mixpap/lavalamp_simulation

Simulation of the dynamics of a lava lamp

lavalamp_simulation

Simulation of the dynamics of a lava lamp

lavalamp_simulation's Issues

Analytical calculation of the environmental temperature

We suppose the following: (a) The lamp is cylindrical and fully isolated from the environment except the cap and the bottom of the cylinder.
(b) The bubble contains much less energy than its environment.
(c) There is no fluid movement

$$ \frac{\partial u}{\partial t} = a\nabla^2u $$ and due to cylidrical symmetry, we employ cylindrical coordinates $\rho,\theta,z$. If we further suppose that the heat transfer in all other axes different than z is negligible, we have $u(\rho,\theta,z,t) = u(z,t) $. $$ \frac{\partial u}{\partial t} = a\frac{\partial^2 u}{\partial^2 z} $$ Next, we suppose that $u(z,t) = T(t)U(z)$ and thsi results: $$ \frac{dT}{dt}T^{-1} = a\frac{d^2 U}{d^2 z} U^{-1}$$ so $$Τ = Τ_0e^{c_1 t} $$ where $c_1$ 𝑐1 is a separation constant and includes a. For the spatial part, we suppose a solution of the form: $$U(z) \sim e^{izc_2} $$ Thus, $$U(z) = c_A e^{izc_2} + c_B e^{-izc_2}$$ Requiring that the phenomenon remains unchangeable over time, we got $c_1 = 0$. $$ u(0,t) = T_0$$ so $$ c_A + c_B = 1$$ Employing the boundary condition $u(h,t) = T_h $, we got $$ u(h,t) = T_0 (-ic_Asin(hc_2) + cos(hc_2))$$ Also, we need to have real solutions, so $c_A = 0$ From the requirement $u(h,t) = T_h$ we obtain $c_2 = 1/hcos^{-1}(T_h/T_0)$ Putting all together, $$ u(z,t) = T_0cos\left(\frac{z}{h} cos^{-1}(T_h/T_0)\right)$$