Lokenath Debnath

Magneto-thermo-elastic plane waves in rotating media

Abstract A study is made of the propagation of plane harmonic waves in an infinite conducting thermo-elastic solid permeated by a primary uniform magnetic field when the entire elastic medium is rotating with a uniform angular velocity. A more general dispersion relation is obtained to determine the effects of rotation, relaxation time and the external magnetic field on the phase velocity of the waves. This result indicates that if the primary magnetic field has a transverse component, then the longitudinal and transverse components of the displacement field are linked together. For the case of low frequency ( Gc ⪡ 1 where κ is the ratio of the wave frequency to the characteristic frequency), the rotation and the thermal relaxation time are found to have no influence on the phase velocity, and the attenuation factor for both finite and infinite electrical conductivity. In the case of high frequency ( κ ⪢ 1), no effect of rotation on the phase speed is observed to the first order of ( 1 κ ), while the relaxation time affects both the phase velocity and the specific energy loss. However, the effects of rotation on the phase speed, the attenuation factor, and the specific energy loss are found only to the second order of ( 1 κ ) in the case of high frequency. Several limiting cases of interest are discussed.

Transform Methods with Applications to Engineering and Operations Research

According to the authors presentation, this 500-page book is the out-growth of successive reelaborations of classroom notes used in a 90-hour course taught at the Politecnico di Milano. The course is primarily addressed to (mature) engineering students at approximately the first or second year graduate level (in the American university system) having familiarity with the basic concepts of linear systems and classical automatic control theory. The aim is to provide control engineers with a working knowledge of the most important modern design techniques of compensa-tors for multivariable linear control systems. The book is addressed not as a research monograph but rather as an effort (probably not yet attempted by others) of giving a unifying and at the same time rather detailed presentation of the most important modern state-space synthesis techniques. Actually, most of the basic recent results (such as pole assignment, various reduced-order observers decoupling, sensitivity theory, and linear optimal control with quadratic performance criterion) are covered in a completely self-contained way. Besides the introduction, the book consists of four chapters, the first three of which (approximately 100 pages each) are dedicated to pole assignment and state reconstructors, decoupling and insensitivity. The last chapter is about 200 pages long and is mainly devoted to optimal control (especially to the linear-quadratic cost problem). Chapter 2 deals with the design of compensators for pole assignment, both in the case of complete information (i.e., the state is totally accessible) and in the partially observable case. After starting and proving Wonhams basic results (using Heymanns simplification in the proof), a detailed discussion of the incomplete information case is given. Luenbergers observers are first introduced, and various reduced-order observations are then discussed, leading to dynamic compensators of order n-rank(C) and n-rank(B) (dual observers of Gopinath). Finally, the author goes through the rather lengthy and technical proofs of the theory of minimal-order observers of Brasch and Pearson. Chapter 3 is dedicated to the decoupling problem. In this part of the book, a noticeable effort is evident to present in an easy readable way the geometrical theories of Basile-Laschi-Marro and Morse-Wonham. The interesting blend of the two approaches presented here may actually be viewed as a personal contribution by the author. Conditions for existence of nondynamic and dynamic compensators to achieve noninteracting control are first carefully discussed, and then the pole assignability for the decoupled blocks is treated. The algorithms for designing these com-pensators are generally to …

On Ekman and Hartmann boundary layers in a rotating fluid

SummaryAn exact solution of the unsteady hydromagnetic boundary layer equation for an incompressible, viscous, electrically conducting rotating fluid with constant properties is derived to investigate the effects of the Coriolis and electromagnetic forces on the flow phenomena. The structures of the associated hydrodynamic and hydromagnetic multiple boundary layers are explicitly determined with several special limiting cases. A comparative study of the structures of the hydrodynamic and hydromagnetic boundary layers is presented.ZusammenfassungEine exakte Lösung der instationären hydromagnetischen Grenzschichtgleichung eines inkompressiblen, viskosen, elektrisch leitenden und rotierenden Fluids mit konstanten Eigenschaften wird hergeleitet, um den Einfluß der Corioliskraft und der elektromagnetischen Kräfte auf Strömungserscheinungen zu studieren. Der Aufbau der zugehörigen hydrodynamischen und hydromagnetischen Grenzschichten wird, einschließlich mehrerer spezieller Grenzfälle bestimmt. Eine vergleichende Untersuchung des Aufbaues dieser Grenzschichten wird angegeben.

Unsteady multiple boundary layers on a porous plate in a rotating system

An initial value investigation is made of the motion of an incompressible, homogeneous, viscous fluid bounded by a porous plate with uniform suction or blowing. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about the z axis normal to the plate, and additionally a nontorsional oscillation of a given frequency is superimposed on the plate for the generation of an unsteady flow in the rotating system. By using the Laplace transform technique, an exact solution of the three dimensional Navier‐Stokes equations for unsteady flow is obtained. The structure of the associated multiple boundary layers is determined. Effects of uniform suction (or blowing) and rotation on the flow phenomena are analyzed. Several known results of interest are found to follow as particular cases of the solution of the problem considered.

Resonant oscillations of a porous plate in an electrically conducting rotating viscous fluid

An unsteady analysis is made of the nontorsionally generated hydromagnetic flow in a semiinfinite expanse of an electrically conducting rotating viscous fluid bounded by an infinite nonconducting porous plate with uniform suction or blowing in the presence of a transverse uniform magnetic field. The structure of the steady and the unsteady velocity distributions and the associated hydromagnetic multiple boundary layers are investigated including the case of blowing and resonant oscillations. Both the steady and the unsteady velocity fields describe the general features of the hydromagnetic boundary layer flows in the rotating fluid including the effects of uniform suction or blowing for all frequencies of oscillations of the plate. Unlike the hydrodynamic situation for the case of blowing and resonance, the hydromagnetic steady‐state solution satisfies the boundary condition at infinity and the associated boundary layers are confined to the plate. The inherent difficulty involved in the purely hydrodynamic problem associated with the case of blowing and the resonant frequency has been resolved in this paper by the addition of the magnetic field.

On a microcontinuum model of pulsatile blood flow

SummaryBased on a microcontinuum model, an initial value investigation is made of the pulsatile blood flow through a rigid circular tube with entrance effects due to the impulsive action of an arbitrary as well as particular pressure gradient. The solutions for the unsteady velocity field as well as the microrotation velocity of the blood red cells are explicitly determined. The asymptotic behavior of the solutions is studied with physical significance. Special attention is given to the investigation of the entrance effects on the structure of the blood flow. It is shown that the entrance effects are reflected in the solutions of the problem through the terms containing the exponential factor exp (−zbδ/V), and the solutions are significantly different from those of the fully developed flow. It is also predicted that the present theory gives rise to a boundary layer phenomenon not present in the previous works in the microcirculation of blood. This new boundary layer is introduced by the influence of the entrance effects. This analysis reveals the generation and propagation of waves in the blood vessels travelling downstream with velocityV/b. These waves are primarily caused by the entrance effects and eventually decay within the boundary layer distance of the orderV/(bδ). An exact as well as an approximate equation for the oscillatory inlet length is derived. Several limiting cases of physical interest are also discussed.ZusammenfassungAusgehend von einem Kontinuumsmodell mit Mikrostruktur, wird eine Untersuchung der pulsierenden Strömung von Blut durch ein starres Kreisrohr zufolge der Impulswirkung von beliebigen und auch speziellen Druckgradienten durchgeführt. Die Lösungen des instationären Geschwindigkeitsfeldes und des Feldes der Mikrorotationsgeschwindigkeit der roten Blutkörperchen werden bestimmt. Das asymptotische Verhalten und die physikalische Bedeutung werden untersucht. Insbesondere untersucht wird der Einfluß des Eintrittseffektes auf die Struktur der Blutströmung. Es wird gezeigt, daß diese Eintrittseffekte durch Glieder mit dem Faktor exp (−zbδ/V) in der Lösung wiedergegeben werden, und daß die Lösungen sich deutlich von denen der Strömung ohne Eintrittseffekt unterscheiden. Diese Theorie führt zu einer durch den Eintrittseffekt hervorgerufenen Grenzschichterscheinung, die in vorangegangenen Arbeiten über die Mikrozirkulation von Blut nicht auftrat. Ferner zeigt sie die Entstehung und Ausbreitung von Wellen von Blutkörperchen, die mit der GeschwindigkeitV/b sich stromabwärts ausbreiten, durch den Eintrittseffekt bedingt sind und mit der OrdnungV/(bδ) abklingen. Eine exakte und eine Näherungsgleichung für die Einströmlänge wird hergeleitet. Verschiedene Grenzfälle von physikalischem Interesse werden diskutiert.

Free convection in hydromagnetic flows in a vertical wavy channel

Abstract A study is made of the free convection in hydromagnetic flows in a vertical wavy channel in the presence of heat source or sink. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved subject to the relevant boundary conditions with the assumption that the solution consists of a mean part and a perturbed part. The zeroth-order, the first order and the total solution of the problem are numerically evaluated for various values of the magnetic parameter, heat source/sink parameter, wall-waviness parameter, and free convection parameter. The velocity and the temperature profiles are graphically represented for these parameters. The qualitative features of the hydromagnetic solution are discussed. A comparison is made between the hydromagnetic and the hydrodynamic solutions. The numerical values of the skin friction and the Nusselt number are tabulated for various parameters involved in the analysis. Special attention is given on the characteristic features of the flow, heat transfer, skin friction and the Nusselt numbers at the walls.

Generation of dispersive surface waves in magnetohydrodynamics

A theory of the generation and propagation of two and three dimensional magnetohydrodynamic surface waves due to an arbitrary as well as special oscillatory pressure distribution acting on the free surface of an inviscid electrically conducting fluid is presented. The unsteady solutions for two as well as three dimensional free surface elevation are obtained explicitly. The asymptotic behaviour of these solutions for large time and distance are determined to describe the qualitative and quantitative nature of the wave motions. The solutions consist of both the steady-state and the transient components. The latter is found to decay asymptotically for large time, and the steady-state is reached. Magnetohydrodynamic surface waves in deep and shallow fluid are dispersive in nature.

Inertial oscillations and multiple boundary layers in an unsteady rotating flow

An asymptotic analysis is made of the unsteady boundary layer flow generated impulsively in an incompressible homogeneous viscous fluid bounded by an infinite porous plate with uniform suction or blowing to describe the manner of the transient approach to the ultimate steady state. It is shown that the initial motion for small times describes the general features of the unsteady boundary‐layer flow and consists of three boundary layers on the plate which are unaffected by rotation. In subsequent large times, the effect of rotation manifests itself through inertial oscillations of frequency 2Ω and the generation of diffused waves propagating outward from the plate with velocity 2(Ων)1/2a1. It is found that the final steady boundary layers are established in a dimensional time (4/Ω) (S4/4E2 + 4)−1/2 which decreases with an increase in the suction parameter. Several limiting cases of interest are recovered from this analysis.

Propagation of electrohydrodynamic surface waves in a conducting fluid

SummaryA comprehensive study is made of the electrohydrodynamic surface wave phenomena in an inviscid, incompressible conducting fluid in the presence of an external steady electric field due to an arbitrary periodic surface pressure distribution. The initial value wave problem has been solved by the Laplace and the generalized Fourier transforms in conjunction with asymptotic techniques — as advanced in a previous work ofDebnath andRosenblat [12]. An explicit steady state solution related to short and long wave approximations is found. The asymptotic behaviour of the transient solution for large time is demonstrated. It is shown that there are two surface wave-trains propagating in the conducting medium, one of which corresponds to the classical surface wave train and the other is originated entirely due to the interactions of the electric field with gravity. The simultaneous effects of the electric field and surface tension on the propagation of small amplitude surface waves are analyzed. The implications of the wave solutions related to very deep and shallow fluids are explored. A discussion of the electrohydrodynamic wave motions with their characteristic features is presented.ZusammenfassungAusführlich untersucht werden die durch beliebigen periodischen Oberflächendruck verursachten Oberflächenwellen einer reibungsfreien inkompressiblen elektrisch leitenden Flüssigkeit unter Einfluß eines äußeren stationären elektrischen Feldes. Das Anfangswertproblem wird durch Laplace- und erweiterte Fouriertransformation und asymptotische Entwicklungen gelöst. Eine explizite Näherungslösung für kurze bzw. lange Wellen wird gefunden, und das asymptotische Verhalten der Übergangslösung für große Zeiten wird erläutert. Es wird gezeigt, daß sich in der Flüssigkeit zwei Züge der Oberflächenwellen ausbilden. Einer entspricht dem klassischen, der andere entsteht durch Wechselwirkung des elektrischen mit dem Schwerefeld. Der gleichzeitige Einfluß des elektrischen Feldes und des Oberflächendruckes auf die Ausbreitung von Oberflächenwellen kleiner Amplitude wird untersucht; ebenso die Lösung für große bzw. geringe Tiefe. Abschließend werden die charakteristischen Eigenschaften elektrodynamischer Wellenbewegungen erörtert.