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Green's identity, shown below, is a very useful tool in analyzing Neumann problems. Consider and ...
Can anyone help with this question please? Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the proble...
Can anyone help with this question please?
Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the problem Uz (x, t)-Au (x, t) + u(x, t) u(x, t = f(x) Y(x, t) E Ω × (0, oo), V(x, t) E 2 x (0, 00), Assume that u(z, t) is a smooth solution and that v(x) is a smooth stationary (i.e., time-independent) solution. Derive a PDE problem for the difference w(x, t)u(x, t)(x) By multiplying...
The Von Karman Momentum Integral (VKMI): dU can be a very powerful tool for generating approximat...
The Von Karman Momentum Integral (VKMI): dU can be a very powerful tool for generating approximate solutions for boundary layer problems. Recall that To is the shear stress at the wall, U00 is the free stream velocity, while 0 and are the momentum and displacement boundary layer thicknesses, respectively. consider a laminar zero-pressure gradient flat plate boundary layer (Le, U” is constant), and assume the following mean profile: u=U,0 sin( ) for y for y > δ(x), 6(x), where δ...
5. The z transform is a very useful tool for studying difference equations. Often difference and ...
5. The z transform is a very useful tool for studying difference equations. Often difference and differential equations are used to describe causal systems and only the causal solution is of interest. This is the "initial condition" problem of a differential equations course. But both difference and differential equations describe more than just the causal system. For instance, "backwards" solutions and "two point boundary value" solutions. One way in which to think about the problem is the ROC of the...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0, u(L,t)=50 where FER is a constant, forcing term Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two u(z,t) = X(z)T(t) + us(z), where the subscript designates the function as the steady limit and does not represent a derlvative. BEWARE: MARKING A STATEMENT TRUE...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y" (t) + 16y'(t) + 68y(t) = –20e-4t u(t), with initial conditions y(0) = -3, and y'(0) = 4. Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y' (t) + 7y(t) = le 4u(t), with initial condition y(0) = 2, Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) , where cis a Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the form constant...
11. Consider a thin, infinitely long rectangular plate that is free of heat sources, as shown...
11. Consider a thin, infinitely long rectangular plate that is free of heat sources, as shown below. For a thin plate, is negligible, and the temperature is a function of x and y only. The solution for this problem is best obtained by considering scaled temperature (ie. 1-T - To, where To is the absolute temperature at T-0) variables, so that the two edges of the plate have "zero-zero" boundary conditions and the bottom of the plate is maintained at...
Consider the steady, incompressible flow of depth h of a liquid of known density ρ and...
Consider the steady, incompressible flow of depth h of a liquid
of known density ρ and unknown viscosity µ down a flat plate as
shown in Figure 1. Air is the fluid above the liquid layer. The
force of gravity is in the vertical direction with acceleration g,
and the plate is at an angle θ with respect to the horizontal.
Assuming the coordinate system as shown, with x aligned with the
flow direction, and y normal to the plate,...
5) ( 40 /oints) Consider the system shown below The weight orali other olycs is nedpble...
5) ( 40 /oints) Consider the system shown below The weight orali other olycs is nedpble Draw free body diagrams of all the objects in the system and Label them Follow the rules we've leamest in slass in rcgarding which obiects should be drawn and what to includs Includo free boty diagram of the whole frame You do NOT need to include dimensions or angles on your free body diagrams. With cach free body diagram, construct all equations noeded to...
Can anyone help with this question please?
Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the problem Uz (x, t)-Au (x, t) + u(x, t) u(x, t = f(x) Y(x, t) E Ω × (0, oo), V(x, t) E 2 x (0, 00), Assume that u(z, t) is a smooth solution and that v(x) is a smooth stationary (i.e., time-independent) solution. Derive a PDE problem for the difference w(x, t)u(x, t)(x) By multiplying...
The Von Karman Momentum Integral (VKMI): dU can be a very powerful tool for generating approximate solutions for boundary layer problems. Recall that To is the shear stress at the wall, U00 is the free stream velocity, while 0 and are the momentum and displacement boundary layer thicknesses, respectively. consider a laminar zero-pressure gradient flat plate boundary layer (Le, U” is constant), and assume the following mean profile: u=U,0 sin( ) for y for y > δ(x), 6(x), where δ...
5. The z transform is a very useful tool for studying difference equations. Often difference and differential equations are used to describe causal systems and only the causal solution is of interest. This is the "initial condition" problem of a differential equations course. But both difference and differential equations describe more than just the causal system. For instance, "backwards" solutions and "two point boundary value" solutions. One way in which to think about the problem is the ROC of the...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0, u(L,t)=50 where FER is a constant, forcing term Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two u(z,t) = X(z)T(t) + us(z), where the subscript designates the function as the steady limit and does not represent a derlvative. BEWARE: MARKING A STATEMENT TRUE...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y" (t) + 16y'(t) + 68y(t) = –20e-4t u(t), with initial conditions y(0) = -3, and y'(0) = 4. Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y' (t) + 7y(t) = le 4u(t), with initial condition y(0) = 2, Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) , where cis a Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the form constant...
11. Consider a thin, infinitely long rectangular plate that is free of heat sources, as shown below. For a thin plate, is negligible, and the temperature is a function of x and y only. The solution for this problem is best obtained by considering scaled temperature (ie. 1-T - To, where To is the absolute temperature at T-0) variables, so that the two edges of the plate have "zero-zero" boundary conditions and the bottom of the plate is maintained at...
Consider the steady, incompressible flow of depth h of a liquid
of known density ρ and unknown viscosity µ down a flat plate as
shown in Figure 1. Air is the fluid above the liquid layer. The
force of gravity is in the vertical direction with acceleration g,
and the plate is at an angle θ with respect to the horizontal.
Assuming the coordinate system as shown, with x aligned with the
flow direction, and y normal to the plate,...
5) ( 40 /oints) Consider the system shown below The weight orali other olycs is nedpble Draw free body diagrams of all the objects in the system and Label them Follow the rules we've leamest in slass in rcgarding which obiects should be drawn and what to includs Includo free boty diagram of the whole frame You do NOT need to include dimensions or angles on your free body diagrams. With cach free body diagram, construct all equations noeded to...
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