wall A liquid layer with thickness & flows along a wall and is in contact with a gas. The gas dissolves in the liquid and undergoes a first-order chemical reaction. The

state is steady. It is assumed that the liquid film is fully developed, implying that is constant and that the speed v does not depend on z. gas The concentration of gas in the liquid is denoted by c. The model for c is based on the strongly simplified equation de 8c = D. -ke, 0≤y≤d, z>0. Oy² (1) dr In here D is the diffusion coefficient and k a measure for the chemical reaction. The term in the lefthand side of (1) represents transport due to convection in the layer. Diffusion is assumed to be negligible in z-direction. Upon scaling, the model reads -1²-²²- Clym1 = 1, ==0 Co= 0, y 1. 1 0≤ y ≤1, z>0, Given: 3 = 4. In model (2), the liquid film spans the y-interval [0, 1]. The wall is at y = 0, which is impermeable for gas. The liquid-gas interface is at y = 1. There, e has the equilibrium value c = 1. The differential equation in (2) is a variant of the instationary heat equation; z plays the role of time. a. For z → ∞ a 'stationary state will develop, i.e., a state with = 0. Determine, analytically, the "stationary' solution estat(y) of (2). Make a graph of estat- On which interval does it hold: estat > 0.1? du dr The flow in the liquid film will satisfy the no-slip condition, Le., v = 0 at y = 0. Away from the wall, the flow can freely develop and will have a nearly constant speed. We denote this speed by . In a first approximation, we may assume that the transition from v = 0 at the wall to = takes place over a distance < 1. From sub-problem a it appears that for 0 0. The discretization of (2) in y-direction is done through a second-order central finite-difference formula. The mesh size is denoted by Ay and the resulting semi-discrete system is written as = Aw+r./nb. Determine A and r. Give details about the implementation of the Neumann boundary condition. c. Show that the eigenvalues A; of A are real and that it holds X; < 0. For the integration in 2-direction of (4) we take the 6-method, given by Un+1 = Un + Ar[(1 – 0) (Aun +™n) +0(AUn+1+n+1)], @[0,1]. In here Az is the step size in 2-direction. d. Determine the order of the local truncation error of (5). We choose 0 = 4. e. Show that (5) is unconditionally stable for 04. Is the method super-stable? f. What is the order of the global discretization error of (5) for 8 = ? The absorption of the gas is determined by a = ly=1. == g. Give a first-order and a second-order accurate (one-sided) finite-difference formula for the computa- tion of a. The corresponding numerical approximations of a are denoted by a₁ and 02. h. Choose Ay = 0.05, Ar = 0.02 and 6 = 1. Plot in a single graph the numerical solution (as a function of y) for 2 =nAr with n = 5, 10, 15, 20, 25. Make tables of an and 02. Discuss the results. i. Choose Az = 0.02, = 1 and define L = 20Ar. We investigate the accuracy in a at z = L. Choose Ay = 0.1, 0.05,0.025. Make tables of an and a₂ for 2 = L. Discuss the accuracy of on and 02. For this, assume that the global discretization error can be written as CAy". Estimate C and p. Discuss the results. j. Add the software you wrote as appendix/appendices to your report.