your second boundary condition is missing a factor $u-u_b$:

$$a\frac{\partial u}{\partial n} - \gamma(u-u_b)(\mathbf v \cdot \mathbf n) + \beta(u - u_b) = 0$$

the coefficients $a$ and $\gamma$ are the same as in the diffusion-convection equation,

$$\frac{\partial u}{\partial t} - a\Delta u + \gamma \mathbf v \cdot \nabla u = f.$$

see, for example, The convective-diffusion equation and its use in building physics (2000).

your second boundary condition is missing a factor $u-u_b$:

$$a\frac{\partial u}{\partial n} - \gamma(u-u_b)(\mathbf v \cdot \mathbf n) + \beta(u - u_b) = 0$$

the coefficient $\beta$ gives the strength of the heat transfer at the boundary; the coefficients $a$ and $\gamma$ are the same as in the diffusion-convection equation,

$$\frac{\partial u}{\partial t} - a\Delta u + \gamma \mathbf v \cdot \nabla u = f.$$

see, for example, The convective-diffusion equation and its use in building physics (2000).

your second boundary condition is missing a factor $u-u_b$:

$$a\frac{\partial u}{\partial n} - \gamma(u-u_b)(\mathbf v \cdot \mathbf n) + \beta(u - u_b) = 0$$

see, for example, The convective-diffusion equation and its use in building physics (2000).

your second boundary condition is missing a factor $u-u_b$:

$$a\frac{\partial u}{\partial n} - \gamma(u-u_b)(\mathbf v \cdot \mathbf n) + \beta(u - u_b) = 0$$

the coefficients $a$ and $\gamma$ are the same as in the diffusion-convection equation,

$$\frac{\partial u}{\partial t} - a\Delta u + \gamma \mathbf v \cdot \nabla u = f.$$

see, for example, The convective-diffusion equation and its use in building physics (2000).