. 2022 Apr 7;12:5882. doi: 10.1038/s41598-022-09859-6

Table 4.

Boundary conditions.

At the inlet	At the outlet	At the substrate bottom surface	At the fluid–solid interface	Other outer surfaces
$u = u_{in}$ $v = w = 0$	$p = p_{0}$	***	$u = v = w = 0$	***
$T = T_{f, i n}$	***	***	$T_{s} = T_{f}$	***
*******	***	$q^{″} = - k_{s} \frac{\partial T_{s}}{\partial n} = c o n s t$	$k_{s} \frac{\partial T_{s}}{\partial n} = k_{f} \frac{\partial T_{f}}{\partial n}$	$q^{″} = - k_{s} \frac{\partial T_{s}}{\partial n} = 0$

At the inlet

At the outlet

At the substrate bottom surface

At the fluid–solid interface

Other outer surfaces

$u = u_{in}$

$v = w = 0$

p = p_{0}

***

u = v = w = 0

***

T = T_{f, i n}

***

T_{s} = T_{f}

***

***

***

q^{″} = - k_{s} \frac{\partial T_{s}}{\partial n} = c o n s t

k_{s} \frac{\partial T_{s}}{\partial n} = k_{f} \frac{\partial T_{f}}{\partial n}

q^{″} = - k_{s} \frac{\partial T_{s}}{\partial n} = 0

where u, v, and w represent the components of the velocity vector in the x, y, and z directions, $u_{in}$ is the fluid inlet velocity, $T_{f, i n}$ is the fluid inlet temperature, $p_{0}$ denotes the atmospheric pressure, q” indicates the heat flux exerted to the bottom surface of the substrate, $T_{s}$ , and $T_{f}$ are the solid and fluid temperatures, and n is the perpendicular direction to the surface.