a. Calculate the velocities of the flows given by the functions: ψ = x/y and ψ...
a. Calculate the velocities of the flows given by the functions: ψ = x/y and ψ = x² - y².
b. See in each case whether or not there is a velocity potential. Indicate the cases where a velocity potential does/does not exist.
Homework Answers
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a. Calculate the velocities of the flows given by the functions:
ψ = x/y and ψ...
a) Derive the general stream function of a potential flow around a cylinder of radius R given the stream functions Y of a uniform flow and a doublet are uniformUy 'doublet where Uis the speed of...
a) Derive the general stream function of a potential flow around a cylinder of radius R given the stream functions Y of a uniform flow and a doublet are uniformUy 'doublet where Uis the speed of the uniform flow and C is the strength of the doublet. (5 marks) b) Find the specific stream function assuming the streamline on the surface of the cylinder is Ψ-0 (5 marks) c) Find the velocities at two points (-3R, 0 and (-2R, 0)....
To create a vertical wall along the y-axis, one can superimpose two source flows of equal...
To create a vertical wall along the y-axis, one can superimpose two source flows of equal strength Q, centered about the origin on the x-axis at (-x0,0) and (x0,0). Given these two source flows: Determine the potential for this flow field in cartesian coordinates. Determine the streamfunction for this flow field in cartesian coordinates. What value of y corresponds to the wall? Determine functions for the cartesian velocities (u,v) along the wall and show that there is no normal (i.e.,...
Given the velocity potential for a 2-D incompressible flow, (x, y) = xy + x2 -...
Given the velocity potential for a 2-D incompressible flow, (x, y) = xy + x2 - y2 (a) Does the potential satisfy the Laplace Equation (i.e. V20 = 0)? What is the physical intepretation of this? (b) Find u(x,y) and v(x,y) (the corresponding velocity field of the flow). (c) Does the stream function y (x,y) exist? If so: (a) Find the stream function. (b) Find the implicit equation of streamline that passes through (x,y) = (1, 2).
1. Which of the following could NOT possibly be wave functions and why? Assume 1-D in each case. ...
1. Which of the following could NOT possibly be wave functions and why? Assume 1-D in each case. (Here C is a normalization constant) If any of these are valid wave functions, calculate C for those case(s). Where is the particle most likely to be? a) Ψ (x) = C x exp (-x) for x > 0. Ψ (x) is zero everywhere else. b) Ψ (x) = C [exp(2x) + exp(-2x)] for all x.
An incompressible fluid flows horizontally in the x-y plane with a velocity given by , and ,...
An incompressible fluid flows horizontally in the x-y plane with a velocity given by , and , where and are in meters and is a constant. Determine the average velocity for the portion of the flow between and if m/s.
a) Find the potential V(x) associated with the wavefunction ψ(x) = Csech(ax) given that its energy...
a) Find the potential V(x) associated with the wavefunction ψ(x) = Csech(ax) given that its energy eigenvalue is zero (i.e. E = 0). b) Plot V(x) and ψ(x) on the same graph.
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and are in such close proximity they can be considered as indistinguishable. wave functions /a(x) and (a) Write down the symmetrised two-particle wave function for the case where the particles are bosons (VB) and the case where the particles are fermions (Vp). (b) Show that the expectation value (xjr2)B,F is given by: (T122) в,F — (а:)a (х)ь + dx x y:(")...
2. Given each of the following functions that maps X-1, 23) to Y= {A, B, C)(20...
2. Given each of the following functions that maps X-1, 23) to Y= {A, B, C)(20 points, 2 parts- 10 points each) b. g(1) C, g(2) -B, g3) A Determine whether the function has an inverse. If it has an inverse, provide it. If it does not, explain why not
y? - 2xy x + y2 if (x, y) + (0,0) 7. Given the piecewise function:...
y? - 2xy x + y2 if (x, y) + (0,0) 7. Given the piecewise function: f(x,y) 0 if (x, y) = (0,0) a) Show that: limf(x,y) does not exist. *(x,y) (0,0) b) Find: fy(0,0). c) Where is f continuous? Where is f differentiable? Explain.
Consider a real-valued function u(x, y), where x and y are real variables. For each way...
Consider a real-valued function u(x, y), where x and y are real variables. For each way of defining u(x, y) below, determine whether there exists a real-valued function v(x, y) such that f(z) = u(x, y) + iv(x, y) is a function analytic in some domain D C C. If such a v(x, y) exists, find one such and determine the domain of analyticity D for f(z). If such a v(x, y) does not exist, prove that it does not...
a) Derive the general stream function of a potential flow around a cylinder of radius R given the stream functions Y of a uniform flow and a doublet are uniformUy 'doublet where Uis the speed of the uniform flow and C is the strength of the doublet. (5 marks) b) Find the specific stream function assuming the streamline on the surface of the cylinder is Ψ-0 (5 marks) c) Find the velocities at two points (-3R, 0 and (-2R, 0)....
Given the velocity potential for a 2-D incompressible flow, (x, y) = xy + x2 - y2 (a) Does the potential satisfy the Laplace Equation (i.e. V20 = 0)? What is the physical intepretation of this? (b) Find u(x,y) and v(x,y) (the corresponding velocity field of the flow). (c) Does the stream function y (x,y) exist? If so: (a) Find the stream function. (b) Find the implicit equation of streamline that passes through (x,y) = (1, 2).
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and are in such close proximity they can be considered as indistinguishable. wave functions /a(x) and (a) Write down the symmetrised two-particle wave function for the case where the particles are bosons (VB) and the case where the particles are fermions (Vp). (b) Show that the expectation value (xjr2)B,F is given by: (T122) в,F — (а:)a (х)ь + dx x y:(")...
2. Given each of the following functions that maps X-1, 23) to Y= {A, B, C)(20 points, 2 parts- 10 points each) b. g(1) C, g(2) -B, g3) A Determine whether the function has an inverse. If it has an inverse, provide it. If it does not, explain why not
y? - 2xy x + y2 if (x, y) + (0,0) 7. Given the piecewise function: f(x,y) 0 if (x, y) = (0,0) a) Show that: limf(x,y) does not exist. *(x,y) (0,0) b) Find: fy(0,0). c) Where is f continuous? Where is f differentiable? Explain.
Consider a real-valued function u(x, y), where x and y are real variables. For each way of defining u(x, y) below, determine whether there exists a real-valued function v(x, y) such that f(z) = u(x, y) + iv(x, y) is a function analytic in some domain D C C. If such a v(x, y) exists, find one such and determine the domain of analyticity D for f(z). If such a v(x, y) does not exist, prove that it does not...
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