Burgers equation

It is, perhaps, less known that the concepts of complementarity and indeterminacy also arise naturally in the theory of Brownian motion. In fact, position and apparent velocity of a Brownian particle are complementary in the sense of Bohr they are subject to an indeterminacy relation formally similar to that of quantum mechanics, but physically of a different origin. Position and apparent velocity are not conjugate variables in the sense of mechanics. The indeterminacy is due to the statistical character of the apparent velocity, which, incidentally, obeys a non-linear (Burgers ) equation. This is discussed in part I. 

In the remaining region t>tu, the thermal region, we apply the mapping onto the Burgers equation (see section 4). In this case the RG-procedure applied to this equation becomes trivial, since there is only a contribution from a single momentum shell and one finds for the correlation length 1 ... 

For K = 0, high temperatures, but weak disorder we adopt an alternative method by mapping the (classical) one-dimensional problem onto the Burgers equation with noise [26]. With this approach one can derive an effective correlation length given by... 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

After the Burgers equation, the numerical analysis of the incompressible boundary layer equations for convection heat transfer are discussed. A few important numerical schemes are discussed. The classic solution for flow in a laminar boundary layer is then presented in the example. 

In the general form, the viscous Burgers equation may be written as... 

When a = 0 and P = 1, the standard nonlinear Burgers equation results ... 

Problem Use the linear Burgers equation for heat convection in a... 

Use the linear Burgers equation for heat convection in a channel where the water is flowing with uniform velocity of 0.1 m/s across the cross section of the channel (boundary layers are neglected). The water is initially at 25°C throughout. At time t = 0 sec, waste heat is continuously rejected at x = 0 m, and the channel is long such that dT/dx = 0 for x > 1 m. The amount of heat rejected is 6.23 W/m2 for t > 0. Using the MacCormack explicit scheme, calculate the first 9 time steps to show the transient temperature distributions. 

Air for ventilation purposes flows through a 10 m insulated duct at 0.75 m/s. Initially, the air is at 25°C. A cooling coil at the entrance cools the air to 15°C. At time = 0, the cooling coil is turned on and the temperature there is maintained constant at 15°C. At the duct exit, the temperature gradient of the air may be assumed unchanging. Use the Burgers equation to model this physical problem, and solve it with an appropriate finite difference scheme. 

The water in a 1.2 m insulated pipe is initially at room temperature, 20°C. At time = 0, cooling water at 0°C enters the pipe at 1 m/s. The entrance of the pipe is maintained at 5°C, and the exit cannot be more than 8°C. Model this practical problem using the Burgers equation, and solve it with an efficient finite difference scheme. 

Yang HQ, Przekwas AJ (1992) A comparative study of advanced shockcapturing schemes applied to Burgers equation. J Comp Phys 102 139-159... 

It is also worth recalling that when simultaneously a and a vanish (we also disregard a ) Equation 4.3 reduces to the Burgers equation,which is known to possess (Taylor-Burgers) shocks. In this case there is an energy balance between the nonlinear a term and the dissipative 2 term. Shocks are also known to possess solitonic-like properties, a phenomenon already discovered long ago by Mach and collaborators (for an historical account see e.g., References 4 and 11). 

Before discussing the Kerr measurements on the solutions, we briefly outline how the size of the polymers was obtained from viscosity and relaxation data. Two well-known expressions which are useful in interpreting these data are the Simha and Burgers equations. The Simha equation (20) relates the intrinsic viscosity of a solution of elongated ellipsoids... 

One can easily show [50] that the stationary equation (87) is satisfied by any solution of the Burgers equation... 

This is the Burgers equation which can be linearized by means of the Hopf-... 

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

There exists a slightly more general class of solutions of (6.2.1) than (6.2.2), and they are obtained by smoothly joining a pair of periodic waves of different q. To find such solutions, we first note that (6.2.1) is equivalent to the Burgers equation (Burgers, 1974)... 

It is also a well-known fact that the Burgers equation (6.2.3) has a family of shock solutions... 

This is essentially the same as the well-known Hopf-Cole transformation which reduces the Burgers equation to a simple diffusion equation (Burgers, 1974). The same transformation reduces (6.3.2) to the linear equation... 

Creep tests with UP mat laminates with various glass fiber-reinforcements subsequent to several months of loading exhibit linear creep profiles for almost 15 years. Figure 6.1 to Figure 6.3. The straight line approximations were performed according to the four-parameter method or Burgers equation. 

In the Burgers equation preferred by Keener and Tyson, the longest time constant of exponential decay is independent of ai, so such observations cannot estimate a. We can estimate ai from the observed dependence of rotation period on twist in the Belousov-Zhabotinsky experiment of Pertsov etal. [46] I estimate it below as 5 x 10 cm/sec. 

Nonlinear Advection. For the nonlinear problem we solve the two-dimensional inviscid Burgers equation... 

Table 4. Results for Burgers equation obtained by (a) ADER2, (b) ADER3, and...

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