conductivity of the porous medium filled with nanofluid can be as modeled as:

$$k\_{\rm mnf} = \varepsilon k\_{\rm nf} + (1 - \varepsilon)k\_{\rm s.} \tag{5}$$

Using the following dimensional variables,

 $\left(\mathbf{X},\mathbf{Y}\right) = \frac{\left(\mathbf{x},\mathbf{y}\right)}{\left(r\_{\mathrm{o}}-r\_{\mathrm{l}}\right)'} \left(\mathbf{U},\mathbf{V}\right) = \frac{\left(\mathbf{u},\mathbf{p}\right)\left(r\_{\mathrm{o}}-r\_{\mathrm{l}}\right)}{a\_{\mathrm{muf}}},\text{ }\mathbf{P} = \frac{p\left(r\_{\mathrm{o}}-r\_{\mathrm{l}}\right)^{2}}{\rho\_{\mathrm{n}}a\_{\mathrm{mnf}}},\text{ }\theta = \frac{T-T\_{\mathrm{o}}}{T\_{\mathrm{l}}-T\_{\mathrm{o}}},\text{ }a\_{\mathrm{mnf}} = \frac{k\_{\mathrm{mnf}}}{\left(\rho\_{\mathrm{p}}\right)\_{\mathrm{nf}}},\text{ }\theta = \frac{T\_{\mathrm{nf}}}{T\_{\mathrm{n}}},\text{ }\mathbf{R}a = \frac{k}{\left(\rho\_{\mathrm{p}}\right)\_{\mathrm{nf}}},\text{ }\theta = \frac{T\_{\mathrm{n}}}{T\_{\mathrm{n}}}$  $\mathbf{R}a = \frac{\mathbf{g}\theta\left(T\_{\mathrm{l}}-T\_{\mathrm{o}}\right)\left(r\_{\mathrm{o}}-r\_{\mathrm{l}}\right)^{3}}{v\_{\mathrm{n}}a\_{\mathrm{mnf}}},\text{ }\mathbf{Pr} = \frac{v\_{\mathrm{nf}}}{a\_{\mathrm{mnf}}},\text{ }\mathbf{Da} = \frac{k}{\mathrm{L}^{2}}$ 

The governing Equations (1)–(4) reduce to a dimensionless form:

$$\frac{\partial \mathcal{U}}{\partial X} + \frac{\partial \mathcal{U}}{\partial Y} = 0 \tag{6}$$

$$
\varepsilon \mathcal{U} \frac{\partial \mathcal{U}}{\partial X} + V \frac{\partial \mathcal{U}}{\partial Y} = -\varepsilon \frac{\partial P}{\partial X} + \varepsilon Pr \left( \frac{\partial^2 \mathcal{U}}{\partial X^2} + \frac{\partial^2 \mathcal{U}}{\partial Y^2} \right) - \varepsilon^2 \frac{Pr}{Da} \mathcal{U} \tag{7}
$$

$$\mathcal{U}\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = -\varepsilon \frac{\partial P}{\partial X} + \varepsilon Pr\left(\frac{\partial^2 V}{\partial X^2} + \frac{\partial^2 V}{\partial Y^2}\right) - \varepsilon^2 \frac{Pr}{Da}U + \varepsilon^2 Ra Pr\theta \tag{8}$$

$$
\hbar \Omega \frac{\partial \theta}{\partial X} + V \frac{\partial \theta}{\partial Y} = \frac{1}{Pr} \left( \frac{\partial^2 \theta}{\partial X^2} + \frac{\partial^2 \theta}{\partial Y^2} \right) \tag{9}
$$

where (*X*, *Y*) are the dimensionless Cartesian coordinates of the geometry, (*U*, *V*) are the dimensionless velocity components, and *θ* and *P* are the temperature and pressure, respectively. *Ra*, *Pr*, and *Da* denote respectively the Raleigh number, Prandtl number, and Darcy number.

The boundary conditions for this problem are as follows:

On the outer cylinder surface:

$$
\theta = 0.\tag{10}
$$

On the inner cylinder surface:

$$
\theta = 1.\tag{11}
$$

On the outer and inner cylinder surfaces:

$$
\mathcal{U} = V = 0.\tag{12}
$$

The local and overall heat transfer rate along the inner cylinder surface are estimated using the local Nusselt number and average Nusselt number, respectively:

$$Nu\_{\rm loc} = \left. \frac{\partial \theta}{\partial N} \right|\_S = \sqrt{\left(\frac{\partial \theta}{\partial X}\right)^2 + \left(\frac{\partial \theta}{\partial Y}\right)^2} \tag{13}$$

$$Nu\_{\text{avg}} = \frac{1}{S} \int\_0^s Nu\_{\text{loc}} \tag{14}$$

where *S* is the non-dimensional length along the inner cylinder surface.
