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Magnetohydrodynamic Stability of Laminar Flow in the Entrance Region of a Parallel-Plate Channel

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Magnetohydrodynamic Stability of Laminar Flow in the Entrance Region of a Parallel-Plate Channel Scholars' Mine Masters Theses Student Theses and Dissertations 1970 Magnetohydrodynamic stability of laminar flow in the entrance region of a parallel-plate channel Thomas Eldon Eaton Follow this and additional works at: https://scholarsmine.mst.edu/masters_theses Part of the Mechanical Engineering Commons Department: Recommended Citation Eaton, Thomas Eldon, "Magnetohydrodynamic stability of laminar flow in the entrance region of a parallel- plate channel" (1970). Masters Theses. 7170. https://scholarsmine.mst.edu/masters_theses/7170 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. MAGNETOHYDRODYNAMIC STABILITY OF LAMINAR FLOW IN THE ENTRANCE REGION OF A PARALLEL-PLATE CHANNEL BY 1~\ I THOMAS ELDON EATON, 1948- A THESIS submitted to the faculty of THE UNIVERSITY OF MISSOURI-ROLLA in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING Rolla, Missouri 1970 T2495 c.l 97 pages Approved by cj,_S,d~~- (advisor) IlL~ 1880:16 ii ABSTRACT The hydrodynamic stability of laminar flow of an electrically conducting fluid flowing in a parallel­ plate channel with an applied transverse magnetic field is investigated. The linear perturbation theory of hydrodynamic stability along with the assumption of low magnetic Reynolds number is applied to the governing equations to derive the governing rnagnetohydrodynarnic stability equation. A finite difference scheme is employed to numerically solve the magnetohydrodynamic stability equation. Neutral stability characteristics of the flow in the entrance region are obtained and presented. The neutral stability characteristics of the fully developed Hartmann flow are also re-examined and compared with those of a previous investigation which utilizes an analytical method of solution. A linearized velocity solution for developing flow is used in the stability calculations. The numerically determined neutral stability results for the fully developed Hartmann flow are in excellent agree- ment with those of the analytical solution. The results presented here for Hartmann flow are believed to be more accurate owing to the more exact nature of the numerical solution. iii It is found that the critical Reynolds number for the developing flow induced by the uniform inlet velocity profile decreases rapidly with axial distance in the entrance region and monotonically approaches the fully developed value at large axial distances. For Hartmann numbers of less than 2, the decrease is monotonic; how­ ever, for larger Hartmann numbers, the critical Reynolds number decreases rapidly and goes below that of the fully developed Hartmann flow somewhere in the entrance region of the channel and then approaches this fully developed value monotonically from below at large axial distances. The critical Reynolds number for the developing flow induced by a parabolic inlet velocity profile approaches the fully developed critical Reynolds number monotonically from a value of 3850 at the inlet for Hartmann numbers of less than 2. For larger Hartmann numbers, the critical Reynolds number over-shoots the fully developed value, approaching it monotonically from above at large axial distances. iv ACKNOWLEDGEMENT The author wishes to express his most sincere gratitude to Professor Ta-Shen Chen for his assistance, advice, and encouragement in this work. Professor Chen has continuously provided vital recommendations and criticisms throughout the investigation. The author is also indebted to Mr. Gwok-Liang Chen and to Mr. Lung-Mau Huang for many discussions con­ cerning the computational aspects of the present investi­ gation. The support of the computational expenses by the Mechanical Engineering Department of the University of Missouri-Rolla is gratefully asknowledged. The author would like to thank Mrs. Connie Hendrix for typing the manuscript. Finally, the author gratefully acknowledges the National Science Foundation for the financial support received during this educational endeavor. v TABLE OF CONTENTS Page ABSTRACT . ii .ACKNOWLEDGEMENT iv TABLE OF CONTENTS v LIST OF FIGURES vii LIST OF TABLES . viii NOMENCLATURE • . ix I. INTRODUCTION . 1 A. General Background . 1 B. Previous Studies . 4 C. The Present Investigation 6 II. FORMULATION OF THE STABILITY PROBLEM 8 A. The Governing Equations . 9 B. The Basic Flow . 13 c. The Stability Equation . 17 D. The Boundary Conditions . 22 III. THE FINITE DIFFERENCE METHOD OF SOLUTION . 24 A. Formulation of the Finite Difference Equations . 24 B. The Eigenvalue Problem . 30 c. The Effect of Step Size on Eigenvalues 32 D. Method of Generating the Neutral Stability Curves . • . • . • 38 vi TABLE OF CONTENTS (Continued) Page IV. THE NEUTRAL STABILITY RESULTS . 41 A. Fully Developed Hartmann Flow . 41 B. Developing Flow - Parabolic Inlet Velocity Profile 44 C. Developing Flow - Uniform Inlet Velocity Profile . 49 D. The Effect of Velocity Profile on the Stability Results . 53 V. CONCLUSION 61 VI. REFERENCES 64 VII. APPENDICES 67 A. Derivation of the Magnetohydrodynamic Stability Equation, Equation (26) 67 B. Tabulation of Numerical Results . 75 VIII. VITA 85 vii LIST OF FIGURES Figure Page 1 Neutral Stability Curves for Fully Developed Hartmann Flow . 42 2 Representative Neutral Stability Curves for the Developing Flow, Parabolic Inlet Velocity Profile 46 3 Variation of the Critical Reynolds Number with Axial Position, Parabolic Inlet Velocity Profile 47 4 Representative Neutral Stability Curves for the Developing Flow, Uniform Inlet Velocity Profile • . • 50 5 Variation of Critical Reynolds Number with Axial Position, Uniform Inlet Profile • 51 6 The Behavior of W, W', and W" Near the Wall for X= 0.002, M = 1 and 4, Uniform Inlet Velocity Profile 56 7 The Behavior of W, W', and W" Near the Wall for X= 0.002, M = 1 and 4, Parabolic Inlet Velocity Profile 59 viii LIST OF TABLES Table Page 1 The Effect of Step Size on Eigenvalues for the Fully Developed Hartmann Flow, M = 10 and 6. ...••... 34 2 The Effect of Step Size on Critical Reynolds Numbers for the Fully Developed Hartmann Flow . • • . • . • . 35 3 The Effect of Step Size on Eigenvalues for M = 4, Uniform Inlet Velocity Profile . 37 4 Selected Step Sizes Used in the Stability Calculations . • • . • . • • . 39 B-1 The Relationship Between X and X 75 B-2 Neutral Stability Characteristics for the Fully Developed Hartmann Flow • • . • 76 B-3 Comparison of Critical Stability Character­ istics for the Fully Developed Hartmann Flow . · · · 78 B-4 Neutral Stability Characteristics for the Developing Flow, Parabolic Inlet Velocity Profile . 79 B-5 Variation of the Critical Wave and Reynolds Numbers with Axial Position, Parabolic Inlet Velocity Profile . • • . • • 81 B-6 Neutral Stability Characteristics for Developing Flow, Uniform Inlet Velocity Profile . 82 B-7 Variation of the Critical Wave and Reynolds Numbers with Axial Position, Uniform Inlet Velocity Profile • . • • • . • • • • • • • 84 ix NOMENCLATURE al,a2,a3 elements of matrix [B) , Equation (38) al,a2,a3 real part of elements of Matrix [A], Equation (40) -+ B magnetic field vector bl,b2,b3 imaginary part of elements of matrix [A] , Equation (40) c wave velocity of disturbances -+ E electric field vector g discrete value of the transformed function of ~ -+ H magnetic field vector H applied magnetic field normal to the flow 0 h ratio of induced to applied magnetic field, H /H X 0 h+ perturbed axial magnetic field X h+ perturbed transverse magnetic field y -+ J electrical current density vector L channel half-he.ight M Hartmann number, ~m H0 L(cr/uf)l/Z N number of steps in finite difference mesh p resultant pressure p pressure of main flow + (' p perturbation pressure R Reynolds number, LU/v magnetic Reynolds number, LU/A Rm r finite difference mesh size or step si.ze t dimensionless time, t*U/L X NOMENCLATURE (Continued) t* dimensional time u resultant axial velocity component u axial velocity component of main flow u + perturbed axial velocity u average axial velocity -+v velocity vector v resultant transverse velocity component v transverse velocity component of main flow v + perturbed transverse velocity w dimensionless axial velocity, u/U X dimensional stretched axial coordinate X dimensionless stretched axial coordinate, (x/L)/R x* dimensional axial coordinate X dimensionless axial coordinate, (x*/L)/R y* dimensional transverse coordinate y dimensionless transverse coordinate, y*/L \1 the del operator gradient the Laplacian operator dimensionless wave number dimensional wave number a. eigenvalues of a. = tan a. J.. J.. J.. central difference operator E: (M ,X) weighting function, Equation (13} xi NOMENCLATURE (Continued) e dimensionless amplitude function of magnetic perturbation, 9*/LH 0 9* dimensional amplitude function of magnetic perturbation a constant, 1/~mcr averaging operator magnetic permeability absolute or dynamic fluid viscosity fluid density a electrical conductivity dimensionless amplitude function of velocity perturbation, ~*/LU dimensional amplitude function of velocity perturbation X dimensionless magnetic stream function, X*/LH0 x* dimensional magnetic stream function dimensionless fluid stream function, ~*/LU dimensional fluid stream function Superscripts ','',''',iv derivatives with respect to y main flow component * dimensional quantity +
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