Science Journal of Applied Mathematics and Statistics
Volume 5, Issue 1, February 2017, Pages: 31-40

Hydromagnetic Turbulent Flow Between Two Parallel Infinite Plates

Kennedy John Mwangi Karimi, Dickson Kande Kinyua

School of Pure and Applied Sciences, Karatina University, Karatina, Kenya

(K. J. M. Karimi)
(D. K. Kinyua)

Kennedy John Mwangi Karimi, Dickson Kande Kinyua. Hydromagnetic Turbulent Flow Between Two Parallel Infinite Plates. Science Journal of Applied Mathematics and Statistics. Vol. 5, No. 1, 2017, pp. 31-40. doi: 10.11648/j.sjams.20170501.15

Received: September 28, 2016; Accepted: November 10, 2016; Published: January 21, 2017

Abstract: In this study we shall investigate hydromagnetic turbulent unsteady flow of an incompressible electrically conducting fluid between two parallel infinite plates. The flow variables such as velocity and thermodynamic properties at every point of fluid vary with respect to time. The effect of an applied transverse magnetic field normal to the main flow direction on the dynamic behavior of the fluid when the lower plate is stationary and the upper plate is impulsively started in opposite direction at constant velocity shall be investigated. Further, we shall investigate how the various parameters such as Peclet Number and Eckert Number affect the flow; in particular, velocity and temperature profiles. A finite difference method shall be used to solve the coupled non-liner and dimensionless partial differential equations governing this problem.

Keywords: Magnetohydrodynamics, Incompressible, Dimensionalization, Temperature Profiles

1. Introduction

1.1. The Background of Magnetohydrodynamics

Magnetohydrodynamics (MHD) is the academic discipline that studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water. The word magnetohydrodynamics (MHD is derived from magneto- meaning magnetic field, and hydro-meaning liquid, and dynamics meaning mechanical properties of fluid. Hydrodynamics on the other hand is the study of fluid flow and the forces that cause the flow in the absence of an electromagnetic field. Fluid is considered to be any matter that undergoes deformation when an external force is applied. In MHD, a current is induced when the fluid conductor moves in magnetic field. This is the central point of MHD theory. As a result, when viscous conducting fluid flow in presence of a transverse magnetic field, electromagnetic forces acts on the fluid particles thereby altering their geometry of motion. The momentum equation describing the motion of fluid in MHD includes body forces, which act on fluid particles from a distance. The applied magnetic field acts on both electronic and ionized atoms to produce dynamic effect. This mass motion in turn produces modification in the electromagnetic field. Consequently we have to deal with complicated coupled system of the partial differential equation representing the flow. The mechanical motion of the system can then be described in terms of a single conducting fluid with hydromagnetic variables of density, velocity and pressure. At low frequency it is customary to neglect the displacement current in Ampere’s law. This is then the approximation that is called magnetohydrodynamics (Jackson 1975). The flow can either be steady or unsteady, where the flow variables such as velocity and thermodynamics properties are independent of time and dependent on time respectively. Turbulent flow occurs when there are disturbances present in a fluid due to a variety of factors such as porous walls, boundary roughness, and variation in the physical properties of fluid motion among others. Under uniform steady flow, particles of the fluid move in straight path. In the case of turbulent flow, the fluid particles cross each other’s paths in disorderly manner varying velocities and pressures. Hartman in (1937) was first to discuss both experimentally and theoretically the hydro magnetic flow between two parallel plates.

1.2. Boundary Layers

When a fluid flows over a stationary surface, e.g. the bed of a river or the wall of a pipe, the fluid touching the surface is brought to rest by the shear stress o at the wall. The velocity increases from the wall to a maximum in the main stream of the flow, as shown in figure 1 below.

Fig. 1. Boundary layer.

1.3. Dimensional Analysis

It is built on the principle of dimensional homogeneity that states that an equation expressing a physical relationship between quantities must be dimensionally homogenous i.e. dimensions of each side of equations must be the same. It affords a means of ascertaining the forms of physical equations from knowledge of relevant variables and their dimensions. It proves a powerful tool in formulating problems that defy analytical solutions and must be solved experimentally.

This method has application in nearly all fields of engineering in particular in fluid mechanics and heat transfer. It is an important tool that presents experimental results in a concise form. In our study, dimensional analysis has been used in the non-dimensionalization of the governing equations. Non-dimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. To non-dimensionalize these equations that describe our flow system appropriately we must do the following:

1)    Identify all the independent and dependent variables.

2)    Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined.

3)    Divide through by the coefficient of the highest order polynomial or derivative term.

4)    Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1.

5)    Rewrite the system of equations in terms of their new dimensionless quantities.

1.4. Literature Review

1.5. Statement of the Problem

In this problem we shall investigate the behavior of a turbulent and unsteady hydromagnetic flow of a viscous conducting fluid between two parallel infinite plates. A uniform magnetic field is applied in a direction normal to the plates. The upper plate is impulsively started in opposite direction to that of the flow parallel to the x-axis at constant velocity U0, while the lower plate is stationery. Our task will be to investigate the velocity profiles and temperature profiles of this turbulent unsteady flow, which has received little attention in previous related research.

1.6. Justification

MHD is a field with a wide range of practical applications particularly in engineering. Scientific research in electricity and magnetism is on a worldwide scale. In many engineering practical applications (e.g. an elevator, ball bearing etc) we encounter conducting fluids flowing between moving boundaries. Our problem is a particular case when the upper boundary moves at a constant velocity and in opposite direction to that of flow and the lower boundary is stationery. Most fluids in engineering are unsteady. This wide range of MHD applications in engineering gives our study a practical framework.

1.7. Objective of the Study

To investigate;

The effects of Eckert Number on both the temperature profiles and the velocity profile.

2. Description of the Flow

Fig. 2. Magnetic field applied orthogonally to the direction of flow.

We consider the turbulent flow of a viscous electrically conducting incompressible fluid, bounded by an infinite horizontal non porous plate in the presence of a strong magnetic field applied orthogonally to direction of fluid flow. A uniform magnetic field H is acting at an angle of 900 to the horizontal plate and the plate is assumed to be electrically non-conducting. Since the plate occupying the plane y=0 is of infinite extent, the physical conditions depends on y only.

2.1. Governing Equations

The ideal MHD equations consist of the continuity equation, the momentum equation, Amperes law and a temperature evolution equation. In the theory of turbulence, Reynolds decomposition (RD) is an important tool. Reynolds decomposition is a mathematical technique to separate the average and fluctuating parts of a quantity. For example, for a quantity u the decomposition would be

(1)

Where denotes the time average of  (the steady component), and  the fluctuating part (or perturbations). The fluctuating part is defined such that its time average equals zero. This method allows us to simplify the Navier-Stokes equations by substituting in the sum of the steady component and perturbations to the velocity profile and taking the mean value. This results to a non-linear equation containing a term known as the Reynolds stresses which gives rise to turbulence.

2.2. Approximation and Assumptions

1)    The fluid is incompressible.

2)    Thermo conductivity, electrical conductivity and coefficient of viscosity are constants.

3)    There is no external applied electric field (E=0).

4)    Compared with the speed of light, the fluid velocity (q) is very small.

5)    The fluid does not undergo any chemical reaction.

6)   The induced magnetic field produced by the motion of the electrically conducting fluid is negligible and hence magnetic Reynolds number is very small.

7)    The plates are insulated

8)    The fluid flow pressure gradient is a constant

9)    The velocity component in the y direction is zero (v=0)

3. Model of the Flow

3.1. Equations Governing Fluid Flow

Conservation of mass The equation of conservation of mass is given by

(2)

Where i=1, 2, 3

Or

(3)

Assuming that density is a constant this equation reduces to

(4)

For a two dimensional flow equation (4) becomes

Or

(5)

For this flow the parallel plates are infinite in length and therefore no flow variable is a function of x. Equation (5) becomes

(6)

For turbulent flow we decompose velocity u into the average component  and the fluctuating velocity component and then take the average

(7)

On averaging we have

(8)

or

(9)

Conservation of momentum

The equation of conservation of momentum is given by

(10)

This equation is based on the Newton’s second law of motion ,that is, the net rate of change of momentum must equal the net sum of forces acting on the fluid. This equation is also known as the Navier-Stokes Equation

The body forces considered in this problem are,  ,that is, the electromagnetic force and gravity respectively. The electromagnetic force is however modified to since the electric field is assumed to be negligible. For a two dimensional flow the flow component in the x direction becomes

(11)

Since the parallel plates are infinite in extend, the velocity profiles at various x- positions depend on y-coordinate and not x, however pressure in this flow is a function of x. Further since there is no flow in the y direction v=0. This equation thus becomes

(12)

In parallel flows the effect of the force of gravity is insignificant and therefore it can be ignored. Equation (12) thus reduces to

(13)

And for pressure gradient equals to a constant this equation simplifies to

Using Reynolds decomposition on this equation yields

(14)

Or

(15)

On averaging and applying the laws of ensemble equation (14) becomes

(16)

Now equation (17) defines Ohm’s law i.e.

(17)

And since  the displacement current is usually negligible at fluid velocity q, the law reduces to

(18)

Thus on substituting (18) in (17) yields

(19)

Conservation of Energy

This equation is the mathematical statement of the principle of conservation of energy. It states that energy cannot be created nor destroyed under normal conditions but can be transformed from one form to another. It is given by

(20)

Where

Cp is specific heat –the amount of heat energy required to change temperature of a body by only one unit

K is the coefficient of thermal conductivity which depends on type of fluid.

Is the material derivative expressed as follows

(21)

Or

(22)

Or

(23)

This term is due to conduction of heat energy

Is the dissipative heat, which can be expressed as follows

(24)

In two dimensions equation (20) becomes

(25)

For parallel flow with the plates infinite in extend, no flow variable is a function of x, and v=0 (since there is no flow in the y direction) equation (25) becomes

(26)

Using Reynolds decomposition on (26) yields

(27)

On averaging equation (27) becomes

(28)

Or

(29)

The final set of equations is thus given by

(30)

and

(31)

3.2. Boundary Conditions

The initial and boundary conditions for this particular flow are

Initial conditions

U (0, 0) =0 (0, 0) =0

Boundary conditions

U (0, y) =0 (0, y) =0

U (t, 0) =0 (t, 0) =0

U (t,) =-U0 T (t,) =T2

3.3. Non-dimensionalization of Equations

We select certain characteristic quantities. The characteristic velocity will be U0. If the temperature difference T-T2 is divided by the maximum possible temperature difference T1-T2, a dimensionless form of the dependent variable may be defined as θ= (T-T2/T1-T2) where T1 and T2 are temperatures next to the plate and free stream temperature and 0 ≤θ ≤1

The independent and dependent variables may then be non-dimensionalised according to

y =

u = u*u0                                  (32)

t = p =p0p*

(

Equation (30) thus becomes

(33)

Simplifying and dropping the asterisks yields

(34)

Dividing by  yields

(35)

Similarly equation (31) can be rearranged as follows

(36)

And on substituting (32) in (36) yields

(37)

On simplifying, dropping the asterisks and dividing by yields

(38)

or

(39)

(40)

On non-dimensionalisation of the initial and boundary conditions and dropping the asterisks we get

U (0, 0) = 0 (0, 0) =0

U (0, y) = 0 (0, y) =0

U (t, 0) = 0 U (t, ) =-1

(t, 0) =0  (t, ) =1

3.4. Important Non-Dimensional Numbers

From equations (35) and (38) the following non-dimensional numbers have emerged.

3.4.1. Hydrodynamic Reynolds Number Re

This is the ratio of inertia force to viscous force. It gives the relative significance of inertia force in fluid flow problem and is expressed as

(41)

When Reynolds number is large, the inertia forces are predominant and the effects of viscosity are negligible and when its value is small, the flow is dominated by viscous force and inertia force can be neglected.

3.4.2. Peclet Number Pe

This is a non-dimensional number Pe defined as

This number is small when viscous force is small while thermal force is large.

3.4.3. The Eckert Number Ec

It is the ratio of kinetic energy of the flow to the thermal energy.

It is expressed as

(42)

3.4.4. Pressure Number

This is a non-dimensional number  defined as

Or

This is the ratio of pressure forces to the inertia force.

4. Method of Solution

4.1. Finite Difference Method

In this study the equations governing fluid flow are non-linear and thus their exact solution is not possible; in order to solve these equations a fast and stable difference method has been developed. The difference method used should be consistent, stable and convergent. A method is convergent if as more grid points are taken or step size decreased, the numerical solution converges to the exact solution. A method is consistent if the truncation error tends to zero as the step size decreases, and stable if the effect of any single fixed round off error is bounded.

We shall now apply a numerical method approach to solve the inhomogeneous partial differential equations. The particular method for this problem will be the central finite difference method, with a uniform time step (m) and spatial step (n). It is defined by expanding f (xo+h) and f (xo-h) using the Taylors series and then solving the two;

(43)

(44)

By subtracting (44) from (43) and for sufficiently small, h yields.

(45)

Similarly by adding (44) to (43) and for sufficiently small, h yields

(46)

In two dimensions the following partial derivatives can be expressed as follows

(47)

Let the mesh point at time t be denoted by . Then the forward difference for the first order derivative with respect to time t will be.

(48)

And thus

(49)

Equations (39) and (40) thus becomes

(50)

Substituting (central difference method)

(51)

(52)

(53)

(54)

Substituting (central difference method)

(55)

(56)

(57)

The two final equations to be solved are thus

(58)

(59)

4.2. Definition of Mesh

In order to give a relation between the partial derivatives in the differential equation and the function values at the adjacent nodal points, we use a uniform mesh. Let the t-y plane be divided into a network of uniform rectangular cells of width m and height n as shown in figure 3 below.

Fig. 3. Uniform rectangular mesh.

Since t-axis is along the infinite horizontal plate then t varies from 0 to infinity. If we set i=21 to correspond to t= and  then we have

Initial and boundary conditions

U (0, 0) = 0 (0, 0) =0

U (0, 2.1) = 0 (0, 2.1) =0

U (2.1, 0) = 0 U (2.1,) =-1

(2.1, 0) =0  (2.1,) =1

The computations are done when m=n=0.1

Reynolds Number Re=10

Pressure Coefficient Pc=0.8

Pecklet Number is 1

Eckert Number varies from 0 to 0.35 to 0.7

5. Results and Discussions

5.1. Tabular and Graphical Results

Table 1. The effect of the Eckert number on velocity.

 Ec=0.1 Ec=0.35 Ec=0.7 0 0 0 0 1 0.42523 1.3543 2.01921 2 0.78487 2.08859 2.75745 3 1.04315 2.50426 3.14721 4 1.27043 2.73271 3.36456 5 1.45 2.90498 3.49561 6 1.56888 2.92693 3.57458 7 1.61894 2.81563 3.61645 8 1.59756 2.59634 3.54562 9 1.5079 2.29903 3.36364 10 1.35856 1.97878 3.11153 11 1.16292 1.59378 2.683244

velocity profiles

Fig. 4. Velocity increases with an increase in Eckert number.

Table 2. The effect of the Eckert number on temperature.

 Ec=0.1 Ec=0.35 Ec=0.7 1 0.075268 0.039551 0.01134 2 0.185963 0.115195 0.05502 3 0.297149 0.214877 0.127943 4 0.430054 0.324874 0.22256 5 0.558765 0.434165 0.319759 6 0.671191 0.551459 0.400204 7 0.776117 0.658201 0.471402 8 0.848983 0.747214 0.528128 9 0.913048 0.824583 0.584311 10 0.963912 0.877218 0.631193 11 0.992123 0.911016 0.67073

Temperature Profile

Fig. 5. Temperature decreases with an increase in Eckert number.

5.2. Discussion of the Results

The effects of the Eckert number on the temperature and velocity profiles.

ii.    An increase in the Eckert number causes an increase in the velocity (figure 4).

iii.   The graph of velocity shows that velocity increases up to a maximum value i.e. the free stream velocity, and then declines to the velocity at which the upper plate is moving.

iv.   An increase in the Eckert number causes a decrease in the temperature (figure 5).

v.   The graph of temperature shows that temperature increases from zero; as the lower plate is maintained at zero temperature, and increases towards the temperature at which the upper plate is maintained.

5.3. Conclusion

A model for the problem was developed carefully putting into considerations appropriate assumptions. A finite difference method was used to solve the resulting equations. From the results it is evident that Eckert number has a fundamental influence in the dynamics of fluid flow and in particular temperature and velocity. It has been shown that an increase in the Eckert number causes an increase in the velocity. Different values of the number were investigated and found to increase velocity as the numbers increased. Velocity however increases up to the free stream velocity, and then declines to the velocity at which the upper plate is moving. It is also evident that an increase in the Eckert number causes a decrease in the temperature. However temperature increases from zero as the lower plate is maintained at zero temperature, and increases in conformity with the boundary conditions.

Nomenclature

 Symbols Quantities B Magnetic field vector (wbm-2) BX, BY, BZ Magnetic flux in x, y, z directions respectively (wbm-2) CP Specific heat at constant pressure (KJ/Kg0C) D Electric displacement vector (Cm-2) Material derivative of velocity ℮ Unit electric charge (C) E Electric field intensity vector (vm-1) with components EX,EY,EZ F Body force (N) H Magnetic field strength with components HX,HY,HZ i, j, k Unit vectors in the x, y, z directions respectively J Current density vector (Am-2) with components JX,JY,JZ K Thermal Conductivity (wm-1k-1) L Characteristic length (m) P Pressure force vector (Nm-2) q Velocity vector with components u,v,w in x,y,z directions respectively Pr Prandtl number (Cpµ/k) Re Hydrodynamics Reynolds number Rσ Magnetic Reynolds number σµeUO L Rh Magnetic pressure number t Time (s) where t t* Dimensionless time T General fluid temperature (K) The steady component of U The fluctuating part of U U0 Characteristic velocity (ms-1) U* Non-dimensional velocity u*,v*,w* Dimensionless velocity components x*,y*,z* Dimensionless Cartesian co-ordinates ρ Fluid density (kgm-3) ρe Electric charge density ( cm-2) µ Coefficient of viscosity (kgm-1s-1) µe Magnetic permeability (Hm-1) σ Electric conductivity (Ω-1m-1) Gradient operator 2 Laplacian operator Φ Viscous dissipation function (s-2) θ Dimensionless fluid temperature

Acronyms and Abbreviations

 MHD Magnetohydrodynamics LHD Left hand side RHD Right hand side RD Reynolds Decomposition

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 Contents 1. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 2. 2.1. 2.2. 3. 3.1. 3.2. 3.3. 3.4. 4. 4.1. 4.2. 5. 5.1. 5.2. 5.3.
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