On the Application of the Fourier Series Solution to the Hydromagnetic Buoyant Two-Dimensional Laminar Vertical Jet

1 High Temperatures Department, Instituto de Innovación en Minerı́a y Metalurgia, Avenida del Valle 738, Santiago, Chile 2 Mechanical Engineering Department, Universidad de Chile, Beauchef 850, Santiago 8370448, Chile 3 Civil Engineering Department, Advanced Mining Technology Center, Universidad de Chile, Blanco Encalada 2002, Santiago 8370449, Chile 4 Mechanical Engineering Department, Universidad de Chile, Beauchef 850, Santiago, Chile


Introduction
The behavior of a two-dimensional symmetric plane jet that spreads out from a nozzle into a fluid at rest is of basic fluid dynamics interest with many applications in engineering.
Several authors have applied the boundary layer equations to jet flows with very good results 1, 2 .The exact analytical solutions found by these authors consider that the size of the nozzle is infinitely small.Crane extends the problem to jets in an atmosphere of different densities 3, 4 .Pozzi and Bassano 5 introduced the Padé approximants to the study of plane jets.Revuelta et al. 6 discussed the idea of the virtual origin of the jet as a first-order correction for the far-field description, with the idea of obtaining better agreement between the observed flow field and the one predicted in 1, 2 when the size of the nozzle is finite.More recently, Wilks et al. studied the development of a jet into uniform and nonuniform external stream 7-9 .Brand and Lahey 10 studied the effect of the buoyancy in axisymmetric and plane jets using the self-similar solution of the boundary layer equations.Sánchez-Sanz et al. 11 studied the hydrogen laminar jet using asymptotic methods.Martynenko et al. 12 presented an analysis of the buoyancy effects on momentum and heat transfer in vertical jets and plumes.The works mentioned above are based on self-similar solutions of the boundary layer equations.The self-similar solutions allow for the transformation of the boundary layer equations into ordinary differential equations 8 .When the width of the nozzle is finite, generally it is not possible to find self-similar solutions to the boundary layer equations; in this case it must resort to numerical solutions 13-15 .In 1992 Yu et al. 16 presented a rigorous numerical solution for free and wall two-dimensional buoyant jets.More recently, Aissia et al. 17 have performed a numerical analysis for the aiding buoyancy on a vertical round jet.In 2007 Ali et al. 18 gave a detailed numerical study of the influence of the Prandtl number on the hydrodynamic and thermal behaviour of a laminar buoyant free jet.Recently, the plane jet problem was revisited by Rosales-Vera and Valencia 19 .
Concerning the application of spectral methods to boundary layer flow problems, Kumar and Yajnik 20 studied a channel flow problem including a sudden expansion, using an expansion in the eigenfunctions of the Poiseuille flow development, and the problem is reduced to solve nonlinear first-order ordinary differential equations that have a tendency to decouple rapidly.In 1982 Plotkin 21 solved the same problem, specifically by applying the Fourier series, getting solutions for a sudden channel expansion, channel entry flow, and slowly diverging cannel, which agree well with the results of previous investigations.More recently, Figueira Da Silva and Cotta 22 solved the entrance channel flow problem by applying integral transform techniques to the boundary layer equations.Khabibrakhmanov and Summers 23 resolved the Blasius equation, using a spectral method with generalized Laguerre polynomials.
In 2004, Rosales and Frederick 24 solved the classical Graetz problem by applying the Fourier series.This solution has the virtue of delivering very precise numerical results for the Nusselt number in the area near the entrance to the channel.Ma et al. 25 develop a Hermite spectral method for parabolic equations in unbounded domains.Recently, Rosales and Valencia 26 studied the Blasius equation by the Fourier series finding a solution with very high accuracy.The same authors applied the Fourier series to the Falkner-Skan equation 27 .The results show that the pseudospectral method has a very high accuracy.A complete review of spectral methods applied to fluid dynamics can be found in 28 .About the pseudospectral methods applied to jet flow, Rosales-Vera and Valencia 19 applied the Fourier series to study the fluid dynamic of the free plane jet.
In this paper, a spectral method using the Fourier series expansions is used to solve the buoyant two-dimensional laminar vertical jet of a conducting fluid in a magnetic field hydromagnetic jet with the width of the nozzle being finite.The method is essentiall, through the Fourier series, to transform the boundary layer equations to a coupled set of nonlinear first-order ordinary differential equations.

Analysis
Consider the steady, laminar, hydromagnetic flow combined with heat transfer by natural convection of a vertical jet flow, where a magnetic field of uniform strength B 0 is applied in the Y direction which is normal to the X axis of the jet direction.
In this case the momentum and energy equations of the hydromagnetic vertical buoyant plane jet flow derived from the Boundary layer approximation are where a is the width of the nozzle, U 0 is the velocity flow in the nozzle, U and V are the velocities in the X and Y directions, T is the temperature flow, and T 0 and T ∞ are the temperature flow in the nozzle and the temperature of the fluid far of the jet, respectively.The Reynolds number is given by Re ρU 0 a/μ, ρ and μ are the density and the viscosity of the fluid, M B 0 a √ σ/ √ μ is the Hartmann number, B 0 is the magnetic field, σ is the conductivity of the fluid, Gr ρ 2 /μ 2 gβ T 0 − T ∞ a 3 is the Grashof number, Ri Gr/Re 2 is the Richardson number, Pr is the Prandtl number, and β is the thermal diffusivity.The boundary conditions are u 0, y u 0 y , u x, y −→ ∞ 0, ∂u ∂y y 0 0, ϑ 0, y ϑ 0 y , ϑ x, y −→ ∞ 0, ∂ϑ ∂y y 0 0.

2.5
The following solution for the velocity field and temperature is considered: where the number of wavelength k i is given by The y dependence is built on cosine functions that automatically satisfy the boundary conditions at the y L and at the symmetry plane for u.These functions form an orthonormal basis, as follows:

2.8
Multiplying 2.8 by cos k p y and integrating from y 0 to L, the following system of equations is obtained: 2.9 where

2.10
The following set of equations p 1, . . ., N is gotten, where N is the number of equations used to approximate the solution with a finite number of terms in the Fourier series.Equation 2.9 can be expressed as follows:

2.11
where 12 Doing a similar analysis to 2.3 , the following set of equations is obtained: Equations 2.11 and 2.13 represent a first-order nonlinear system of equations.The functions f and g, which account for the axial dependence of velocity field and temperature field, can be determined as follows.

Numerical Solution
Equations 2.11 and 2.13 are discretized in the variable x, being the following system of algebraic equations:

3.1
The numerical solution of the systems 2.11 and 2.12 is for each station x.To solve the system of algebraic equations for the coefficients f p x and g p x , where the values of the coefficients f p x − Δx and g p x − Δx are known from the station x − Δx , the solution is found through an iterative method for coefficients of the Fourier series, defining the following vectors:

3.2
So, the functions f i x and g i x of the Fourier series may be recursively determined by the following equations: G n f n , g n ,

3.3
where the Jacobian matrixes are given by and the vectors f n , g n represent the values of f, g in the n iteration.The iterations for f n , g n from 2.11 may be repeated until a n 1 − a n ≤ ε, for some prescribed error tolerance ε.
The boundary condition at x 0 is considered a uniform profile and a profile of parabolic shape.
In the uniform initial profile, one has

3.5
In the space of Fourier, the vectors f i x 0 and g i x 0 are obtained from the previous condition:

3.6
In the parabolic initial profile, one has

3.7
In the space of Fourier, the vectors f i x 0 and g i x 0 are obtained from the previous condition:

Free Plane Jet
The case of the free plane jet is obtained when M 0, Ri 0, and Pr / 0. Figure 1 shows the velocity field obtained for a uniform initial velocity profile with Pr 1 and Re 1250, in different x position.
In this case, the velocity field is not coupled with the temperature field.In the far field, the velocity field is the similarity solution of Bickley 1 , where the centerline velocity is given by u c x C/x 1/3 , where C 0.4543 for a rectangular initial profile Bickley 1, Figure 2 with Ri 0. The case of a parabolic initial profile using, Fourier series was reported by Rosales-Vera and Valencia 19 .

Buoyant Vertical Jet
The case of a buoyant two-dimensional laminar vertical jet is obtained when M 0, Ri / 0, and Pr / 0. The Reynolds number was kept constant Re 1250 to validate the numerical results in 18 .Figure 2 shows the evolution of the centerline velocity profile as function of the x distance from the nozzle exit for several Richardson numbers with a uniform initial velocity profile.The Richardson number varies from 0 to 0.2, considering the forced convection case.This, in reality, with Re 1250, considers Gr between 0 and 312500 for Pr 0.71 air .The centerline velocity u distribution obtained from the Fourier series solution solid line is compared with the solution reported in 18 using finite differences points .The results coincide very well.
For the field near to the exit of the nozzle, the jet velocity remains constant with a value equal to the initial velocity, which represents the length of the potential core.For Richardson number less than ∼10 −3 , the length of the potential core is x ∼ 10 Re 2h.When Ri > 10 −3 , the length of the potential core is drastically reduced.In this case the effect of the buoyancy is present from distances x ∼ 0.1 Re 2h very near to the nozzle.
Figure 3 shows the evolution of the dimensionless centerline temperature profile as function of the x distance from the nozzle exit, for several Richardson numbers.For the field near to the exit of the nozzle, the centerline temperature remains constant with a value equal to the initial temperature.This is the length of the potential core of the temperature field, and no dependence on the Richardson number exists.The length of this region is x ∼ 10 Re 2h.remains constant with a value equal to the initial velocity, as in the case of pure buoyant jet.However as x is increased, the centerline velocity of the jet is considerably shortened by increasing the Hartmann number.The effect of the magnetic field is effective to start at x ∼ Re 2h.The deceleration of the jet makes the decrease of temperature in the jet axis slower, weakening the heat transfer between the colder jet and the cold atmosphere Figure 5 .However the effect of the magnetic field on the temperature is less intense than in the velocity field.
Figure 6 shows the effect of the magnetic field in the velocity profile in x 0.8 far field , for several values of Hartmann number.The magnetic field intensifies so the central velocity of the jet decreases making the velocity profile wide and less intense.Then, if the magnetic field continues to increase, it will reach the point where the buoyancy vanishes.This is because the Hartmann number reached a value such that M 2 ∼ Gr/ Re; considering in particular the case Pr ∼ 1, 2.1 -2.3 with initial conditions 2.5 become the equations for a free jet with the solution u x, y θ x, y ; then, in this case the magnetic field completely cancels the effect of buoyancy.This phenomenon is shown in Figure 6, where the dashed line represents the velocity profile for a free jet.The profile is very similar to the velocity profile obtained for M 3, where M 2 ∼ Gr/ Re.  Figure 7 shows the effect of magnetic field on the temperature profiles for x 0.8.The temperature profile is widening, for a fixed distance x, to the extent that the magnetic field increases.The temperature in the jet is more homogeneous, and the core temperature, as mentioned above, will generate increasing jet cooling.
Results obtained in the previous case remain valid for other Richardson numbers.Figure 8 shows the evolution of the centerline velocity profile as function of the x distance from the nozzle exit for Ri 0.1 and 0.01.In the figure, the continuous line represents a uniform initial velocity profile and the dashed line represents a parabolic initial profile.When increasing the Richardson number, the central velocity increases due to increased jet buoyancy.However, the centerline velocity is also reduced when the magnetic field is increased, reducing the effect of jet buoyancy.Concerning the influence of the initial velocity profile on the development of the central jet velocity, Figure 8 shows that there are differences only in areas very close to the inlet, since for relatively small distances x ∼ 0.8 the central velocity acquires the asymptotic solution.Both cases, the uniform and the parabolic profile, converge to a solution of the far field where the initial profile has no influence.becomes narrow, so that the jet maintains its center temperature for longer distances, slowing the cooling of the jet.Thus the buoyancy stays longer and central jet velocity is increased slightly.
Figure 12 shows the evolution of the centerline temperature profile as function of the x distance from the nozzle exit for Ri 0.01 and 0.1.The results show a similar result in Figure 9; when the Hartmann number increases, the temperature tends to fall more slowly, slowing the cooling jet.This effect is reduced when the Richardson number is increased.For example, for Ri 0.1, the effect of the magnetic field is much less important than in the case of Ri 0.01.

Conclusion
This paper demonstrates that the Fourier series is a suitable method to solve the boundary layer problem applied to jets.The hydromagnetic buoyant two-dimensional laminar vertical jet was analyzed.In particular, we studied the combined effect of the Hartmann number Mathematical Problems in Engineering  and the Prandtl number, showing that the magnetic field could vanish the effect of natural convection.The results obtained show a high accuracy.The present method can be easily extended to more complex flows, for example, jets with heat and mass transfer including multiphase flow and combustion problems.

Figure 4 :
Figure 4: Centerline velocity profile as a function of the distance from the nozzle exit for several magnetic numbers with uniform initial velocity profile.

Figure 5 :
Figure 5: Centerline temperature as a function of the distance from the nozzle exit for several magnetic numbers with uniform initial velocity profile.

1 M = 2 M = 3 Figure 6 :
Figure 6: Effect of the magnetic field in the velocity profile in x 0.8 far field , for several values of the Hartmann numbers with a uniform initial velocity profile.The dashed line represents the velocity profile for a free jet.

Figure 7 :
Figure 7: Effect of the magnetic field in the temperature profile in x 0.8 far field , for several values of Hartmann numbers with a uniform initial velocity profile.

Figure 10 := 9 Figure 11 :
Figure 10: Centerline velocity profile as function of the distance from the nozzle exit for several Hartmann numbers, Richardson number 0.1 and 0.01, and Prandtl number 9.

Figure 12 :
Figure 12: Centerline temperature profile as function of the distance from the nozzle exit for several Hartmann numbers, Richardson number 0.1 and 0.01, and Prandtl number 9.