# Glimpses of P.L. Bhatnagar

A man of simple ways and high principles, indefatigable in his pursuit of excellence, for whom mathematics was his very life

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There were hardly any who came in contact with prabhu lal bhatnagar (1912–1976) and went away unimpressed. Here was a man bubbling with enthusiasm, fervour, and love for mathematics. When he was giving a lecture, he looked restless, pacing up and down, trying to convey to his audience the beauty and order he saw in mathematics. His loud and clear voice, speed, and complete command over the subject matter of his talk used to keep his audience spellbound. To him mathematics was more than a subject—it was his very life. He loved, breathed, and lived mathematics, and above all, he tried to make others appreciate this too. To his students he was more than a teacher; he was a guru in the true sense. To his colleagues he was an idealist, with no room for compromise. There were only two ways—the right and the wrong, and the right won every time. To his admirers he was almost superhuman and he could do no wrong.

### Early Life and Education

Prabhu Lal Bhatnagar was born on 8 August 1912 in Kota, Rajasthan, the second of five sons. His parents belonged to a well-connected family which had been advisers to the rulers of the princely state of Kota. His forefathers were given the name “Rai Dwarkdas” and had donated all their wealth and property to the Lord Mathuradhish temple at Kota. It was Bhatnagar’s grandfather who gave him his first lessons in arithmetic, giving him problems to work on mentally. Young Bhatnagar showed promise from very early on and nothing came between him and his studies. He went to school first at Kota, then the Government School at Rampura and later to Herbert College, Kota. He secured first rank in the Intermediate examination and was encouraged by the Principal and Director of Education, Lala Daya Kishan Gupta, to go for further studies to Jaipur. It was during this time that his father passed away; Bhatnagar was just over twenty years of age at that time. But he supported himself and his family with the scholarship he was a regular recipient of from matriculation onwards.

His talent became further evident in 1934 when he completed his B.Sc. degree at Maharajah’s College, Jaipur which was affiliated, at the time, to Agra University. Bhatnagar obtained the highest marks that year in Mathematics and Chemistry in the B.Sc. examination of Agra University, and for this he was awarded the Krishna Kumari Devi gold medal and Umang Lakshmi Kanti Lal Pandya gold medal, respectively. For the first rank in B.Sc., he was awarded the Maharajah Fateh Singh gold medal.

Bhatnagar went on to do his M.Sc. at the same college, where K.L. Varma was one of the teachers for whom Bhatnagar had profound regard. In 1936, for his outstanding performance in the Master’s degree examinations of Agra University, Bhatnagar was awarded the Lord Northbrook gold medal by Maharajah’s College, Jaipur. The young Bhatnagar must have come to the crossroads at this stage. Most brilliant young men of that time aspired to appear for the Indian Civil Services (ICS) examination, which was the entry to a lucrative, powerful profession for a chosen few. However, Bhatnagar was no ordinary young man. In spite of pressures and coaxing from friends and relatives, he took the advice of his teacher, K.L. Varma, who had urged him to take up research as a career at Allahabad University.

### At Allahabad, Delhi and Harvard

Allahabad University was at that time at the pinnacle of its glory. A.C. Banerji, a Wrangler from Cambridge, was the Head of the Department of Mathematics. (At the University of Cambridge, a “Wrangler” is a student who is awarded first-class honours in the third year of the University’s undergraduate degree in mathematics.) There, Bhatnagar first worked with B.N. Prasad on the summability of Fourier and allied series. His research work at Allahabad, from 1937–39, included the solution of second-order linear ordinary differential equations by the Laplace transform technique. Two of these have been incorporated in the famous book of Erich Kamke, Differentialgleichungen, Vol. 1 [4]. These are:

1. $y'' = (a^2 x^{2n} - 1)y$

The solution for $a\,n\,x^{n-1} > 1$ is

$y = C_1\int_{-1}^{1} +\, C_2 \int_{1}^{\infty}e^{-\lambda t}(t-1)^{\mu-\nu}(t+1)^{\mu + \nu} dt$

with $\lambda = \frac{a}{n+1}x^{n+1}, \mu = - \frac{n+2}{2(n+1)}, \nu = \frac{x^{1-n}}{2a(n+1)}.$

2. $y'' = (4a^2 b^2 x^2 e^{2bx^2} - 1)y$

with solution

$y = \int e^{-\xi t} (t-1)^{\eta-\zeta} (t+1)^{\eta + \zeta} dt$

where $\xi = ae^{bx^2}, \eta = \frac{1-2bx^2}{4bx^2}, \zeta = \frac{e^{-bx^2}}{8ab^2 x^2}$, and the limits of integration are so chosen that, at these points, the integrand multiplied by $(t^2 -1)$ must vanish.

Both these results were published in collaboration with A.C. Banerji [5].

After this, Bhatnagar’s research interests slowly shifted to astrophysics. This was inspired by his coming in contact with M.N. Saha, who was already famous for his work in physics and astrophysics. Bhatnagar began to work on the spiral nebula and the tidal theory of planetary formation. For his thesis titled On the Origin of the Solar System, under the supervision of A.C. Banerji, Bhatnagar obtained his D.Phil. degree in Mathematics in 1939. He also won the E.G. Hill Memorial prize for the best research in the Faculty of Science, Allahabad University, during 1937–39.

He was then invited to join St. Stephen’s College, Delhi, by its Principal, S.N. Mukherjee. Here Bhatnagar spent the better part of the next 16 years of his life, first as a Lecturer and later as Head of the Department of Mathematics and concurrently a Reader in Mathematics at Delhi University. One could perhaps say that Bhatnagar was in full bloom during those years. He indulged his interest in astrophysics, working both independently and in collaboration with D.S. Kothari. The result was a spate of publications from 1939 to 1946, the highlight of which was the theory of anharmonic pulsations of Cepheids and white dwarf stars. This work brought Bhatnagar international recognition and caught the attention of the scientific community, especially the astronomers. While the pulsation phenomenon had been observed only in “supergiant” stars, Bhatnagar saw no theoretical reason why it should not occur in denser stars and even in white dwarfs. He felt that a nova outburst associated with the sudden collapse of a star could leave the white dwarf pulsating. The pulsation, once started, could last for a period comparable to $10^3$ years. The period of pulsation (assuming a “homogeneous” model) for the fundamental mode is given by

$P = \left( \frac{9h^3}{16 \pi m^{3/2} H^{5/2} G^2} \right) \frac{1}{\mu^{5/2}M} \approx \frac{M_{\odot}}{M}\, \mathrm{seconds}$

where $M$ is the mass of the white dwarf, $h$ the Planck’s constant, $m$ the electron mass, $H$ the proton mass, $G$ the gravitational constant, $\mu$ the mean molecular weight, and $M_{\odot}$ the solar mass. However, Bhatnagar realized that the pulsation period for a white dwarf was too small to be directly observable and so the existence of pulsation in white dwarfs had to be looked for through its secondary effects. In later years, high-speed techniques have discovered such pulsations, and substantially verified his calculations.

In an attempt to explain the observed skewness in the velocity–time curve of the Cepheid variables (the curve had an asymmetrical form rather than being a symmetrical sine-wave), Bhatnagar and Kothari derived exact expressions for the period $P$ of oscillation and the times $t_1$ and $t_2$, which are parts of the period where the radius of the star is greater and lesser than the equilibrium radius $R$, respectively. They observed that if $\gamma$, the ratio of specific heats, were taken to be $5/3$, then the observed skewness would demand a semiamplitude of oscillation which was almost equal to the radius of the star. This was inconsistent with observation, which gave a value for the semiamplitude as $0.1R$. A rough calculation showed that the observed skewness would arise for a semiamplitude of $0.1R$ only if $\gamma$ were comparable to $10$. The complex computer calculations of Robert Christy later showed that the skewness was essentially a surface phenomenon and that the equations used by Bhatnagar converged too slowly to be able to handle surface layers.

At St Stephen’s, a busy schedule of over 20 hours of teaching per week did not dampen Bhatnagar’s enthusiasm for research. For his work on astrophysics, he was awarded the D.Sc. degree from Allahabad University in 1947. His interest in stellar structures and interiors led him to the study of rarefied gases and ionised media. This was a harbinger of the monumental work he was to do a few years later.

In 1951, Bhatnagar went to Harvard University, Cambridge, Massachusetts as a Fulbright scholar for two years, where he lectured on the mathematical theory of non-uniform gases. This handsome tall scholar from India was often mistaken in the university corridors for a student. Once he took his place at the lecture rostrum, though, the students realized that he was indeed a senior faculty member!

### The BGK Model

In this period, together with D.H. Menzel and H.K. Sen, he wrote a book, Stellar Interiors [6]. The Boltzmann equation had captured Bhatnagar’s attention at that time. The complicated integrals which described the collision effects were far too difficult to handle. His passion for simplification led to the emergence of the BGK (Bhatnagar, Gross, Krook) model. This gives a simple, yet a very realistic, Boltzmann-like equation, which has since been used as an alternative to the Boltzmann equation in solving problems in rarefied gas dynamics, plasma physics, and kinetic theory itself. The classic paper of Bhatnagar, E.P. Gross, and M. Krook published in the Physical Review in 1954 [7] is one of the most widely referred papers in plasma physics and is still very extensively used. The BGK model is more than just pedagogical textbook material at graduate level—today it has emerged as a basis of developing numerical methods to solve a wide variety of problems. It includes kinetic schemes for simulating compressible flows, and Lattice Boltzmann methods for simulating compressible and incompressible flows on computers. There are also several numerical simulations based directly on kinetic theory with particle simulations that use the BGK model, for example, in rarefied gas dynamics.

In order to highlight the key features of the BGK model, especially as it appears in modern day research, let us first motivate, very briefly, the central role played in physics by the ideas of Ludwig Boltzmann. Now, there are two ways to describe a fluid in motion. One is via kinetic theory, in which we track the motion of the constituent molecules on a length scale of the order of the mean free path of the molecules. In this theory, the position $\mathbf{x}$ and velocity $\mathbf{v}$ vectors of each of the molecules are the quantities of interest. Kinetic theory is also important in rarefied gases where the mean free path becomes very large—for example, in the calculation of the forces and pressures acting on a space shuttle as it re-enters the earth’s atmosphere from outer space. The second way to describe fluid motion is via continuum theory, in which we look at fluid motion on length scales much larger than the mean free path of the molecules. At length scales which we encounter in a laboratory, the atmosphere, the oceans or in stellar models, the individual molecules are no longer important and the state of the fluid is described in terms of some average properties of fluid elements containing billions, or perhaps even a much larger number, of molecules. These properties are mass density $\rho(\mathbf{x}, t)$, fluid velocity $\mathbf{u}(\mathbf{x}, t)$, fluid stress or pressure $p(\mathbf{x}, t)$ and other quantities such as internal energy density of the fluid at a point $\mathbf{x}$ and time $t$. A comparative example, where the above considerations are key, would be the calculation of the lift and the drag forces on the wings, fuselage or even the entire geometry of a supersonic modern-day fighter jet.

The BGK model is more than just pedagogical textbook material at graduate level—today it has emerged as a basis of developing numerical methods to solve a wide variety of problems

For our present purposes though, we shall stick to the understanding of fluid flows via kinetic theory. Of course, the laws of fluid dynamics, the Navier–Stokes equations, are fully consistent with the continuum limits of the equations of kinetic theory. In fact, the two viewpoints—viewing a flowing fluid as an ensemble of interacting molecules as in kinetic theory, or alternatively, as seen by the likes of Euler, Bernoulli, Navier and Stokes, as a continuum in motion—complement each other. This dichotomy between the continuum and the discrete is in some sense reflective of a deep basic question that has confronted human thought for millennia. If all matter is made of atoms, can macroscopically observed properties such as pressure, temperature and viscosity be finally traced back to interactions between these atoms?

Our current understanding of the microscopic origins of all macroscopically observed phenomena owes a lot to the Austrian physicist Ludwig Boltzmann, who laid the foundations of kinetic theory. Boltzmann dealt with an ensemble, or a collection of a large number of particles. Realizing that it was futile to attempt a deterministic analysis of this ensemble, he thought that one should study only their average property—that is, study them probabilistically. He did so using probability distribution functions, which are quantities that live in higher-dimensional spaces owing to the fact that every particle which is part of the ensemble has six numbers associated with it—three for its position and three for its velocity. If there are $N$ particles in the ensemble, the ensemble as a whole requires $6N$ pieces of information for its complete description. But this is the picture at only one instant of time. If one needs to know the dynamical evolution of such an $N$-particle ensemble, one needs an equation to track its evolution, snapshot by snapshot. This can be done using Liouville’s equation, which employs the notion of a “phase space”, where a single point, for its full description, needs all the above mentioned six pieces of information. Therefore, a curve traced by such a point in phase space is actually a geometrical way of tracking the dynamical evolution of a particle. The collection of all such curves, for all the particles for all times, is the geometrical way of capturing the entire system’s dynamical evolution. Considering the mind-boggling amount of bookkeeping required to track each of the particles in the ensemble, and also reasoning that it is not terribly important to track every one of those particles, Boltzmann employed probabilistic reasoning to make sense of the enormous amount of ceaseless microscopic activity. He realised the importance of collisions between particles as the key dynamical property.

James Clerk Maxwell too had been studying collisions around the 1860s. The key idea is that a system returns to its equilibrium state after collisions between its constituent particles. Maxwell had some early results in this area, but Boltzmann went much further, his work being valid in situations that are very far from thermal equilibrium. It is in such situations that one finds the work of Bhatnagar very relevant. In 1954, while at Harvard, he realized that Boltzmann’s collision integrals, or collision operators, as they are called, are not easy to compute with. Meanwhile, computers were becoming available in the 1950s, and there was a need for advances in tackling the unwieldy and computationally intensive Boltzmann collision operators. In collaboration with Gross and Krook, Bhatnagar provided an approximation that was founded on the idea that the rate at which the journey towards a Maxwellian distribution occurs, say inside a star or for the air in a room, is proportional to the frequency of molecular collisions. This assumption not only makes it easier to handle the collision operators, but also looks physically justifiable. This, briefly, is the BGK Model.

Let us see what all this means mathematically. When the density of a gas becomes sufficiently low and the mean free path of molecules is no longer negligibly small compared to a characteristic dimension of the flow geometry, results of continuum fluid dynamics require correction. This becomes more and more important as the degree of rarefaction increases. Continuum dynamics must then be replaced by the kinetic theory of gases, and the Navier–Stokes equations by the Boltzmann equation:

$\frac{\partial f}{\partial t} + \mathbf{v}\cdot\frac{\partial f}{\partial \mathbf{x}} = Q(f,f), \label{boltzmann1}$

which gives the evolution of an “expected number density” $f(\mathbf{x},\mathbf{v},t)$ in the phase space described by $(\mathbf{x}, \mathbf{v})$. That is, $f(\mathbf{x}, \mathbf{v}, t) \, d\mathbf{v}\, d\mathbf{x}$ is the expected number of molecules which at time $t$ lie in the “volume” element between $(\mathbf{x}, \mathbf{v})$ and $(\mathbf{x}+d\mathbf{x}, \mathbf{v}+d\mathbf{v})$ in the six-dimensional phase space. Here $Q(f,f)$ is the collision operator or collision integral and represents the time rate of increase of the expected number of molecules (per unit volume in the phase space), due to binary collisions of molecules. In equation \eqref{boltzmann1}, the dependence of $f$ on only the velocity $\mathbf{v}$ is reflected, while its dependence on $\mathbf{x}$ and $t$ has been suppressed.

For our purposes, the expression of $Q(f,f)$ is not important; it is a very involved quadratic integral operator, which acts via its dependence on the velocity. Further, it is symmetric—that is, $Q(f,g) = Q(g,f)$. What is important to mention here is the property that in a local thermodynamic equilibrium state, where the molecules are still moving, $f$ satisfies $Q(f,f)=0$.

Formally: In local thermodynamic equilibrium at position $\mathbf{x}$ and time $t$, the function $f$ satisfying $Q(f,f)=0$, is given by Maxwellian $f = M$, where

$M({\mathbf{v} -\mathbf{u}}) = \frac{\rho(\mathbf{x},t)}{({2RT(\mathbf{x},t)})^{-3/2}}\exp{\left(-\frac{|\mathbf{v} -\mathbf{u}|^2}{2RT}\right)},$

with $R$ as the Boltzmann constant, $\rho$ fluid density, $T$ temperature and $\mathbf{u}$ the macroscale fluid velocity at $(\mathbf{x},t)$.

As far as the validity of the Boltzmann equation goes, it is not necessary to assume the gas flow to be rarefied. It is valid for much denser gases and has been used to compute the transport coefficients, namely, the coefficients of viscosity, diffusion and thermal conduction, in order to complete the conservation equations of continuum mechanics.

The Boltzmann equation, however, is an extremely difficult nonlinear integro-differential equation, which, to this day, continues to pose formidable problems to mathematicians, physicists and engineers alike. But whenever it has yielded, some really deep results have emerged. One of them is: A quantity $H$, associated with the gas as a whole, is a non-increasing function of time (this is the celebrated $H$-theorem of Boltzmann) and is steady only when the distribution function $f$ is Maxwellian. Now, Maxwell’s basic hypothesis in his 1860 paper was that the innumerable collisions between the molecules in a gas would, instead of tending to equalize the velocities of all the molecules in the gas, produce a statistical distribution of velocities in which any velocity may occur but with a certain definite probability. The existence of such a unique equilibrium distribution to which any other distribution would evolve towards was a departure from previous thinking on the subject. As a corollary of this hypothesis, it follows that if the molecules of a gas already possess a Maxwellian distribution, then further collisions leave the distribution unchanged. If the gas has a configuration whose statistical distribution deviates from the Maxwellian distribution, it will achieve the Maxwellian distribution via collisions. However, molecular collisions do not cease once the Maxwellian is reached—they do continue indefinitely, but the distribution of velocities remains the same. The journey from any arbitrary distribution to the Maxwellian proceeds so as to reduce the value of the $H$-function to its minimum. The idea of a local Maxwellian is that the distribution of velocities is Maxwellian within an arbitrarily chosen control volume.

Another result to be noted here is regarding the collision process of two molecules, which gets completed over a very short distance compared to the mean free path of the molecules. After the collision, the total mass, momentum and thermal energy of the participating molecules remain the same as those before the collision. (The thermal energy is just the kinetic energy for the simplest molecules—that is, molecules for which the state can be described by three space coordinates and three velocity components.) Thus, for any collision process, there are five independent functions $\psi_{\alpha}(\mathbf{v})$, $\alpha= 0, 1, 2, 3, 4$, such that $\int \psi_{\alpha}(\mathbf{v}) Q(f, f) d\mathbf{v} = 0$, where the integration is over the whole three-dimensional velocity space $\mathbf{v}$. Hence $\psi_0 = 1$, $(\psi_1, \psi_2, \psi_3) = \mathbf{v}$ and $\psi_4 = \mathbf{v}^2$. The functions $\psi_{\alpha}$ are called elementary collision invariants, and it can be proved that a general collision invariant for the simplest molecules is a linear combination of the above mentioned five elementary collision invariants.

One of the major shortcomings in dealing with the Boltzmann equation is the complicated nature of the collision integral. Bhatnagar, along with his collaborators, Gross and Krook, realized that much of the detail of the two-body interaction, which is contained in the collision term, is not likely to influence significantly the values of many experimentally measured quantities and that a coarser description may be obtained by replacing the true collision term $Q(f,f)$ by a simpler one $J(f)$, which retains only the qualitative and average properties of $Q(f,f)$. This led to the BGK model in which $J(f)$ is assumed to possess the following two main features of the true collision term $Q(f,f)$: (i) The elementary collision invariants of $Q(f,f)$ retain their properties with respect to $J(f)$ too—that is, $\int \psi_{\alpha} J(f) d\mathbf{v} = 0$, $\alpha = 0, 1, 2, 3, 4$. (ii) The collision term expresses the tendency to approach a Maxwellian distribution ($H$-theorem).

Bhatnagar, Gross and Krook took the second feature into account by assuming that each collision changes the distribution function $f(\mathbf{v})$ by an amount proportional to the departure of $f(\mathbf{v})$ from the Maxwellian $M(\mathbf{v})$. That is,$J(f) = \frac{1}{\nu}[M(\mathbf{v}) - f(\mathbf{v})]$, where $\nu$ is independent of $\mathbf{v}$, and the other parameters in $M(\mathbf{v})$ are determined from the fact that at point $\mathbf{x}$ and time $t$, $M(\mathbf{v})$ must have the same density, velocity and temperature of the gas as that given by $f(\mathbf{v})$.

The principal theoretical advantage of employing the BGK operator is the ability to deduce the macroscopic conservation laws for mass, momentum and energy for any problem. Another advantage, especially in the present era of high performance computing, is the ability to treat important scientific problems of current interest within reasonable time and computational cost. What is interesting here is that the BGK operator, despite being intrinsically strongly nonlinear, still allows us to solve problems of technical importance quite admirably. Contrast that with the near impossibility of finding analytical solutions to the same problems via the full Boltzmann equation. A classic example here is the relaxation to equilibrium of a gas in the spatially homogeneous case [10]. In this case, an arbitrary distribution function depending only on the velocity is given, and we wish to study the evolution of this system with time. While this demonstrates the BGK operator’s practicality and thus its indispensability to fluid dynamicists, what is also equally pleasing to the theorist—in addition to the deduction of the macroscopic conservation laws mentioned earlier—is the ease with which the associated $H$-theorem can be proved. The BGK operator is therefore not only here to stay, but is a major landmark in the larger edifice of theoretical and computational fluid dynamics.

The BGK model contains the most basic features of the Boltzmann collision integral; however, it is not derived from the Boltzmann equation. Therefore, the discovery of the BGK model has led to systematic investigations for deducing models of increasing accuracy. Undoubtedly, the BGK model is one of the most important scientific contributions by an Indian mathematician.

In Bhatnagar’s own words, “Reward of research is the joy of creation”

On his return to India from Harvard, Bhatnagar exemplified two of his convictions which he held throughout his life. The first was to reach out to as many people as he could, and help them pursue research. He believed that since he had the capacity and the gift to do independent research, he should inspire others and introduce them to the joys of scientific research. In his own words, “Reward of research is the joy of creation.” Besides, every scientist had social obligations to fulfill. The second of his convictions was that the research of tomorrow has to be done by a group. The more people there are working cohesively on various aspects of the same theme, the more fruitful the findings would be. In 1953, Bhatnagar began to gather a band of colleagues and other teachers from Delhi University and encouraged them to take up research. As local secretary of the annual conference of the Indian Mathematical Society in December 1953 at Delhi, he used every opportunity to draw others into the scientific fold. Just as the scientific activity was picking up, there was a change in Bhatnagar’s life.

### At IISc, Bangalore

In 1950 he was elected fellow of the National Institute of Sciences (now INSA, Indian National Science Academy). In 1955, he was elected fellow of the Indian Academy of Sciences and, later in 1967, fellow of the National Academy of Sciences, India. As his stature rose, he was much sought after as a professor. In January 1956, the Indian Institute of Science, Bangalore invited him to join as the first Professor of the newly created Department of Applied Mathematics. It was initially intended that Bhatnagar give lectures on various topics in mathematics in the already established departments of science and engineering at the Institute so that the scientific community there could benefit from his presence. It was not intended that the Department grow in size or scope, but that it remain very much a one-man affair.

This, however, was against all that Bhatnagar believed in or cherished. He was bubbling with enthusiasm for scientific research, and he had to transmit this to others. Slowly he began to gather young research students from all parts of India. In order that the department be truly a department of applied mathematics, Bhatnagar realized he would have to widen his scientific research. Besides kinetic theory of gases and plasma physics, he initiated research in fluid dynamics, including boundary layer theory, magnetohydrodynamics, and the theory of non-Newtonian fluids. The choice of these subjects was inspired by the fact that the engineering departments at the Institute could verify experimentally the theoretical results obtained by his group. He desired that as far as possible, research activity at the Institute should be a cooperative effort. Acutely aware of the fact that the work of the future would involve nonlinear effects and discontinuous solutions, he initiated work in shock propagation and the theory of nonlinear waves. He was also convinced that computers would play a dominating role in research in the years to come and encouraged the use of computer-oriented numerical techniques and the study of mathematical logic along with other basic mathematical topics.

At IISc, Bhatnagar initiated research in fluid dynamics

His work in this period in the area of non-Newtonian fluid flows deserves special mention, as it was a relatively new area at that time and many questions were yet to be answered. The famous experiments of Weissenberg and Merrington [11] had shown that fluid behaviour could not always be explained by the Newtonian stress-rate-of-strain relation

$\tau_{ij} = -p \delta_{ij} + \mu e_{ij}$

where $\tau_{ij}$ is the stress tensor, $e_{ij}$ is the rate-of-strain tensor, $\delta_{ij}$ is the Kronecker delta, $p$ is the pressure and $\mu$ is the Newtonian coefficient of viscosity. When a fluid is subjected to shearing flow between two coaxial cylinders, the inner of which is at rest and the outer rotating, then, in general, the level of the free surface close to the inner cylinder drops. However, in certain highly viscous fluids, it was found that the fluid close to the inner cylinder was forced inwards against the centrifugal force and upwards against the force of gravity, so that the fluid rose at the inner cylinder. This is the Weissenberg effect. Such phenomena showed that in addition to the shear stress components, which were predicted by the Newtonian theory, there were additional normal stress components not predicted by the theory. This led to a spate of fluid models generally referred to as “non-Newtonian” fluids. Among these models were: (i) those which took account of the elasticity of the fluids—“viscoelastic fluids”, (ii) those which were governed by higher order stress-rate-of-strain relations—“power law fluids”, (iii) those with cross viscosity, (iv) those with couple stress and microrotations. In order to classify and distinguish between various non-Newtonian fluid models and calculate the innumerable constants associated with these models, Bhatnagar undertook a study of secondary flows. He realized the importance of studying shearing flows in these fluids, and together with his doctoral students published a large number of papers on flows between rotating boundaries, like flat plates, cylinders, spheres, cones and plates. This was acknowledged by C. Truesdell and W. Noll in their book The Non-Linear Field Theories of Mechanics, originally published in 1965 [8].

Besides suggesting how to classify various non-Newtonian fluids, Bhatnagar also realized that these different models, though apparently based on widely differing phenomenological considerations, behaved very similarly in shearing flows. In fact, the Oldroyd model, the Walter model and the Rivlin–Ericksen model (with a certain restriction on its parameters) could be described in shearing flow by the same expressions with a suitably defined common non-Newtonian parameter. Bhatnagar also showed that the normal stresses, which were present in rotating shearing flows of non-Newtonian fluids, opposed the centrifugal forces, in general. This leads to a phenomenon peculiar to non-Newtonian fluids, namely, that even when the boundary rotates in the same sense, the secondary flows can “break” and there can be reversal of flows, provided the normal stresses are strong enough to overcome the effects of the centrifugal forces.

In the period 1960–1965, Bhatnagar spent many months abroad at the Harvard College Observatory at Cambridge, Massachusetts. During this period, he developed a severe problem in the spinal region of the lower back. He was unable to walk long distances or keep standing for a long time. This did not dampen his spirit. In spite of the fact that he usually paced up and down impatiently during a lecture, he got used to delivering talks while seated in a chair. The walk to the department and back to his house was a painful process and he would have to rest in between. It was a familiar sight to see him seated on the steps en route with a knot of students around him, discussing some research problem quite unmindful of the pain in his back. Surgery in the United States corrected his spinal problem and soon he was seen to run and dash around campus.

Bhatnagar was my teacher and the research supervisor of my Ph.D. thesis. But it did not end there. As it was with all of his students, his concern for my welfare and academic progress was so great that unknowingly and gradually I was drawn closer towards him. I vividly recall my first meeting with him. It was 9:30 am, on 9 April 1965. I had just arrived from Calcutta and I was waiting in the office of the Department of Applied Mathematics, IISc, to meet the much admired Professor whom I had not seen before. Then suddenly a tall handsome person passed by very swiftly and Srinivasamurthy (Secretary in the office) whispered, “He is the Professor.” After ten minutes I was facing Bhatnagar and C. Devanathan in the discussion room where I was subjected to examination for more than two hours. This first meeting with him created in me an image of a great mathematician who was deeply interested in mathematics as well as in his students.

Whenever a distinguished visitor came to the Applied Mathematics Department at IISc, Professor Saheb (that is how we referred to him) used to call all the members (most of whom were his students) in the seminar room for academic discussion and, while introducing them to the visitor, apart from mentioning the research work of each of them, he used to say proudly, “Here is a mini-India; you will find here a person from every State of India.” It was a fact, and no other department of IISc could boast of such a “cosmopolitan” group. By April 1969, the Department of Applied Mathematics at IISc had grown in stature. About 25 of his students had either completed or were in the process of completing their work for a Ph.D. degree. The department had grown in size to over ten faculty members working in various areas of applied mathematics.

### At Jaipur and Shimla

Bhatnagar was not one to rest on his laurels. With an overwhelming desire to serve a larger academic community, he decided to leave it all behind and move to Jaipur, where he was offered the Vice-Chancellorship of the University of Rajasthan. It was a time of student unrest and tumult. Bhatnagar with his strong belief in the innate honesty and goodness of man, especially of the younger generation, was never ruffled. He talked to the students, cajoled, coaxed and convinced them on major contentious issues. He prided himself on the fact that anyone could walk into his office and talk to him. However, this was not his life’s work and it pained him that he found very little time for research; so he decided to relinquish office at the end of two years.

In December 1969, I went with my wife, Mandra, to Jaipur where Bhatnagar was then the Vice-Chancellor of the Rajasthan University. He arranged for our stay in his own bungalow and it was here that I first got a glimpse into his personal life. He used to get up 4:30 am (the minimum temperature then was about zero degree Celsius), take a brisk walk for about half an hour in the garden and then used to study in the library till 7:30 am. At 7:30 am, he would take tea with all members of the family and the guests, himself preparing and serving each cup. At this time he would discuss familial and general matters, and would be found in a most jovial mood. His witticisms would cheer us up. Then after bath and breakfast, he would become busy with matters related to university administration. Again in the night he would find time to work on research problems till 11 pm.

In May 1971, at the invitation of the newly formed Himachal Pradesh University at Shimla, he joined as Senior Professor and Head of the Department of Mathematics. Soon after, he left to spend a year as a Visiting Professor at the University of Waterloo in Canada. It was a special occasion, because for the first time he was going abroad with his wife, Anand Kumari, his soulmate for over forty years. To him, family always meant the “extended” family—his students, colleagues and coworkers included. His wife and children hardly saw him during the week, because there were so many of his “extended” family clamouring for his time and attention. It was always “open house” at the Bhatnagars’, with gracious hospitality.

To Bhatnagar, family always meant the “extended” family—his students, colleagues and coworkers included

At Shimla, Bhatnagar again set out from scratch to build a department of mathematics. He was appointed a UGC National Lecturer in 1972. He also gave the N.R. Sen Memorial lecture sponsored by the Calcutta Mathematical Society that year. While he was at Meerut on a lecture assignment in January 1973, he heard about the passing away of his wife at Kota. After this tragic incident, Bhatnagar was very unhappy and alone at Shimla. His appointment as member of the Union Public Service Commission (UPSC) came as a welcome change and he moved to Delhi in October 1973. In Delhi, he lived in a large house close to India Gate. However, he was very lonely. Of his four sons, Rakesh and Brijendra were abroad and the others, Vinay and Kamal, were working outside Delhi. His only daughter, Kalpana, was busy with her own growing family at Bhopal. As usual, he devoted his entire energy to his work. At heart, he was a mathematician and a teacher and UPSC could not hold him long. When the Mehta Research Institute (MRI) of Mathematics and Mathematical Physics at Allahabad was to take shape, Bhatnagar accepted the offer to be its first Director in July 1975. (MRI is now Harish-Chandra Research Institute, named after the mathematician Harish-Chandra.)

### At Mehta Research Institute

After the death in January 1973 of his wife, who used to look after every aspect of the household, he became completely broken and his personal life was filled with great sadness. In that state of depression his food intake was reduced to only once a day, and his health began to deteriorate rapidly. After six months he asked himself, “Why am I wasting my life?” Then he took courage and decided not to continue to brood over what he could not get back; he decided to live once again a life dedicated to the service of mathematics and his country. Gradually his health too stabilised to a considerable extent. But his personal life at Allahabad was far from comfortable and sometimes loneliness used to be unbearable for him. He compounded his inner discomfort by sacrificing his personal comforts for the sake of the MRI. He would not utilize for personal use the car given to him as the Director, and he did not allow the MRI to even appoint a watchman for the Director’s residence. Once or twice, when his personal servant left the job, he did every household chore himself, and stubbornly refused any help from the Institute employees. Once on a Saturday evening a member of the administrative staff and a Research Fellow of the MRI told him that they would come to his residence at 11 am the next day. It was apparent that they wished to offer help for his personal work. The next day Professor Saheb personally finished all the household work very early. When the two persons reached his residence at 11 am, he was sitting comfortably working on a research problem. After receiving them personally, he himself prepared tea for everyone, and even offered his visitors some food. All that they did was spend a few delightful and memorable hours with him.

Bhatnagar worked ceaselessly to nurture MRI

Bhatnagar was very critical of the poor quality of research in mathematics and its applications in India. He attributed the poor standards of research and teaching to the existence of persons in key positions in the universities who were more interested in retaining their positions, their powers and a host of personal benefits unchallenged than in devoting their valuable time for study. In December 1975 he convened a Regional Conference on the Development of Integrated Curriculum in Mathematics for developing countries in Asia. The conference was at Bharwari, near Allahabad. MRI was still in its infancy at the time. Bhatnagar conducted the proceedings with finesse—looking after the foreign and Indian guests, arranging the lectures, initiating discussions—all in this tiny little place. Only he could have done it.

In May 1976, the indefatigable Bhatnagar conducted a one-month course on “Hyperbolic Systems of Partial Differential Equations and Nonlinear Waves”. With only one faculty member to help him, he kept up a furious pace of lectures starting from a simple linear wave, going through the Burger’s and Korteweg-de-Vries equations in detail, the method of inverse scattering, group velocity in nonlinear systems and its Lagrangian formulations. In his concluding talk during this course, he advised the research students to revolt against supervisors who were incapable of guiding their research students properly and who kept them only to get some joint research papers. He personally told me once, “I am more anxious to get devoted bright workers in the Institute than to obtain financial support for it.”

Let me mention here a beautiful book, which Bhatnagar wrote in just six months based on the course mentioned above and from the reprints of recent articles on the Korteweg-de-Vries equation, which I had carried with me from IISc, Bangalore to MRI during my stay there in 1975–76. This book is Nonlinear Waves in One-dimensional Dispersive Systems, published in the series “Oxford Mathematical Monographs”, and edited by I.G. MacDonald and R. Penrose [9]. The importance of the book was realized by M.J. Lighthill, who had written a foreword for the book before Bhatnagar passed away and added in it later: “I had written the above words before the deeply regretted and untimely death of Professor Bhatnagar on 5 October 1976, when the world of applied mathematics suddenly lost one of its most respected figures. After the shock of this great loss had subsided, I felt anxious to ensure that Professor Bhatnagar’s last book would receive the wide circulation that it richly merits.” The book received good reviews, was immediately translated into Russian and it was used as a textbook in USSR. No less a person than V.E. Zakharov, who has contributed significantly to the development of the theory of solitons, told me: “I was surprised to see the first book (a good book) on the subject from a country where no contribution to the theory of solitary waves was made.”

As the tempo of his research activities increased, Bhatnagar began to get more and more international recognition. He was a member of Commissions 27 (Variable stars) and 43 (Magnetohydrodynamics) of the International Astronomical Union, and a corresponding member of the International Commission on Plasma Physics, appointed by the International Union of Pure and Applied Physics. In 1967 he was invited by the Royal Society as a Distinguished Visiting Scientist and he delivered two lectures on his research work on Slip Flows and Secondary Flows at the Imperial College, London. In 1968 he was invited to the International Congress on Rheology at Kyoto in Japan.

At home, too, the 1960s were a very busy period for Bhatnagar. He was, among other things, President of the Mathematics Section of the Indian Science Congress (1962), the Indian Mathematical Society (1964, ’65, ’68), the Physical Sciences Section of the National Academy of Sciences (1969), the Congress of the Indian Society for Theoretical and Applied Mechanics (1971), and the Association of Mathematics Teachers of India (1968–76). In his immediate neighbourhood, he formed the Bangalore Mathematical Association and under its auspices, apart from lectures, introduced Mathematics Olympiads on the lines of those held in East European Countries. This was the first-ever Mathematics Olympiad held in India and it is a tribute to the foresight of Bhatnagar that almost thirty years ago he saw the need for them in India. Later, the National Board for Higher Mathematics began conducting the Indian National Mathematics Olympiads all over the country. Today, Olympiads in many subjects, including mathematics, are organised by the Homi Bhabha Centre of Science Education, TIFR and students are trained for International Olympiads. In fact, training programmes in India for the International Mathematical Olympiads started in 1986 at IISc, from where Bhatnagar had begun the Mathematics Olympiad activity in 1968. For his service to the nation, Bhatnagar was fittingly awarded the Padma Bhushan on 26 January 1968.

He was a UNESCO expert in Egypt for a three-month period in 1969, a member of the Executive Council of the International Conference on Mathematics Education at Lyon in 1969, a leader of the Indian delegation to the lndo–US Binational Conference on Mathematics Education and Research at Bangalore in June 1973, and a member of the International Commission on Mathematical Instruction at Vancouver in August 1974.

In spite of recurring spells of pain and giddiness, Bhatnagar went abroad in 1976. He had been leader of the Indian delegation to the XIV International Congress of Theoretical and Applied Mechanics at Moscow in 1972 and was now attending the XV Congress at the Hague, Netherlands. He also attended the International Conference on Mathematics Education at Karlsruhe in West Germany.

In all that is said of Bhatnagar the scientist, one cannot overlook Bhatnagar the man. He was a man of simple ways and high principles. He believed in the existence of God, but at the same time, he felt that this belief was not wholly rational. The rituals of “bribing” deities for purely selfish reasons were obnoxious to him. Even the thought of bargaining with his employers for his own benefit was repulsive. He said, “I would not like to bargain with an institution that I am going to serve.” All his life he indulged in writing poetry. He never published his poems, but those who have heard him recite his poetry were impressed by his creativity and found it an unforgettable experience. Humility was another of his characteristics. If he found that he were in the wrong, he never hesitated to apologize to his students or anyone else whom he felt he had wronged. Perhaps the following words of Bhatnagar summarize his qualities as a human being: “What I cherish most in my life is peace and humility. Peace within ensures peace outside and wise decisions in all matters. Humility enables one to understand the other man’s point of view and creates an atmosphere of affection all around you.”

One incident would suffice to illustrate his humility. In April 1975, he was in Madras to conduct some interviews as a Member of the UPSC. After ascertaining his programme, V.G. Tikekar and myself informed him that we would like to meet him and that we would arrive in Madras by a particular train. On reaching Madras we thought that we would be able to meet him only in the evening as during the day he would be busy conducting the interviews. We went to the railway canteen and when we were about to finish our lunch, Tikekar got up from his chair and said “Professor Saheb has come!” and rushed towards him. During the lunch-break, Professor Saheb had rushed to the station to receive us; he had searched for us and had located us in the canteen. This is how he would receive anyone even if it meant inconvenience or personal discomfort to him. Likewise, he was most hospitable in his house. He never had a watchman manning his gate. Anybody could meet him without any formality.

On 5 October 1976, a month after his return from abroad to India, Bhatnagar passed away very quietly. He was to go to Delhi that afternoon. He had complained of chest pain and had gone for a medical check-up at 9:30 am. The doctors found nothing wrong with him and even told him that he could go to Delhi that afternoon. A few minutes after leaving the hospital, alone in the back seat of his car, he had a massive heart attack and passed away. His body was consigned to flames on the banks of the Ganga. The most important weekly magazine of India at that time, The Illustrated Weekly of India paid tribute to him with a two-page article by the well-known science writer Jagajit Singh.

A man of his calibre needs no epitaph. His life is an example in itself and speaks far more than what anybody can write about him. His was a life of simple honesty, high ideals, and high thinking. May his tribe increase.

acknowledgements This article was prepared and edited by Sudhir Rao and Nithyanand Rao, based on [1], [2], and [3], with kind permission from the respective authors—Phoolan Prasad, V.G. Tikekar and Renuka Ravindran—and publishers—The Association of Mathematics Teachers of India, the Indian Mathematical Society and the Indian National Science Academy.

### References

• [1] P. Prasad. A Few Glimpses of Professor P.L. Bhatnagar. Mathematics Teacher (India). 1978. 14:82–86
• [2] V.G. Tikekar, P. Prasad, R. Ravindran. P.L. Bhatnagar (1912–1976), Biographical Memoirs of Fellows of the Indian National Science Academy. 1989. 14:116–134
• [3] For a more detailed treatment of the BGK model, see: P. Prasad. P.L. Bhatnagar and the BGK Model. 19th P.L. Bhatnagar Memorial Award Lecture – 2006, delivered at the 72nd Annual Conference of the Indian Mathematical Society at Jabalpur, 27–30 December 2007
• [4] E. Kamke. Differentialgleichungen. Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen. 1977. Vieweg+Teubner Verlag|Springer/Fachmedien Wiesbaden GmbH, Wiesbaden. ISBN 978-3-663-05926-4. DOI 10.1007/978-3-663-05925-7
• [5] A.C. Banerji and P.L. Bhatnagar. On the Solution of Certain Types of Differential Equations. Proceedings of the National Academy of Sciences, India. 1938. 8:85–87 and 87–91
• [6]D.H. Menzel, P.L. Bhatnagar and H.K. Sen. Stellar Interiors. Chapman & Hall. 1963. International Astrophysical Series Vol. VI. Online; accessed 27 March 2017:
• [7] P.L. Bhatnagar, E.P. Gross, and M. Krook. A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems. 1954. Phys. Rev. 94(3):511–525. DOI 10.1103/PhysRev.94.511
• [8]C. Truesdell and W. Noll. The Non-Linear Field Theories of Mechanics. Ed. S.S. Antman. 2004. Springer-Verlag Berlin Heidelberg. Softcover ISBN 978-3-642-05701-4. DOI 10.1007/978-3-662-10388-3
• [9] P.L. Bhatnagar. Nonlinear Waves in One-dimensional Dispersive Systems. Oxford University Press. 1980. ISBN 978-0198535317
• [10] C. Cercignani. Methods of Solution of the Boltzmann Equation for Rarefied Gases, pp.3–104 in Rarified Gas Flows Theory and Experiment. Ed. W. Fiszdon. CISM 224. 1981. Springer-Verlag Wien. DOI:10.1007/978-3-7091-2898-5
• [11]A good description of the Weissenberg and Merrington effects can be found in the abstract of the paper: S.L. Rathna, P.L. Bhatnagar. Weissenberg and Merrington effects in Non-Newtonian Fluids. Journal of the Indian Institute of Science. 1963. 45(3). These effects can also be seen, respectively, in the following YouTube videos (online; accessed 23 March 2017): https://youtu.be/npZzlgKjs0I and https://youtu.be/KcNWLIpv8gc
Phoolan Prasad, a student of P.L. Bhatnagar, is a retired professor of mathematics at the Indian Institute of Science, Bangalore. He gratefully acknowledges that parts of this article have been sourced with permission from a joint article [2] he wrote with V.G. Tikekar and Renuka Ravindran, also students of P.L. Bhatnagar.