Which one of these phenomena would the PDE utt = c2uxx be the appropriate governing equation for?A. Diffusion of one gas into anotherB. Oscillation of a tensioned stringC. Steady-state temperature distribution in an insulated steel bar

Task DetailsThroughout this assignment, all variables are non-dimensionalised. The heat transfer system can beapproximated by the following partial differential equation (PDE) in time and space:𝜕Θ𝜕𝑡 + 𝑐 𝜕Θ𝜕𝑥 − 𝑑 𝜕 ! Θ𝜕𝑥 ! = 0, 𝑥 ∈ [0, 1], 𝑡 ∈ [0, 3], (1)where Θ is a non-dimensional temperature, 𝑡 is time, 𝑥 is a spatial coordinate with inlet at 𝑥 = 0 andoutlet at 𝑥 = 1, 𝑐 is a convection speed, and 𝑑 is a positive thermal diffusivity. Boundary conditionsare specified below, and they are different for tasks 1 and 2. The following initial condition applies:Θ(𝑥, 𝑡 = 0) = 𝜏 cos(2𝜋 𝑘 𝑥) , (2)where 𝑘 is an integer wave number. The PDE defined by equations (1) and (2), should be solved by aMATLAB script and/or functions using a suitable Runge-Kutta time-integration scheme and finite-difference spatial discretization of 𝑥 (do not exceed 4th order in time and space).1. (a) Take 𝑑 = 0, 𝜏 = 1 and apply periodic boundary conditions in 𝑥 as well as convectionspeeds c ∈ [0.1,0.3]. Run a total of 𝑁 simulations for 𝑘 = 2, each one with a different c, andrepeat the same for 𝑘 = 3, then store the temperatures Θ(𝑡 = 3, 𝑥 = 1), respectively.Choose appropriate spatial and temporal step sizes, and hence CFL number, and keep thischoice for all simulations, respectively, and briefly explain this choice in the written report,considering numerical stability and accuracy.

In the solution of the heat equation, which describes the diffusion of temperature T(x,t) in a slender rod of length L, you are given the boundary conditions T(0,t) = 273 K and T(L,t) = 298 K. This is an example of a homogeneous problem.Group of answer choicesTrueFalse

Aerospace Engineering (AE)Page 9 of 35Organizing Institute: IIT KanpurQ.13 Consider the one-dimensional wave equation0u ut x + =  forx−    ,0t  .For an initial condition2( ,0) xu x e−= , the solution at1t = is(A)2( 1)( ,1) xu x e− −=(B)1( ,1)u x e−=(C)2( ,1) xu x e−=(D)2( 1)( ,1) xu x e− +=Q.14 A two-dimensional potential flow solution for flow past an airfoil has a streamlinepattern as shown in the figure. Which of the following conditions is additionallyrequired to satisfy the Kutta condition?(A) Addition of a source of strength0Q (B) Addition of a source of strength0Q (C) Addition of a circulation of strength0  (counter-clockwise)(D) Addition of a circulation of strength0  (clockwise

The homogeneous form of the heat equation requires the boundary conditions to be identically zero.Group of answer choicesTrueFalse

Exercice - 1 Discrétisation de l'équation de la chaleur 1DConsidérons le problème monodimensionnel de la conduction de la chaleur dans une barre de 1mde longueur. Le champ de température T (x, t) vérie l'équation de la chaleur :∂T∂t = α ∂2T∂x2où α est la diusivité thermique.Á cette EDP s'ajoute deux conditions aux limites aux extrémités de la barre T (0, t) = Tg etT (1, t) = Td ainsi qu'une condition initiale T (x, 0) = T0.L'intervalle [0, 1] est discrétisé en N n÷uds de coordonnées xi régulièrement espacés. Notons∆x le pas d'espace. Le temps est discrétisé en intervalles de pas constant ∆t avec M noeud. NotonsT ni la température au n÷ud i à l'instant n.1. Calculer les pas ∆x et ∆t. Préciser la natures des conditions aux limites.2. Écrire la formulation discrétisée centrée.3. Écrire la forme matricielle du problème4. Écrire un script MATLAB qui implémente la méthode explicite, et qui trace les prols àdivers instants.5. Comparer la solution théorique Tth et numérique Tnum avec les deux schéma par rapportau temps.Exercice - 2 Considérons le problème bidimensionnel instationnaire de la conduction de lachaleur dans un domaine rectangulaire [−L1, L1] × [1, L2] × [0, tmax]. La fonction U (x, y, t) vériel'équation de Laplace :∂U∂t = α ∂2U∂x2 + α ∂2U∂y2U (−L1, y, t) = Ug , U (L1, y, t) = UdU (x, 1, t) = Ub, U (x, L2, t) = UhU (x, y, 0) = u0(x, y) = U0Le domaine de calcul est discrétisé en N × P n÷uds et M n÷uds par rapport au temps. Onsupposera que les pas d'espace dans chaque direction ∆x, ∆y et du temps ∆t sont constants. Lafonction discrète au n÷ud (xni , ynj ) sera notée U nij = U (xni , ynj ).1. Calculer les pas ∆x, ∆y et ∆t. Préciser la natures des conditions aux limites.2. Écrire la formulation discrétisée.3. Écrire la forme matricielle.4. Écrire un script MATLAB qui implémente la méthode explicite, et qui trace les prols àdivers instants.1