CHAPTER FOUR Application of Generalized Burgers equation 67
4.1 INTRODUCTION Exact solutions are presented for Burgers equation in a finite layer connected to an underlying semi-infinite medium of different conductivity and diffusivity by Philip and et al 39. A constant-flux boundary condition is assumed at the surface. This has direct application to steady rainfall on layered field soils. At large times, a travelling wave profile develops in the deep layer and the concentration in the upper layer approaches a non-trivial steady state. Water flux and potential energy are continuous across the interface but the concentration gradient may show a marked discontinuity. The inviscid limit of the stochastic Burgers equation emphasizing geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence was done by Neate and Truman 40. They have shown that for small viscosities there exists a vortex filament structure near to the Maxwell set. They have discussed how this vorticity is directly related to the adhesion model for the evolution of the early universe and include new explicit formulas for the distribution of mass within the shock. The dynamics of an upwardly propagating flame front in a vertical channel, a Burgers equation for the flame front interface was studied in Rakib, Sivashinsky 41 using a weak thermal expansion approximation. In a particular parameter regime and under various physical assumptions, the dimensionless flame-front interface y= y(x, t) was found Rakib, Sivashinsky 41. The asymptotic results were compared with corresponding full numerical results using a transverse method of lines approach Ascher, 68
Robert, Russell 42 by Xiaodi and Michael 43. This method is based on replacing the time derivative by a difference approximation and the solving the resulting boundary value problems in space. Problem [I]: Application in nuclear fusion reactor 4.2 INTRODUCTION Many researchers are interested in magneto-hydro-dynamics MHD since the last century due to its applications. For example, MHD steam plants and MHD generators are used in the modern power plants. The basic concept of the MHD generator is to generate electrical energy from the motion of conductive liquid that is crossing a perpendicular magnetic fluid. Carnot efficiency is improved by the presence of MHD unit. Another example is the MHD pumps and flow meters. In this type of pumps, the electrical energy is converted directly to a force which is applied on the working fluid. MHD separation in metal casting with superconducting coils is another important application. A very useful proposed application which involves MHD is the lithium cooling blanket in a nuclear fusion reactor. The high temperature plasma is maintained in the reactor by means of a toroidal magnetic field. The liquid-lithium circulation loops, which will be located between the plasma and magnetic windings, are called lithium blankets. The lithium performs two functions: it absorbs the thermal energy released by the reaction (and subsequently used for power generation) and it participates in 69
nuclear reactions in which tritium is produced. The lithium blanket is thus a very important reactor component. On other hand, the blanket will be acted upon by an extremely strong magnetic field. Consequently, to calculate the flow of liquid metal in channels or pipes situated at different angles to the magnetic field, and to determine the required pressure drop, heat transfer, etc., knowledge of the appropriate MHD relationships will be necessary. Magnetohydrodynamics (MHD) has been studied since the 19 th century, but extensive investigations in this field accelerated only at the beginning of the 20 th century. The first theoretical and laboratory studies of MHD flows in pipes and ducts were carried out in the 1930s. Williams published results of experiments with electrolytes flowing in insulated tubes. The tubes were placed between the poles of a magnet, and the potential difference across the flow was measured using wires passed through the walls. Hartmann and Lazarus made some very comprehensive theoretical and experimental studies of this subject. They performed their experiments with mercury which has an electrical conductivity 1, 00, 000 times greater than that of an electrolyte. This made it possible to observe a wider range of phenomena than in the experiments by Williams. In particular, Hartmann and Lazarus were able to investigate the change in drag (friction) and, indirectly, the suppression of turbulence caused by magnetic field. Hartmann obtained the exact solution of the flow between two parallel, non-conducting walls with the applied magnetic field normal to the walls. Shercliff 19, in 1956, has solved the problem of rectangular duct, from which he noticed that for high Hartmann numbers M the velocity distribution consists of a uniform core with a boundary layer near the walls. 70
This result enabled him to solve the problem for a circular pipe in an approximate manner (a first approximation which gives rise to errors of order M -1 ) for large M assuming walls of zero conductivity and, subsequently, walls with small conductivity. In 1962, Gold and Lykoudis has obtained an analytical solution for the MFM flow in a circular tube with zero wall conductivity while in 1968, Gardner and Lykoudis have acquired experimentally some results for circular tube with and without heat transfer. The MFM flow is also examined numerically by Al-Khawaja et al. for the case of circular tube with heat transfer and for the case of uniform wall heat flux with and without free convection. The solution for MFM square duct flow is obtained using spectral method by Al-Khawaja and Selmi for the case of uniform wall temperature. Also, the MFM combined free-and-forced convection duct flow was considered by many researchers. Chang & Lundgren considered the effect of wall conductivity for this problem. Gold analytically solved the MHD problem in a circular pipe with zero wall conductivity. His solution was an infinite series of Bessel functions, which was approximated for large M with the first few terms. For the same problem, Shercliff 20 used the second approximation (which gives rise to errors of order M -2 ) to get the solution for large M. Gardner used Gold s solution to evaluate the exact solution for temperature profile, which turned out to be very complex. Then, he approximated velocity profile for small to moderate M with a polynomial form from which he calculated the Nusselt number Nu. For large M, he used Gold s approximation to determine Nu. Gardener and Lykoudis experimentally studied MFM turbulent pipe flow in a transverse magnetic 71
field with and without heat transfer. Gardener and Lo tried to solve the problem of a circular pipe flow with combined forced-and-free convection analytically using a perturbation technique in which the solutions were generated in inverse powers of the Lykoudis number, Ly. They obtained only the distribution of stream function and azimuthal velocity for some small Hartmann numbers. M. Weiss studied a nonlinear two-dimensional magnetoconvective flow in a Boussinesq fluid with a series with a series of numerical experiments. Tabeling and Chabrerie analyzed the secondary laminar flows in annular ducts of rectangular cross-section subjected to a constant axial magnetic field. They considered the cases for high M and treated the equations of flow by a perturbation method involving an infinite series expansion. In addition, some researchers investigated the case of nonuniform magnetic field. Petrykowski and Walker examined the liquid-metal flows in rectangular ducts having electrically insulating top and bottom walls and perfectly conducting sides and in the presence of strong, polar, non-uniform, transverse magnetic field. The presented solutions for the boundary layers adjacent to the sides those are parallel to the magnetic field. Singh and Lal have calculated numerically the temperature distribution for steady MHD axial flow through a rectangular pipe with discontinuity in wall temperature. Mittal, Nataraja and Naidu obtained a numerical solution of the equations governing the flow of an electrically conducting, viscous, compressible gas with variable fluid properties in the presence of a uniform magnetic field. They analyzed the velocity and temperature distributions for subsonic and supersonic flows as these occur in these duct of an MHD generator. Setayesh and Sahai studied numerically the effect of temperature- 72
dependent transport properties on the developing magnetohydrodynamic flow and heat transfer in a parallel-plate channel whose walls are held at constant and equal temperatures. In addition, the problem of the combined free-and-forced convection in horizontal tubes in the absence of magnetic field was investigated considerably in the 1960s. Morton solved the problem of laminar convection in uniformly heated horizontal pipes at low Rayleigh numbers Ra using a perturbation method to obtain a formula for Nusselt number Nu with is valid only for ReRa=3,000. Here, Re and Ra are Reynolds and Rayleigh numbers based on diameter, respectively. Mori, Futagami, Tokuda and Nakamura analyzed the same problem experimentally for air but for high Ra, and they noticed that Nusselt numbers would be about twice as large as those calculated by neglecting the effect of the secondary flow caused by buoyancy at ReRa= 4 x 10 5. They concluded that buoyancy has little effect on velocity and temperature fields in turbulent flow. The critical Reynolds number (laminar-turbulent transition) was, however, affected by the secondary flow. Later, Mori and Futagami, investigated this problem theoretically on a fully developed laminar flow. On the assumption of a boundary layer (by making the velocity and temperature distributions are affected only by viscosity and thermal conductivity) along the tube wall and by use of the boundary-layer integral method, they obtained (after assuming the velocity and temperature fields are affected only by the secondary flow in the core region) the relations between Nusselt number and ReRa(=10 4 ) for Prandtl number Pr not far from unity. Faris and Viskanta examined this problem analytically using a perturbation method. They presented 73
approximate analytical solutions as well as average Nusselt numbers graphically for a range of Prandtl and Grashof numbers of the physical interest. Eckert and Peterson measured the temperature profile along the vertical diameter and calculated Nusselt number as a function of Peclet number Pe for the problem of the heat transfer to mercury in laminar flow through a horizontal tube with a constant heat flux. Siegwarth and Hanratty, measured the fully developed temperature field and axial velocity profile for Prandtl number Pr=80 at the outlet of a long horizontal tube which is heated electrically. They also solved this problem by finite difference techniques to obtain the secondary flow pattern as well as the temperature field and axial velocity field. Newell and Bergles, formulated a numerical investigation of the effects of free convection on fully developed laminar flow in horizontal circular tubes with uniform heat flux. They obtained solutions for heat transfer and pressure drop, with both heating and cooling, for water with two limiting tube-wall conditions: low thermal conductivity(glass tube) and infinite thermal conductivity. They found that the infinite-conductivity tubes exhibits higher Nu and friction factor f than the glass tube, with Nu being over five times the Poiseuille value at Grashof number (based on the difference of the wall and bulk mean temperatures) ~ 10 6. Yousef and Tarasuk, investigated experimentally the influence of free convection due to buoyancy on forced laminar flow of air in the entrance region of a horizontal isothermal tube for a narrow range of Grashof numbers (based on logarithmic mean temperature difference ) from 0.8 x 10 4 to 8.7 x 10 4. That same year, Hishida, Nagano and Montesclaros, published numerical solutions without the aid of a large Prandtl number assumption for combined 74
free-and-forced laminar convection in the entrance region of a horizontal pipe with uniform wall temperature. Chou and Hwang, studied numerically, without the aid of the large Prandtl number assumption, the Graetz problem with the effect of natural convection in a uniformly heated horizontal tube by a relatively novel vorticity-velocity method. They showed the variations in local friction factor and Nusselt number with Rayleigh number for Prandtl number Pr=5, 2 and 0.7. Rustum and Soliman, investigated numerically the steady, fully-developed, laminar, mixed convection in horizontal internallyfinned tubes for the case of uniform axial heat input and circumferentially uniform wall temperature. At Pr=7 and for modified Grashof number varies from 0 to 2 x 10 6, they obtained numerical results which include the secondary flow (velocity) components, axial velocity and temperature distributions, wall-heat flux, friction factor and average Nusselt number for different fin geometries. Finally, Al-Khawaja, Agarwal, and Gardner considered numerically the problem of MFM combined-free-and-forced convection pipe flow using modified third-order-accurate upwind scheme to handle the problem of high Grashof number. However, for high Hartmann number, they refined the mesh near the boundary. 4.3 ASSUMPTIONS: The problem considered herein is one of the forced convection in a horizontal, circular pipe of radius a in a uniform, vertical, transverse magnetic field B 0. A homogeneous, incompressible, viscous, electricallyconducting fluid flows through a horizontal circular pipe and is subjected to 75
a uniform surface temperature and a uniform surface heat flux. In conjunction with defining this problem, the following assumptions are made: (a) All fluid properties are constant (the fluid considered is incompressible) and independent of the temperature. (b) The pipe is sufficiently long that it can be assumed the flow and heat transfer are fully developed and entrance and exit effects can be neglected. Further, it can be deduced that none of the variables except pressure and temperature vary linearly with axial direction. (c) The contributions of viscous and Joulean dissipation in the energy equation are small and can be neglected. This assumption has been shown to be applicable to a similar problem when no external electric field is imposed on the flow. (d) The induced magnetic field produced as a result of interaction of applied field, B 0, with either main or secondary flow, will be assumed negligibly small compared to B 0. This assumption follows from the fact that the magnetic Reynolds number based on the flow is much smaller than unity under conditions found in typical applications. 4.4 DERIVED EQUATIONS: For incompressible Newtonian liquid metal fluid and steady state conditions, the modified Navier-Stokes equations in Burgers form under the effect of magnetic field body force in vector forms is u t = θ u xx u u x + B 0 (4. 3. 1) 76
We consider the different cases of applied magnetic field as under Case i B 0 = e t cos πx u t = θ u xx u u x + e t cos πx 2 u(x, 0) = cos(πx)/2; x ϵ R 2 in equation 4.3.1 we get the following form ; x ϵ R; t > 0, 4.5 SOLUTION FOR B 0 = e -t cos (πx)/2 USING HPM Table 4.1: Numerical solutions of Burgers equation in nonhomogeneous form obtained for ϑ = 1 at different times x -1 t = 0.01 0.0008 t = 0.001 0.0008 t = 0.0001 0.0008-0.8 0.6038 0.6171 0.6185-0.6 1.1489 1.1732 1.1757-0.4 1.5836 1.6147 1.6178-0.2 1.8651 1.8985 1.9018 0 1.9655 1.9965 1.9997 0.2 1.8737 1.8994 1.9019 0.4 1.5974 1.6162 1.6180 0.6 1.1628 1.1747 1.1758 0.8 0.6123 0.6180 0.6186 1 0.0008 0.0008 0.0008 77
u(x,t) 4.6 FIGURES: 2 1.8 1.6 Numerical solutions for non-homogeneous form of Burgers' equation t=0.01 t=0.001 t=0.0001 1.4 1.2 1 0.8 0.6 0.4 0.2 0-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 x Fig 4.1: Figure showing numerical solutions for non-homogeneous Burgers equation 78
u(x,t) Numerical solutions for non-homogeneous form of Burgers' equation 1.98 1.96 t=0.01 t=0.001 t=0.0001 1.94 1.92 1.9 1.88 1.86 1.84 1.82 1.8-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 x Fig 4.2: Figure showing numerical solutions for non-homogeneous Burgers equation 79
Case ii B 0 = e t sin πx in equation 1 we get the following form u t = θ u xx u u x + e t sin πx ; x ϵ R; t > 0, u x, 0 = sin πx ; x ϵ R 4.7 SOLUTION FOR B 0 = e t sin (πx) USING HPM Table 4.2: Numerical solutions of non homogeneous Burgers equation obtained for ϑ = 1 at different times. x 0 t = 0.01 0 t = 0.001 0 t = 0.0001 0 0.1 0.5852 0.6143 0.6175 0.2 1.1148 1.1687 1.1747 0.3 1.5387 1.6091 1.6169 0.4 1.8150 1.8926 1.9010 0.5 1.9158 1.9912 1.9991 0.6 1.8293 1.8949 1.9016 0.7 1.5619 1.6130 1.6182 0.8 1.1385 1.1729 1.1763 0.9 0.6005 0.6176 0.6192 1 0.0016 0.0016 0.0016 80
u(x,t) 4.8 FIGURES: 2 1.8 1.6 Numerical solutions for non-homogeneous form of Burgers' equation t=0.01 t=0.001 t=0.0001 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Fig 4.3: Figure showing numerical solutions for non-homogeneous Burgers equation at different times 81
u(x,t) Numerical solutions for non-homogeneous form of Burgers' equation 1.95 t=0.01 t=0.001 t=0.0001 1.9 1.85 1.8 1.75 1.7 1.65 0.4 0.45 0.5 0.55 0.6 x Fig 4.4: Figure showing numerical solutions for non-homogeneous Burgers equation at different times 82
4.9 INTERPRETATION: The problem considered here is a duct with electrically conducting fluid and with two heat transfer limits. The problem is analyzed numerically when a uniform transverse magnetic field is applied to the duct. The assumption of laminar flow is mostly valid in MFM flows since the turbulences will be damped out due to the opposing force induced in the flow. The fluid mechanic part of this problem was considered extensively and the results were shown using the spatial method. Also the heat transfer results for only uniform temperature boundary conditions were shown. In this problem, we consider the two heat transfer limits as a uniform heat flux and temperature boundary condition were taken using Homotopy Perturbation method, and the software package Matlab is used to achieve this approach. The result obtained for the case of constant temperature condition agrees very well with the references. 83
Problem [II]: Application in Traffic flow 4.10 INTRODUCTION: Consider the flow of cars on a highway and let ρ(x, t) denote the density of cars and f(x, t) the traffic flow. We will also consider ρ* to be the restriction of ρ to a certain range,0 ρ ρ max, where ρ max is the value at which cars are bumper to bumper. 38 Since cars are conserved, the density of cars and the flow must be related by the continuity equation ρ t + f x = 0 (4.1) Obviously, the first expression in which one thinks for the flow is f = vρ where v is the velocity. However, it turns out that in order to reflect the fact that drivers will reduce their speed to account for an increasing density ahead we should suppose that f is a function of the density gradient as well. A simple assumption is to take f ρ = ρ v ρ D ρ where D is a constant. (4.2) x We are assuming also that the velocity v is a given function of ρ : On a highway we would optimally like to drive at some speed v max (the speed 84
limit perhaps) but with heavy traffic we slow down, with velocity decreasing as density increases. The simplest relation that is aware of this is v ρ = v max ρ max ρ max ρ (4.3) Substituting (4.2) and (4.3) into (4.1) gives ρ t + d dx v max ρ max ρ max ρ ρ = D ρ 2 (4.4) x 2 Scaling through v max = x 0 t 0, ρ = ρ max ρ, x = x 0 x and t = t 0 t results in ρ t + 1 ρ ρ x = ερ xx with ε = D v max x 0 and 0 ρ 1. (4.5) Equation (4.5) is Generalized Burgers equation which can be solved by Homotopy Perturbation Method as follows: 85
4.11 SOLUTION OF THE PROBLEM USING HPM. Table 4.3: Numerical solutions of Generalized Burgers equation for u(x, 0) = exp(-2(x-1) 2 ) and u (-1, t) = u (3, t) = 0 obtained for ϑ = 1 at different times. x -1 t = 0.01 0.0000 t = 0.001 0.0000 t = 0.0001 0.0000-0.5 0.0004 0.0002 0.0001 0 0.0348 0.0200 0.0185 0.5 0.4129 0.3723 0.3683 1 0.8194 0.9820 0.9982 1.5 0.4275 0.3738 0.3685 2 0.0416 0.0206 0.0185 2.5 0.0005 0.0002 0.0001 3 0.0000 0.0000 0.0000 86
u(x,t) 4.12 FIGURES: 1 0.9 0.8 Graph of traffic flow with i.c. as exp(-2(x-1) 2 ) at different times t=0.0001 t=0.001 t=0.01 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-1 -0.5 0 0.5 1 1.5 2 2.5 3 x Fig 4.5: Figure showing numerical solutions for homogeneous form of Generalized Burgers equation at different times. 87
4.13 INTERPRETATION: The problem considered here is traffic flow densities as a result of disturbance occurred due to unpredictable reason. Initial condition shows the normal distribution for different traffic flow conditions. The fluid mechanic part of this problem was considered as an individual car extensively taken as a fluid molecule and then the results were shown using the spatial method. In this problem, we consider the uniform heat diffusion with the conservation of the traffic density. As a part of the model requirement we take a boundary condition and solve using Homotopy Perturbation method and the software package Matlab is used to achieve this solution. The result obtained for the particular initial condition agrees very well with the observations. 88
CHAPTER FIVE CONCLUSION 89
In the above research work, we have found the solution of Burgers equation and Generalized Burgers equation by Homotopy Perturbation method using initial condition u(x,0) = sin πx and boundary conditions u(0, t) = u(1, t) = 0. Thereafter we have found the general form of both Burgers and Generalized Burgers equation, so that by taking any initial condition and boundary conditions one can find the solution of both these equations. The same has been solved by taking initial condition u(x, 0) = cos(πx)/2 and boundary conditions u(-1, t) = u(1, t) = 0. The numerical solutions of Burgers and Generalized Burgers equations are found through MatLab software. Algorithms and programs have been depicted. Later on their graphs and interpretations have been mentioned. In chapter 4, problems related to application of Burger s and Generalized Burgers equation in industries as well as real life problems have been solved using HPM. In problem 1, we have shown the application of non homogeneous Burgers equation in nuclear fusion reactor whereas problem 2 shows the solution of traffic flow model using homogeneous Generalized Burgers equation, each with the help of Homotopy Perturbation Method under different initial and boundary conditions. Overall we conclude that the result obtained for the particular initial condition agrees very well with the observations and the results. Therefore the numerical solution of Burgers and generalized Burgers equation using Homotopy Perturbation method gives significant results with various industrial problems. 90
Scope of future research: In the current research we have worked on one dimensional Burgers and Generalized Burgers equation. Further the work can be extended on the same for 2 or more dimensions. We also confirm that the iterative process in Homotopy Perturbation method so that the method gives the reasonable response for a wide range of time interval. 91