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Chapter 5: Q5.3-27E (page 261)

Generalized Blasius Equation. H. Blasius, in his study of the laminar flow of a fluid, encountered an equation of the form y'''+yy''=(y')2-1. Use the Runge鈥揔utta algorithm for systems with h = 0.1 to approximate the solution that satisfies the initial conditions y(0)=0,y'(0)=0,y''(0)=1.32824. Sketch this solution on the interval [0, 2].

Short Answer

Expert verified

The result can get by the Runge-Kutta method, and the result is y(2)=1.6001.

Step by step solution

01

Transform the equation

Here the equation is y'''+yy''=(y')2-1.

The system can be written as:

x1=yx2=y'=x'1x3=y''=x'2

The transform equation is:

x'1=x2x'2=x3x'3=-x1x3+x22-1

The initial conditions are:

x1(0)=y(0)=0x2(0)=y'(0)=0x3(0)=y''(0)=1.32824

02

Apply the Runge-Kutta method

For h=0.1

t

Y

T

Y

0

0

1.1

0.599

0.1

0.00647

1.2

0.69515

0.2

0.0252

1.3

0.79515

0.3

0.0553

1.4

0.89926

0.4

0.0957

1.5

1.0072

0.5

0.1456

1.6

1.1189

0.6

0.20407

1.7

1.234

0.7

0.27032

1.8

1.3526

0.8

0.34363

1.9

1.4747

0.9

0.4233

2

1.6001

1

0.50882

03

Graph

Therefore, the value of y(2)=1.6001.

Thus, this is the required result.

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Most popular questions from this chapter

In Problems 3 鈥� 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

D2-1u+5v=et,2u+D2+2v=0

Use the result of Problem 31 to prove that all solutions to the equation\({\bf{y'' + }}{{\bf{y}}^{\bf{3}}}{\bf{ = 0}}\)remain bounded. [Hint: Argue that \(\frac{{{{\bf{y}}^{\bf{4}}}}}{{\bf{4}}}\) is bounded above by the constant appearing in Problem 31.]

In Problems 3鈥�6, find the critical point set for the given system.

dxdt=y2-3y+2,dydt=(x-1)(y-2)

Sticky Friction. An alternative for the damping friction model F = -by鈥� discussed in Section 4.1 is the 鈥渟ticky friction鈥� model. For a mass sliding on a surface as depicted in Figure 5.18, the contact friction is more complicated than simply -by鈥�. We observe, for example, that even if the mass is displaced slightly off the equilibrium location y = 0, it may nonetheless remain stationary due to the fact that the spring force -ky is insufficient to break the static friction鈥檚 grip. If the maximum force that the friction can exert is denoted by m, then a feasible model is given by

\({{\bf{F}}_{{\bf{friction}}}}{\bf{ = }}\left\{ \begin{array}{l}{\bf{ky,if}}\left| {{\bf{ky}}} \right|{\bf{ < }}\mu {\bf{andy' = 0}}\\\mu {\bf{sign(y),if}}\left| {{\bf{ky}}} \right| \ge {\bf{0andy' = 0}}\\ - \mu {\bf{sign(y'),ify'}} \ne 0.\end{array} \right.\)

(The function sign (s) is +1 when s 7 0, -1 when s 6 0, and 0 when s = 0.) The motion is governed by the equation (16) \({\bf{m}}\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - ky + }}{{\bf{F}}_{{\bf{friction}}}}\)Thus, if the mass is at rest, friction balances the spring force if \(\left| {\bf{y}} \right|{\bf{ < }}\frac{\mu }{{\bf{k}}}\)but simply opposes it with intensity\(\mu \)if\(\left| {\bf{y}} \right| \ge \frac{\mu }{{\bf{k}}}\). If the mass is moving, friction opposes the velocity with the same intensity\(\mu \).

  1. Taking m =\(\mu \) = k = 1, convert (16) into the firstorder system y鈥� = v (17)\({\bf{v' = }}\left\{ \begin{array}{l}{\bf{0,if}}\left| {\bf{y}} \right|{\bf{ < 1andv = 0}}{\bf{.}}\\{\bf{ - y + sign(y),if}}\left| {\bf{y}} \right| \ge {\bf{1andv = 0}}\\{\bf{ - y - sign(v),ifv}} \ne 0\end{array} \right.\) ,
  2. Form the phase plane equation for (17) when v 鈮� 0 and solve it to derive the solutions\({{\bf{v}}^{\bf{2}}}{\bf{ + (y \pm 1}}{{\bf{)}}^{\bf{2}}}{\bf{ = c}}\).where the plus sign prevails for v>0 and the minus sign for v<0.
  3. Identify the trajectories in the phase plane as two families of concentric semicircles. What is the centre of the semicircles in the upper half-plane? The lower half-plane?
  4. What are the critical points for (17)?
  5. Sketch the trajectory in the phase plane of the mass released from rest at y = 7.5. At what value for y does the mass come to rest?

In Problems 3鈥�6, find the critical point set for the given system.

dxdt=y-1dydt=x+y+5
See all solutions

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