Riemann surface
matlab

Problem 1.2: A cylindrical storage tank of diameter $D$ contains a liquid at depth (or head) $h(x, t)$. Liquid is supplied to the tank at a rate of $q_i\left(\mathrm{~m}^3 /\right.$ day) and drained at a rate of $q_0\left(\mathrm{~m}^3 /\right.$ day). Use the principle of conservation of mass to arrive at the governing equation of the flow problem.

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Solution: The conservation of mass requires time rate of change in mass = mass inflow – mass outflow
The above equation for the problem at hand becomes
$$\frac{d}{d t}(\rho A h)=\rho q_i-\rho q_0 \quad \text { or } \quad \frac{d(A h)}{d t}=q_i-q_0$$
where $A$ is the area of cross section of the tank $\left(A=\pi D^2 / 4\right)$ and $\rho$ is the mass density of the liquid.

Problem 1.3: Consider the simple pendulum of Example 1.3.1. Write a computer program to numerically solve the nonlinear equation (1.2.3) using the Euler method. Tabulate the numerical results for two different time steps $\Delta t=0.05$ and $\Delta t=0.025$ along with the exact linear solution.

Solution: In order to use the finite difference scheme of Eq. (1.3.3), we rewrite (1.2.3) as a pair of first-order equations
$$\frac{d \theta}{d t}=v, \quad \frac{d v}{d t}=-\lambda^2 \sin \theta$$
Applying the scheme of Eq. (1.3.3) to the two equations at hand, we obtain
$$\theta_{i+1}=\theta_i+\Delta t v_i ; \quad v_{i+1}=v_i-\Delta t \lambda^2 \sin \theta_i$$
The above equations can be programmed to solve for $\left(\theta_i, v_i\right)$. Table P1.3 contains representative numerical results.

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