ANALYTICAL SOLUTION FOR FREE PARABOLIC FLOW OVER A FLAT PLATE UNDER A MAGNETIC FIELD

1 * Shaban Aghamirzai

1 Department of Mechanical Engineering , Qaemshahr Branch, Islamic Azad University, Qaemshahr, Mazandaran, Iran

Corresponding Author: Shaban Aghamirzai

ABSTRACT

Keywords : Curved flow, flat plate, magnetic field, collocation method.

INTRODUCTION

The study of fluid flow under the influence of a magnetic field is referred to as magnetohydrodynamics (MHD). When a conducting fluid moves within a magnetic field, an electric current is induced inside it. The interaction between this induced current and the magnetic field produces a resistive force opposing the motion, known as the Lorentz force. The flow over a surface carries a certain amount of momentum within the boundary layer. Applying

a magnetic field causes the Lorentz force and interlayer stresses to act as a resistance against the flow in the boundary layer, thereby reducing its momentum. As a result, the pressure gradient changes, leading to alterations in the boundary layer flow.

Most fluid mechanics problems are intrinsically nonlinear. Apart from a few exceptions, most do not have exact solutions. With the emergence of high-memory computers and their advancements, solving linear problems has become increasingly easier, yet obtaining exact solutions for nonlinear problems remains difficult. Generally, three main methods exist for solving engineering equations and problems: experimental, numerical, and analytical. Experimental approaches allow investigation of problems as they occur in reality, but require advanced equipment and extensive modeling. Moreover, challenges such as laboratory environmental effects, measurement errors, and high costs limit their widespread use. Numerical methods enable complex mathematical problems to be formulated for computational solutions. This allows the study of complex physical and nonlinear problems but comes with drawbacks including numerical errors, programming difficulties, high costs, and lengthy solution times. Additionally, purely numerical approaches make it hard to develop an overall understanding of nonlinear problems. Analytical methods have gained considerable attention among researchers recently. Although approximate analytical solutions do not exactly satisfy the problem, they can yield acceptable results by applying appropriate approximations and enabling thorough analysis.

The governing equations for Newtonian fluid flow (Navier-Stokes equations) are highly nonlinear and complex. Moreover, the surrounding environment affects fluid behavior. Therefore, examining fluid dynamics in magnetic fields has become significantly important as an applied science in diverse fields such as military, food processing, pharmaceuticals, and notably medicine. Magnetic fields influence many natural and human-made flows. Natural examples include the Earth’s remarkable geomagnetic field generated by fluid motion in its core, the solar magnetic field responsible for solar flares and sunspots, and the galactic magnetic field thought to impact star formation in interstellar clouds. In industrial and engineered systems, magnetohydrodynamic forces are commonly used in thermal and refrigeration energy devices, pumps, fluid agitation, and the flotation of molten metals. Magnetohydrodynamics (MHD) concerns the interaction between fluid flow and magnetic fields. Non-magnetic fluids with electrical conductivity, such as molten metals, hot ionized gases (plasmas), and strong electrolytes, fall within this category.

The passage of a free rotating flow over a flat plate, which has a fixed rotation axis perpendicular to the surface, may cause the curvature of the streamlines inside the boundary layer to differ from those in the free stream due to the presence of a vertical pressure gradient. This scenario is similar to the flow passing over turbomachinery blades. Therefore, because of the velocity reduction within the boundary layer, the curvature of the boundary layer flow must increase to balance this pressure gradient. The velocity component of the boundary layer perpendicular to the free stream is called the secondary or cross flow. This secondary flow depends on the curvature of the primary streamlines and the pressure gradient.

In August 1904, Ludwig Prandtl, aged 29, delivered a ten-minute lecture on the boundary layer at a conference in Heidelberg, Germany, introducing a new concept that revolutionized the understanding and analysis of fluid dynamics. His boundary layer theory deserved a Nobel Prize, but the Nobel Committee, which focused on classical physics, showed reluctance to award him. In a paper published the following year, he explained that viscous fluid flow along a wall forms a very thin layer near the surface where velocity gradients perpendicular to the flow are very large [1].

In 1942, Hans Alfén was the first to study magnetohydrodynamic flow, examining the motion of charged particles influenced by the Sun’s gravitational, magnetic, and electric forces. He found that the Sun’s magnetic field had a much greater effect on ions and electrons than its

gravitational force, suggesting that during planet formation, electromagnetic forces could have been more significant than mechanical forces [2].

Several studies have investigated incompressible boundary layer flow over a flat plate in the presence of magnetic fields, demonstrating the effects of magnetic forces on the flow [3–7]. Loose [8] studied laminar incompressible boundary layer flow over a flat plate with a parabolic free stream and found no boundary layer separation despite a strong pressure gradient. Wagner and Oszarek [9] experimentally and analytically examined the overall pressure gradient’s role in producing secondary flow inside a nozzle, showing that increasing the Reynolds number enhances the secondary flow velocity. Torne et al. [10] analytically analyzed the rotating boundary layer over a disk, demonstrating that for small angles, the asymmetric rotating flow near the edge resembles Blasius flow with minimal reverse flow effects; however, for larger angles, the reverse flow inside the boundary layer is significant. Wiedemann [11] analytically studied three-dimensional boundary layer flow and derived nonlinear ordinary differential equations. The boundary layer thickness fully developed downstream, and transverse flow inside the boundary layer was not observed. Stradza et al. [12] compared two methods for calculating three-dimensional boundary layer flow over supersonic wings: an analytical two- dimensional assumption and a numerical three-dimensional finite element solution. The analytical approach provided acceptable cross-flow estimates.

Bhattacharya and Pope [13] conducted an analytical study of the boundary layer and energy equations with the cross-flow, simplifying the nonlinear governing equations using similarity solutions. They showed that thermal dissipation increases the Prandtl number and reduces heat transfer rates. Feng and Li [14] analytically studied three-dimensional boundary layer cross flow with laminar flow and a moving boundary, revealing velocity profiles over oscillating plates differ significantly from Stokes’ oscillating plate model. Arangra et al. [15] investigated cross-flow effects at low Reynolds numbers and found even small amounts of cross-flow can significantly affect transport at low Reynolds numbers. Hossein et al. [16] conducted experimental and theoretical studies of the rotating boundary layer on a long rotating cone under axial flow, showing that flow instability transitions to a spiral Görtler-type vortex flow.

Most scientific problems and phenomena, such as magnetohydrodynamic (MHD) fluid flows, are inherently nonlinear and are described by nonlinear equations. With the advent of high- memory computers and their continual advancement, solving linear problems has become increasingly easier and more convenient. However, obtaining exact analytical solutions for nonlinear problems remains a challenging task. Generally, an ideal analytical method should first be able to effectively generate approximate expressions and second, guarantee that these approximations maintain acceptable accuracy across the entire range of physical parameters. These two criteria serve as suitable standards for evaluating the advantages and limitations of various analytical approaches for solving nonlinear problems [17].

In general, there are three main approaches to solving engineering equations and problems: experimental, numerical (computational), and analytical methods. Analytical methods can be further divided into exact analytical solutions and approximate analytical solutions. Experimental methods allow the investigation of problems closer to real-world conditions. However, they require advanced equipment and modeling, and face challenges such as laboratory environmental effects, measurement errors, and high costs, which limit their frequent use. Although numerical solutions offer clear advantages over analytical ones— provided their validity is confirmed—and can solve nonlinear equations within a computational environment relatively easily, they yield discrete data points rather than continuous curves. Completing these data sets often requires significant time and expense. Moreover, numerical results generally provide a limited comprehensive understanding of nonlinear equations. Numerical methods also encounter difficulties in problems where nonlinear equations do not have unique solutions or have multiple solutions.

Given the above considerations, the present study aims to develop an analytical solution for free parabolic flow over a flat plate in the presence of a magnetic field.

Materials and Methods

Figure 1 illustrates the steady, incompressible flow profile of a parabolic flow over a flat plate in the presence of a magnetic field.

Figure 1: Problem Schematic – Velocity Distribution in the Presence of a Magnetic Field

To work with the differential form of the fundamental equations, the symbols u, v, and w are used to denote the velocity components in the x , y, and z -directions, respectively. Consider an infinitesimally small control volume in the shape of a rectangular cuboid fixed in the xyz

coordinate system. This control volume corresponds to a general flow V ( x , y , z , t ) , measured relative to the xyz frame. According to the principle of mass conservation, the net outflow rate of mass through this control volume equals the rate of mass decrease within the volume. Therefore, the continuity equation is expressed as:

which, for two-dimensional incompressible flow, is given by:

Figure 2 : Infinitesimal Control Volume

If an infinitesimal particle with mass dm , having a rectangular cuboid shape at time t , is part of the velocity field V ( x , y , z , t ) , then in an inertial reference frame, according to Newton’s

second law, it is expressed as:

where T is the stress tensor, B is the body force, V is the velocity, and ρ is the fluid density, which are defined in the Cartesian coordinate system as follows:

Therefore, the momentum equation in the presence of a magnetic field for an incompressible Newtonian fluid is examined as follows [18].

These correspond, respectively, to the continuity equation, the momentum equation, and Ohm’s

law. The total magnetic field is denoted as B  B  b . When the fluid is non-inductive, the

magnetic Reynolds number is much smaller than one, allowing the induced magnetic field

to be neglected [19]. Therefore, B  B . According to Ohm’s law, equation (6), in the absence

of an external electric and neglecting the induced electric field, can be written as:

where J is the magnetic current density and σ is the electrical conductivity. For a parabolic

flow over a flat plate subjected to a magnetic field B applied in the Z-direction, the magnetic current density and Lorentz force are given, respectively, by:

When the velocity exists in the direction of the parabolic flow (x, z), the Lorentz force is given by:

Therefore, the momentum equations, with the addition of the Lorentz force, can be written as follows [20]:

These correspond, respectively, to the momentum equations in the x and z directions. The boundary conditions related to the governing equations are described as follows:

Normal velocity and tangential velocity The normal velocity V n and tangential velocity V t are defined as follows:

1. 

Therefore, g(z) will be:

Findings

To verify the accuracy and validity of the results, they were compared with numerical solutions as well as with previously reported results for special cases of the problem. Subsequently, the effects of varying different parameters on the velocity profile were examined.

Figure 3: Comparison of the hybrid solution with reference [18] for the special case without the presence of a magnetic field

Figure 4: Comparison of analytical results with numerical results for different values of the magnetic parameter N

The curves in Figure (3) show the results obtained by the hybrid method compared with reference [20] for the special case when the magnetic field is absent. Additionally, Table (2)

presents the analytical results alongside the numerical results corresponding to Figure (4) in the presence of the magnetic field.

Table 2: Differences between the results of the combined analytical and numerical methods for when N=
	N=0.0			N=0.1			N=0.2			N=0.4
	C.M.	Num.	Diff.	C.M.	Num.	Diff.	C.M.	Num.	Diff.	C.M.	Num.	Diff.
0. 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0	0.000 0
0. 4	0.132 9	0.133 0	0.000 1	0.077 4	0.077 4	0.000 0	0.044 4	0.044 5	0.000 1	0.017 1	0.017 1	0.000 0
0. 8	0.265 0	0.265 1	0.000 1	0.155 8	0.155 9	0.000 1	0.090 3	0.090 3	0.000 0	0.035 3	0.035 3	0.000 0
1. 2	0.394 3	0.394 4	0.000 1	0.235 7	0.235 7	0.000 0	0.138 7	0.138 7	0.000 0	0.055 7	0.055 7	0.000 0
1. 6	0.517 2	0.517 5	0.000 3	0.316 9	0.317 1	0.000 2	0.190 8	0.190 7	0.000 1	0.079 6	0.079 6	0.000 0
2	0.630 6	0.630 7	0.000 1	0.399 2	0.399 3	0.000 1	0.247 2	0.247 2	0.000 0	0.108 3	0.108 3	0.000 0
2. 4	0.729 8	0.730 0	0.000 2	0.481 3	0.481 4	0.000 1	0.308 9	0.308 8	0.000 1	0.143 3	0.143 3	0.000 0
2. 8	0.812 6	0.812 6	0.000 0	0.561 8	0.561 9	0.000 1	0.375 9	0.375 7	0.000 2	0.186 3	0.186 4	0.000 1
3. 2	0.879 1	0.879 2	0.000 1	0.639 3	0.639 1	0.000 2	0.448 2	0.448 0	0.000 2	0.239 5	0.239 4	0.000 1
3. 6	0.922 5	0.922 3	0.000 2	0.710 9	0.711 2	0.000 3	0.525 0	0.524 9	0.000 1	0.304 1	0.304 2	0.000 1
4	0.964	0.956 6	0.000 2	0.776 8	0.776 9	0.000 1	0.605 7	0.605 2	0.000 5	0.382 8	0.382 6	0.000 2
4. 4	0.977 2	0.977 0	0.000 2	0.835 3	0.835 4	0.000 1	0.687 8	0.687 5	0.000 3	0.476 2	0.476 2	0.000 0
4. 8	0.988 8	0.988 9	0.000 1	0.886 5	0.886 4	0.000 1	0.769 8	0.769 9	0.000 1	0.585 9	0.585 7	0.000 2
5. 2	0.995 2	0.995 3	0.000 1	0.930 2	0.930 3	0.000 1	0.850 3	0.850 5	0.000 2	0.710 7	0.710 9	0.000 2
5. 6	0.998 4	0.998 5	0.000 1	0.967 7	0.967 8	0.000 1	0.927 5	0.927 6	0.000 1	0.849 9	0.850 0	0.000 1
6	1.000 0	1.000 0	0.000 0	1.000 0	1.000 0	0.000 0	1.000 0	1.000 0	0.000 0	1.000 0	1.000 0	0.000 0

From Table (2) and Figure (4), it can be concluded that in this problem, the results obtained from the hybrid analytical method show good agreement with those from the numerical solution method.

Investigation of the Effect of Various Parameters on the Velocity Profile Using the hybrid analytical method, the dimensionless tangential velocity profile and energy ratio for different values of the magnetic parameter, the initial flow line slope at the start of the plate, and the curvature parameter of the parabolic streamlines are illustrated in the following figures.

Figures 5-a to 5-e show the effect of the magnetic parameter on the tangential velocity for

different flow line slope angles at the leading edge of the plate   0 and various parabolic

flow line curvatures  In Figures 5-a and 5-b, where the flow line curvature is 0 and 20 degrees, the tangential velocity increases more rapidly with the increase of the magnetic parameter. However, as the flow line curvature rises to 40, 60, and 80 degrees, the growth of tangential velocity decreases with increasing magnetic parameters, although its maximum velocity remains higher than the free stream velocity. Figures 6-a to 6-e illustrate the effect of the

magnetic parameter on

  30

for different  values. When the curvature of the streamlines

is zero, the tangential velocity grows faster as the magnetic parameter increases. However, with an increase in the streamline curvature to 20 degrees, as the magnetic parameter reaches N=0.4, the growth rate of the tangential velocity becomes so large that reverse flow occurs. With a further increase in the curvature of the streamlines to 40 degrees, increasing the magnetic parameter causes the growth of tangential velocity to begin decreasing. This trend continues at angles of 60 and 80 degrees, although the maximum velocity still exceeds the free stream velocity. Additionally, Figures 7-a to 7-e examine the effect of the magnetic parameter for

different   60

values at  At zero curvature, the tangential velocity grows faster with

increasing magnetic parameter. As the curvature increases, reverse flow becomes observable at 20 and 40 degrees. From 60 degrees onward, the growth rate of tangential velocity begins to decrease, but at 80 degrees, the tangential velocity increases again with the magnetic parameter, surpassing the free stream velocity.

,

Figures 8-a to 8-c show the effect of the curvature of the parabolic streamlines  for the

magnetic parameter

  0 , 30, 60 and the slope of the streamline at the leading edge of the

plate at N=0.2, respectively. With an increase in the magnetic parameter, the maximum tangential velocity increases. In Figure 8-c, when the curvature of the lines is 0 and 60 degrees, a reverse flow occurs; however, increasing the curvature to 80 degrees or changing it to 20 or 40 degrees eliminates the reverse flow.

Figure 9 shows the effect of the magnetic parameter on the ratio of kinetic energy inside the

boundary layer to that of the free stream, for the slope of the streamline at the leading edge of

the plate

  60

and the curvature of the parabolic streamlines

 40 . As the magnetic

parameter increases, this ratio exceeds one, indicating an increase in kinetic energy within the boundary layer.

CONCLUSION

In this study, the influence of various parameters on the parabolic velocity distribution of fluid flow over a flat plate under a magnetic field was examined using a hybrid analytical method. The results showed that as the magnetic parameter increases, the tangential velocity grows faster and becomes flatter. At certain flow curvature angles, this leads to an increase in the maximum velocity within the boundary layer. Furthermore, increasing the flow curvature angle parameter also raises the tangential velocity and eliminates flow reversal observed at small angles. An increase in the magnetic parameter enhances the kinetic energy within the boundary layer relative to the free stream. Based on these findings, it is recommended to investigate the flow of a second-order viscoelastic fluid over a porous inclined flat plate.

REFERENCES

1. Jr, J.D.A., Ludwig Prandtl’s Boundary Layer, in Physics Today. 2005, American

   Institute of Physic. p. 7.

2. Alfvén, H., On the cosmogony of the solar system. Stockholms Observatoriums Annaler, 1942. 14: p. 2.1-2.33.

3. Abbas, Z., M. Sajid, and T. Hayat, MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel. Theoretical and Computational Fluid Dynamics, 2006. 20(4): p. 229-238.

4. Andersson, H., MHD flow of a viscoelastic fluid past a stretching surface. Acta Mechanica, 1992. 95(1-4): p. 227-230.

5. Mukhopadhyay, S., Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation. Ain Shams Engineering Journal, 2013. 4(3): p. 485-491.

6. Mukhopadhyay, S., G. Layek, and S.A. Samad, Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity. International journal of heat and mass transfer, 2005. 48(21-22): p. 4460-4466.

7. Xu, H., S.-J. Liao, and I. Pop, Series solutions of unsteady three-dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate. European Journal of Mechanics-B/Fluids, 2007. 26(1): p. 15-27.

8. LOOS, H.G., A Simple Laminar Boundary Layer with Secondary Flow. Journal of the Aeronautical Sciences 1955. 122(1): p. 6.

9. W. B. WAGNER, J.A.O., An Investigation of the Corner Secondary Flows Generated in Planar Nozzles. Journal of Fluids Engineering, Trans. ASME, 1974: p. 24.

10. M. Toren, A.S., M. Ungarish, Rotating Flow Over a Disk Sector. Journal of Applied Mechanics 1982. 49: p. 6.

11. Weidman, P.D., New solutions for laminar boundary layers with cross flow. Zeitschrift

    f¨ur angewandte Mathematik und Physik ZAMP, 1997. 48: p. 16.

12. Peter Sturdza, V.M.M., Ilan M. Kroo, Boundary Layer Calculation for Preliminary Design Of Wings In Supersonic Flow. American Institute of Aeronautics & Astronautics 1999: p. 11.

13. Pop, K.B.a.I., Heat transfer for boundary layers with cross flow. Chin. Phys. B V, 2014.

    23(2): p. 6.

14. Fang, T. and F.L. Chia-fon, Three-dimensional wall-bounded laminar boundary layer with span-wise cross free stream and moving boundary. Acta mechanica, 2009. 204(3- 4): p. 235.

15. Uranga, A., et al. Preliminary investigation into the effects of cross-flow on low Reynolds number transition. in 20th AIAA Computational Fluid Dynamics Conference. 2011.

16. Hussain, Z., et al., The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream. Journal of Fluid Mechanics, 2016. 788:

    p. 70-94.

17. Liao, S., An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 2010. 15(8): p. 2003-2016.

18. Weller, H.G., et al., A tensorial approach to computational continuum mechanics using object-oriented techniques. Computers in physics, 1998. 12(6): p. 620-631.

19. Roidt, M. and RD.Cess, An Approximate Analysis of Laminar Magnetohydrodynamic Flow in the Entrance Region of a Flat Duct. Journal of Applied Mechanics, 1962. 29(1):

    p. 6.

20. Loos, H.G., A Simple Laminar Boundary Layer with Secondary Flow. Journal of the Aeronautical Sciences, 1955(1): p. 6.