Book of Abstract:
“Flow and transport in porous media with applications”
Prof. K. Muralidhar
Department of Mechanical Engineering
Indian Institute of Technology Kanpur; Kanpur 208106
Abstract:
Flow, heat, and mass transfer in porous media are encountered in a great many fields of engineering: civil,
chemical, mechanical and petroleum, to name a few. A wide variety of flow patterns can be realized in
the porous medium. These are controlled by the flow regime - for example, steady or unsteady, laminar
or turbulent, fluid interfaces in multi-phase flow, non-equilibrium phenomena in the transported variables
such as pressure, temperature and concentration, chemical reactions, and phase change. Additionally, it is
well-established that flow patterns and transport depend intricately on the variability of the pore space and
the solid matrix.
Strategies employed for studying transport phenomena in a non-uniform porous medium include
the following:
1. Formulate the governing equations for an inhomogeneous, anisotropic region. Properties such as
permeability and dispersion are interpreted here as second order tensors that have built-in
dependence on spatial location.
2. Permeability and dispersion are treated as random variables with a well-defined average that can
be a function of space. These are associated with appropriate statistics that account for
fluctuations in the physical properties and possible spatial correlation, apart from correlations
among the properties themselves.
While approach 1 leads to a deterministic model that can be solved by a numerical technique, approach 2
leads to a stochastic model that must be solved several times to generate realizations of the resulting flow
and scalar fields.
One of the major difficulties encountered in the two approaches referred above pertains to the
sensitivity of the predicted variables on the specification of pore-scale variation (deterministic or
statistical). For example, a refinement of the computational grid will permit greater details of the pore
structure to be prescribed and yield a completely new solution. Yet another difficulty is experienced when
the model parameters have to be determined from laboratory or field-scale experiments. The scale of
measurements may be such that variations on a smaller scale are completely masked. The difficulties
become compounded when transport occurs over a wide range of length scales or over an extremity of
pore scales. The best one can accomplish is to determine parameters in such a way that a few limited
goals of the mathematical model are fulfilled, though the estimations may not strictly conform to every
possible variation in the pore geometry. The subject of hierarchical modeling of transport in porous media
is in its infancy.
The present talk is introductory and deterministic modeling of flow and transport in porous media
will be discussed. Applications relate to enhanced oil recovery, regenerators, recovery of gas hydrates,
and coil embolization in medical treatment. Research directions in this field will be highlighted.
“Dynamics of a radially expanding liquid sheet”
Prof. Mahesh S Tirumkudulu
Department of Chemical Engineering
Powai, IIT-Bombay
Abstract:
A fundamental understanding of the break-up of bulk liquids and subsequent drop formation is important
in areas as diverse as spray coating, combustion, and in biomedical devices such nebulizers. One such
route of atomization involves radial liquid sheets generated by laminar jet impingement which eventually
break-up into fine droplets. Radial liquid sheets are formed by head-on impingement of liquid jets, where
the liquid sheet spreads out radially in a plane perpendicular to direction of the jets. We focus on the
dynamics of break-up of such liquid sheets when the point of impingement is subjected to perturbations
via acoustic forcing of controlled sound intensity and frequency to identify regimes of accelerated and
violent sheet break-up. Stability equations are derived from the inviscid flow equations for a radially
expanding sheet that govern the time-dependent evolution of the two liquid interfaces. The analysis
accounts for the varying liquid sheet thickness while the inertial effects due to the surrounding gas phase
is ignored. When the sheet is excited at a fixed frequency, a small sinuous displacement introduced at the
point of impingement grows as it is convected downstream suggesting that the sheet is unstable at all
Weber numbers (ratio of inertia to surface tension forces) in the absence of the gas phase. Asymptotic
analysis of the sinuous mode for large frequencies shows that the disturbance amplitude diverges
inversely with the distance from the edge of the sheet. The varicose waves (thickness modulations), on the
other hand, are neutrally stable at all frequencies and are convected at the speed of the liquid jet. To test
the predictions of the theory, we used laser induced fluorescence technique (LIF) to determine the
thickness variation and deflection of the sheet for varying sound pressure levels and frequencies. In the
absence of the disturbance, the measured liquid sheet thickness varies inversely with radial distance from
the point of impingement for the smooth sheet and matches well with theoretical predictions. When the
perturbations are introduced, sheet undergoes flag like motion though the thickness modulations appear to
negligible. The measured wave speed, wavelengths and growth rates of the sinuous waves match with
those predicted by the theory confirming that the sinuous waves are unstable and that a simple theory that
ignores the inertia of the surrounding gas phase is sufficient to capture the main aspects of the sheet
dynamics.
“Effect of specimen size on strength in concrete under axial compression”
Prof. ArghyaDeb
Department of Civil Engineering,
IIT Kharagpur
Abstract:
In the absence of significant plastic deformations, the effect of specimen size on nominal strength of
tensile specimens is well known. In case of quasi-brittle materials such as concrete, where specimen size
plays an important role in determining whether LEFM is applicable, Bazant and his collaborators have,
since 1981, formulated a size-effect law for concrete specimens subjected to tensile stresses. While this
law has been found to be a reasonable fit to experimental data in concrete flexural members, its extension
to concrete compression members (again by Bazant and co-workers) is somewhat controversial. In this
presentation, the existing state of experimental knowledge and conclusions therefrom, for unconfined
concrete specimens subjected to uniaxial compression, will be summarized. Next, the case for an
energetic size effect, amenable to deterministic treatment, will be presented – along with some recent
numerical results. The effect of specimen size on strength in concrete columns subjected to triaxial
compression will be discussed. Here, the existing experimental data seems to confirm that no size effect
exists. However some recent numerical results seem to show the existence of a significant size effect in
weakly confined concrete members, which however becomes insignificant as confining pressure is
increased. What is also interesting is that the deterministic size effect in weakly confined systems may
actually be higher than in unconfined systems. Tentative explanations for this behavior will be presented.
“Non-­linear dynamics of non-­smooth systems using smoothening schemes”
Prof. P. Chandramouli
Department of Mechanical Engineering
Indian Institute of Technology Madras, Chennai 600036
Abstract:
Dry friction damping is an effective way of reducing the resonant amplitudes inturbine bladed system
where the inherent structural damping is very small. One method ofinducing damping in such systems is
by providing shrouds at the tip of the blades. In this case,the shrouds constrain the blade motion not only
along the contact plane but also along thedirection normal to the contact plane. The tangential motion
along the contact plane results infriction between the contact surfaces and the interface undergoes stick
slip motion. The motionnormal to the contact plane results in intermittent separation and impact of the
contactingsurfaces.
In this talk the nonlinear dynamics of such dry friction damped systems is investigated withapplication to
the vibration damping in shrouded turbine blade systems. A contactelement proposed by Yang and Menq
is used to model the contact interface, the stick slip motioninduced by friction and the intermittent
separation and impact due to variable normal load. The dynamics of these systems are governed by
differential equations of discontinuous naturewhich are treated as Filippov systems for smoothening the
dynamics. It is shown that such smoothening enables the solution of these kind of systems for situations
where even numerical integration has convergence issues. Assuming periodic motion of the interface the
transition times corresponding to different states are computed and the hysteresis plots generated. A
multi-harmonic balance procedure in combination with path following based on an arc length
continuation is used to generate the periodic solutions.
“Teaching of rotations, Euler angles, and angular velocity”
Prof. AnindyaChatterjee
Mechanical Engineering
Indian Institute of Technology
Kanpur 208016
Abstract:
I will discuss the way in which I prefer to teach these topics, which differs somewhat from the approach
followed in most textbooks. While the end results are of course the same, I have found that my students
have developed some level of comfort with writing equations of motion for moderately complex systems
of rigid bodies. The approach I follow is to introduce Euler's theorem, find a rotation tensor for vectors
subjected to arbitrary rotation about an arbitrary axis, develop the composite rotation matrix for rotations
described by Euler angles, and motivate the concept of angular velocity from there. The advantage of the
present approach lies largely in the way in which actual details of trigonometric manipulations are moved
into the background, so that a more compact and conceptual presentation can be made. This approach
leads easily into computer implementations as well (both numeric and symbolic).
“Polycrystal plasticity models with grain interaction”
Prof. Sivasambu Mahesh
Department of Aerospace Engineering,
Indian Institute of Technology, Madras.
Abstract:
This talk will begin by explaining the concept and need for polycrystal models of metal plasticity. The
development of such models goes back to 1940s by the likes of Rodney Hill and Ekkehart Kroner. The
key ideas behind these models will be outlined, and the present model, which accounts for grain
interactions in a simplified way will be introduced. Applications of the model to two manufacturing
processes: equal channel angular pressing, and pilgering will be discussed.
“Non-Associated Flow Models and Effects on
Macroscopic Failure Mechanisms”
Dr. VikranthRacherla
Department of Mechanical Engineering
IITKharagpur
Abstract:
The physical basis of conventional macroscopic plasticity theories for crystalline solids is a simple slip
mechanism, one that is controlled only by the shear stress on the slip plane in the direction of slip. This is
somewhat, exceptional behavior since in all but FCC lattices the core structures of screw dislocations tend
to be three dimensional (non-planar) and, as a result, slip depends upon so-called non-glide stresses. The
effects of non-glide stresses result in non-associative flow behavior in both single and polycrystals.
Frictional materials such as rock and sand are typically modeled as non-associated flow as well.
In this work, the effects of non-glide stresses are shown to persist at macroscopic scales and strongly
affect deformation behaviors. For example, the critical pressure at which cavitational instabilities occur
and critical necking strains are significantly affected by non-associated flow. The structure of constitutive
models is studied in detail, and new relations are proposed. In classical, rate-independentm, nonassociated flow models uniqueness and stability of solutions to incremental boundary value problems can
be lost
even at small strains. However, in a corner theory, a deformation theory, or a ratedependent theory we
demonstrate that uniqueness and stability can be guaranteed at small deformations. Non-associated flow is
shown to significantly alter strain localization in thin sheets (sheet necking), and this class of problems is
studied in detail. The effects are most prominent near the plane strain loading state. To investigate the full
three dimensional nature of instabilities in sheet deformation, a finite element analysis is carried out for
nearly rate-insensitive response using an implicit dynamics scheme. This led to the discovery of “strain
bursts” as a consequence of non-associated flow, particularly for deformations near the plane strain
loading state.
“Instability and transition in flow through deformable tubes and channels”
Prof.V Shankar
Department of Chemical Engineering
Indian Institute of Technology
Kanpur 208 016
Abstract:
Fluid flow through conduits made of soft, deformable walls is encountered in many biological flows, and
more recently, in microfluidic device applications. The dynamics of fluid flow past deformable solid
surfaces is qualitatively different from that of rigid surfaces due to the coupling of fluid flow with the
deformation in the solid medium. In particular, this `elasto-hydrodynamic' coupling could affect
the stability of laminar flow in tubes and channels made of deformable solids. These instabilities could be
potentially exploited to induce secondary flows which aid in improving mixing in micro-scale
flows. Our theoretical studies based on linear stability analysis have uncovered new types of instabilities
for the case of Newtonian fluid flow through deformable tubes and channels, which are absent in their
rigid counterparts. This talk will provide an overview of the subject and a summary of the main results in
this area.
“Gas flow in complex microchannels”
Prof. AmitAgrawal
Department of Mechanical Engineering,
Indian Institute of Technology Bombay, Powai, Mumbai 400076
Abstract:
Microscale passages are an intricate part of any microfluidic device. These devices have potentially large
applications in health-care, defense, electronic and agriculture industries. Gas flow behavior at small
scales is governed by an additional non-dimensional parameter – the Knudsen number. The flow physics
at high Knudsen number is however not well understood. Studies have shown that new effects such as:
velocity and temperature jump at the solid-fluid interface and compressibility effect at low Mach
numbers, are present at micro-scale involving gas flow. Recent experiments have shown that the Nusselt
number is anomalously low with gas flow, and that the available theoretical analyses are unable to
correctly predict its value.
In this talk, I will first present theoretical results for isothermal gas flow in microchannel. However, since
the validity of the Navier-Stokes equations to high Knudsen number flows is questionable, we will
introduce and extend the solution to Burnett equations. Burnett equations are higher-order continuum
equations, with non-linear constitutive relations, and our solution is the first known analytical solution of
Burnett equations for any configuration.
In the next part, I will present experimental and numerical evidence from our lab to elucidate the nonintuitive flow behavior when the passage is complex. We will consider passages involving a sudden
change in cross-sectional area or a bend. Unlike conventional flow, the absence of flow separation is
noted at the junction in a passage involving a sudden change in the cross sectional area. In contrast, flow
separation is noted even at very low Reynolds numbers in the presence of a bend. Detailed results for
these configurations will be presented, along with a discussion on the possible reasons for the change in
flow behavior.
“Aeroelastic Model for Prediction of Blade Loads, Blade Response and Vehicle Response of
a Helicopter in General Maneuver”
Prof. C.Venkatesan
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Abstract:
ELICOPTERS operate in a very complex dynamic and aerodynamic environment. Determination of
distribution on the rotor is a basic necessity for an improved design of rotorcraft. The
Hairloads
designer is usually concerned with the periodic loads occurring in steady-state straight or
maneuvering flight. The periodic aerodynamic environment of the helicopter rotor produces high
oscillatory loads in the blades, hub, and control system. Oscillatory loads are a major factor in rotor
design as they cause vibration and fatigue in the helicopter. Hence, a major design goal in helicopters is to
reduce the oscillatory loads on the rotor while improving performance. Accurate prediction of helicopter
loads and response requires the development of a multi-disciplinary comprehensive analysis program.
Helicopter loads and response is an aeroelastic problem as it involves interaction of the structural, inertia
and aerodynamic operators. This work describes the development of a comprehensive helicopter
aeroelastic analysis including the rotor-fuselage coupling; presents some validation studies; examines the
flight dynamics of the helicopter in a steady, level, turning maneuver and finally, examines the effect of
rotor blade tip geometry on loads and control response. The helicopter modeled is a conventional one
with a hingeless single main rotor and single tail rotor. The blade undergoes flap, lag, torsion and axial
deformations and is modeled using beam finite elements. Tip sweep, pretwist, precone, predroop, torque
offset and root offset are included in the model. Aerodynamic model includes Peters-He dynamic wake
theory for inflow and the modified ONERA dynamic stall theory for airloads calculations. The complete
6-dof nonlinear equilibrium equations are solved for analyzing general flight conditions including steady
maneuver. Validation studies presented include comparison of experimental data with the analysis results
pertaining to (i) structural dynamics of swept-tip blades, (ii) whirl tower test and (iii) steady forward
flight trim state of a helicopter. Results of a study showing the effects of the blade geometric parameters
on the performance, response and loads of the rotor are also given.
For steady level turns, which is the simplest of maneuvers, it is shown that mathematically, three
independent approaches exist for the solution of trim equations. The three approaches correspond to
setting any one of the three parameters, namely, the lateral acceleration, the sideslip or the track angle
equal to zero. Results from the three different solution procedures are presented. It is shown that these
three approaches provide different trim quantities leading to several interesting observations. For
performance improvement and for loads and vibration reduction, new generation helicopter advanced
geometry rotor blades have incorporated swept and anhedral tips. With addition of these features to the tip
of the rotor blade, it has become necessary to study their effects on the aeroelastic behavior of the rotor
system. Rotor blades with sweep and anhedral at the blade-tip region experience bending torsion and
bending-axial coupling effects. Swept tips influence blade dynamics because they are located at regions
of high dynamic pressure and relatively large elastic displacements. The effects of tip-sweep and tipanhedral on the trim characteristics and vehicle response to pilot inputs shall also be presented.
“Research into stochastic structural dynamic testing and reliability model updating”
Professor C S Manohar
Department of Civil Engineering
Indian Institute of Science
Bangalore 560 012
[email protected]
Abstract:
It has become feasible in recent years to instrument civil engineering structures to measure their
performance under operational conditions. These measured responses provide an opportunity to examine
and update mathematical models for structural behavior. The resulting problems constitute inverse
problems with attendant challenges in dealing with over determined set of equations, important influence
of imperfections in measurements and mathematical models, and structural nonlinearities. Besides, the
developments in testing hardware and combining them with computational resources have opened newer
possibilities in structural testing for vibratory loads. The talk addresses problems of dynamic qualification
testing of highly reliable systems functioning in random environment (such as nuclear power plant
components under earthquake loads) and updating reliability models for the structures based on measured
structural performance.
Random vibration testing with controlled samples
The first part of the talk discusses a novel experimental test procedure to estimate the reliability of
structural dynamical systems under excitations specified via random process models. The samples of
random excitations to be used in the test are modified by the addition of an artificial control force. An
unbiased estimator for the reliability is derived based on measured ensemble of responses under these
modified inputs based on the tenets of Girsanov’s transformation. The control force is selected so as to
reduce the sampling variance of the estimator. The study observes that an acceptable choice for the
control force can be made solely based on experimental techniques and the estimator for the reliability
can be deduced without taking recourse to mathematical model for the structure under study. This permits
the proposed procedure to be applied in the experimental study of time variant reliability of complex
structural systems which are difficult to model mathematically. Illustrative example consists of a multiaxes shake table study on bending-torsion coupled, geometrically non-linear, five-storey frame under
uni/bi-axial, non-stationary, random base excitation.
Time variant reliability model updating in instrumented dynamical systems
The problem of updating the reliability of instrumented structures based on measured response under
random dynamic loading is considered. A solution strategy within the framework of Monte Carlo
simulation based dynamic state estimation method and Girsanov transformation for variance reduction is
developed. For linear Gaussian state space models the solution is developed based on continuous version
of Kalman filter while for nonlinear and (or) non-Gaussian state space models, bootstrap particle filters
are adopted. The controls to implement Girsanov transformation are developed by solving a constrained
nonlinear optimization problem. Numerical illustrations include studies on a multi degree of freedom
linear system and nonlinear systems with geometric and (or) hereditary nonlinearities and nonstationary
random excitations.
“Mechanics of Fibrous Composites: Teaching, Research and New Directions”
Prof. P M Mohite
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Abstract:
Composites find wide applications in high speed vehicles due to their high strength to weight ratio and
tailorable properties. Apart from aerospace, automobile and naval industry composites are used in various
other industries, including civil engineering applications. The field of composite materials is growing
very fast. Thus, to keep abreast with this field the study of mechanics of these materials is vital for the
engineers and researchers working in this field. A course on the mechanics of composites, either as a
compulsory or an elective, must be included in the curriculum of Applied Mechanics, Aerospace, Civil,
Mechanical and Production Engineering.
The first part of the present talk is devoted to the teaching of the mechanics of fibrous composites
at IIT Kanpur. The in-depth topics covered in this course will be presented. In this, the student projects
during a course, that use various concepts learnt, will also be presented in brief. The second part is
devoted to the special topics in composites like micromechanics, damage mechanics and testing of
composites at various (micro-meso-macro) scales. The third and the last part of the talk is devoted to the
discussion of issues in composites like modeling of thermoplastic composites, increasing the life of
composite structures, natural fibre composites, waste disposal, etc. The current research activities and
facilities available at IIT Kanpur in this field will also be presented briefly. Discussion on effective
pedagogy, augmentation of laboratory facilities and research directions will be encouraged.
“Challenges in Flight control Design, Development and testing”
Prof. Vijay Patel
DRDO
Abstract:
Light Combat Aircraft (LCA) is a single engine, tail-less, delta wing, supersonic fighter aircraft which is
designed to be aerodynamically unstable in its longitudinal axis. In order to stabilize the airframe and
achieve the desired performance over the entire flight envelope it incorporates a quad redundant full
authority fly-by-wire (FBW) flight control system (FCS). The control laws in addition to guaranteeing
stability, optimize the aircraft performance and pilot handling qualities for various combinations of
external store configurations. LCA has flown more than 2000 flights including trainer and naval variants
without encountering any major faults in the flight safety critical fly-by-wire control software. This talk
narrates flight control law development cycle including parameter identification to update aero- database
in order to expand flight envelope. In this talk fundamentals of control law design for an unstable fighter
aircraft will be discussed and the mathematical challenges in order to achieve certain performance goals
will be presented.
“Dealing with stress”
Prof.AnuragGupta
Department of Mechanical Engineering
Indian Institute of Technology, Kanpur
Abstract:
We will discuss the concept of stress--ubiquitous in the mechanics of continuous media--from several
conceptual and historical viewpoints. For instance, we will consider stress as an abstract notion describing
the mutual action between two bodies in contact, or alternatively as derived from the intermolecular
forces. We will also spend some time in tracing the historical evolution of the relevant ideas, in an attempt
to understand the conceptual difficulties which surround stress. The talk will end by emphasizing the
importance of rightfully dealing with stress for an improved life-time and strength.
“Optimal strategies for throwing fast”
Madhusudhan Venkadesan
Assistant Professor
National Centre for Biological Sciences(NCBS)
Abstract:
Several morphological features of humans are attributed to evolutionary selection for long distance
running and high speed throwing, both important for hunting. In this talk, I will provide a brief
background to human evolution, before talking about throwing. Humans are unique among primates in
their ability to throw with remarkable accuracy at high speeds. Throwing involves the entire body, with a
cascade of energy flow that starts at the legs and ends at the fingertips, leading to a fast, yet accurate
throw. Experiments with human subjects suggest hypotheses on which morphological features help in
generating such high speeds. We use numerical optimization on these hypothetical models to find control
strategies that maximize throwing speed. The optimum is predictive of how a baseball pitcher throws;
transmitting power produced at the hip using elastic energy storage at the shoulder and kinetic energy
transfer across multiple segments, resembling an elastically loaded whip.
"From Finite Elements to a Virtual Experimentation Station for damaging
composites"
Prof. C S Upadhyay
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Kanpur - 208016
Abstract:
Why do we compute? Computations are carried out to obtain an understanding of the underlying physics
of a phenomenon - as captured by the mathematical model used. Implicitly, it is assumed that the model is
very good, i.e. it is a good representation of the physics. Finite element computations then focus on
getting good solutions to a model, with an ultimate comparison of the results obtained with experimental
data or with other computations.
As computing power has grown, detailed computations are possible. Exotic models of material behavior
can be solved for with guaranteed accuracy. Hence, a natural extension of computations is to understand
the underlying physics in greater detail. This understanding can influence the way we do experiments.
Experiments can be better designed, at different length-scales, to get the insight into how things happen.
An example is the ongoing research in damage of composites. The initial attempts were direct extensions
of Mises failure criterion and definition of a failure locus (yield surface), together with a plasticity type
approach to post-yield behavior. Laminates were tested and fairly good comparison was obtained for the
initiation of damage (first ply failure theories).
Progressive damage was still tricky, and laminate dependent. All this meant several expensive and
elaborate tests had to be conducted to get the damage parameters. Good computations, together with a
better understanding of micro-mechanics led to the continuum damage models that could not only
predict onset, but also growth, of damage using experimental data for two specific laminates. The damage
parameters thus obtained were enough to predict the onset and growth of damage for any lamination
sequence. Further developments at IIT Kanpur have brought this process down to tests on individual
fibers and matrix material, and using the material parameters of these constituents determining the onset
and growth of damage for any laminate. This brings in the exciting prospect of carrying out systematic
finite element computations on suitable three-dimensional representative volume elements to obtain the
behavior of an individual ply (or the interface), for any combination of fiber and matrix material (and
for any volume fraction), to get predictions about the onset and growth of damage in any laminate.
Further, a clear insight into the damage modes is also obtained. Thus limited laminate level
experimentation is required, only to validate the model. This brings in a paradigm of a virtual
experimentation station that works along with specific experiments to give a complete picture of behavior
of damaging composites under monotonic or cyclic loading.
“Non-dimensionality in Slender Elastic Objects”
Prof. G. K. Ananthasuresh,
Mechanical Engineering,
Indian Institute of Science, Bangalore
Abstract:
In this talk, we will discuss non-dimensional analysis of slender elastic objects that undergo large
deformations. First, it will be noted that there exist two independent size scales, one that decides the
overall size and another that decides the cross-section size. For a fixed topology and shape of the object
and fixed relative proportions of the constituent members of the object at the two size scales, it will be
shown that the nonlinear response of the system can be non-dimensionalized against suitably defined nondimensional parameters. These parameters subsume the geometry, the loads, and the material
characteristics of the object. The non-dimensionalized responses considered include displacement,
stiffness, stress, and natural frequency.
The implication of the newfound non-dimensionality and its practical utility in simulation and design of
elastic structures and compliant mechanisms comprising slender elastic segments will be discussed with
examples. The talk will end with speculation about similar non-dimensional parameters for general elastic
objects.
“Understanding Reynolds Transport Theorem”
Prof. Naveen Tiwari
IIT Kanpur, UP -208016
Abstract:
Reynolds Transport Theorem is a logical way of deriving a balance equation for any conserved quantity.
It can be used to obtain governing equations for mass, momentum, energy, species, and other quantities.
In this lecture, Lagrangian and Eulerian coordinates will be discussed in context of describing the motion
of a fluid “particle”. The derivation of Reynolds Transport Theorem starting from fundamentals of
kinematics will be discussed. Its application to transport of heat, mass, and momentum will be illustrated.