3. (7 points) Let u(x, y) be the steady-state temperature u(r, y) in a rectangular plate...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
1. Consider a thin rectangular plate in the ry-plane, the figure. The PDE describing the temperature of the plate is the heat equation shown in as 0 xa, 0< y < b, t>0. D + at where u(x, y, t) is the temperature at point (x, y) diffusivity at time t andD> 0 is the thermal (a) Suppose that the solution to the PDE (once we impose initial and boundary con ditions) reaches equilibrium when t o, that is there...
2. Find the steady-state temperature u(r,0) in a semicircular plate of radius r 2 if [10<0< π/2 u(2,0) and the edges 0 = 0 and 0 = T are insulated. 0 /20 T пл sin 2 1 2 + cos(n0) Ans: u(r,0) п
Problem 2(30 points) Consider the steady-state temperature distribution in a square plate with dimensions 2 m x 2 m. There is a heat generation of ġ(x.y)=6x [W/my], and the thermal conductivity of k=1[W/(m-°C)]. The temperature on the top boundary is given by a piecewise function, f(x), which is defined below. x(4- x²)+10 0<x<1 | x(4- x?) + 20, 1<x<2 The bottom boundary is insulated. The temperatures on left-handed and right-handed boundary are maintained at constants 10[°C] and 20 [°C] as...
Consider the steady temperature T (2,y) in a rectangular plate that occupies 0 <<< 9 and 0 <y<5, which is heated at constant temperature 150 at 9 and 0 along its other three sides. (a) For separation solutions T(1,y) = F(x)G(y), you are given that admissible F(1) are the eigenfunctions Fn (1) = sinh(An I) for n=1,2,... and G(y) are the eigenfunctions Gn(y) = sin(Any) for n=1,2,... A for In = (b) The solution is the superposition T(z,y) = an...
9. Find the steady-state temperature distribution in the plate shown below y= x и —D () insulated и 3D Т 1/2 1 и 3D 0 9. Find the steady-state temperature distribution in the plate shown below y= x и —D () insulated и 3D Т 1/2 1 и 3D 0
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T<02T Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T
Steady, 2-D conduction takes place in a rectangular solid having height (in the y-direction), H, and width (in the X-direction), W. The bottom of the rectangle, from x=0 to x-W, is insulated. The left boundary x-0, 0 y s H, is held at a temperature of 500 K, as is the top boundary, y=H, 0 s x sW. The right boundary, x=W and 0 Sys H, has a steady temperature distribution given by Ty,x W) 500 (1-sin(ny/H) Find the temperature...
Steady, 2-D conduction takes place in a rectangular solid having height (in the y-direction), H, and width (in the X-direction), W. The bottom of the rectangle, from x-0 to x=W, is insulated. The left boundary, x0, 0 s ys H, is held at a temperature of 500 K, as is the top boundary, y=H, 0 sxs W. The right boundary, x=W and0 Sys H, has a steady temperature distribution given by TOy,x W) 500 (1-sin(ny/H) Find the temperature distribution, T(x,...
Steady, 2-D conduction takes place in a rectangular solid having height (in the y-direction), H, and width (in the X-direction), W. The bottom of the rectangle, from x-0 to x-W, is insulated. The left boundary, x0, 0 ys H, is held at a temperature of 500 K, as is the top boundary, y=H, 0 sxs W. The right boundary, x=W and0 Sys H, has a steady temperature distribution given by TOy,x W) 500 (1-sin(ny/H) Find the temperature distribution, T(x, y),...