Shreyas Mandre

Dimensional analysis

Sun, 23 Aug 2026 00:00:00 +0000

Almost every quantity in engineering and the sciences carries units, and that single fact constrains the form any physical law can take. These notes develop that constraint into a working method: how to choose a system of dimensions, how to reduce a relationship to its dimensionless parameters, and what Buckingham’s Pi theorem does and does not tell you. The final chapter puts it to work on experimental data and on scale models.

MSCA Postdoctoral Fellowships 2026

Sun, 23 Aug 2026 00:00:00 +0000

Applications close on 9 September 2026. I can act as the Cambridge supervisor or host this scheme requires — write to me before the deadline if you intend to apply.

MSCA Postdoctoral Fellowships 2027

Sun, 23 Aug 2026 00:00:00 +0000

Applications close on 8 September 2027. I can act as the Cambridge supervisor or host this scheme requires — write to me before the deadline if you intend to apply.

Newnham College Research Fellowship (Science, Mathematics, Engineering or Psychology)

Sun, 23 Aug 2026 00:00:00 +0000

A three-year stipendiary Research Fellowship at Newnham College, open across science, mathematics, engineering and psychology. Applications close 23:59 on Thursday 3 September 2026, for a Fellowship starting 1 October 2027.

Summer internships I can host

Sat, 22 Aug 2026 09:02:00 +0100

Most summer research schemes — departmental, national, or run by your own university — require a host who agrees to take you before you can apply. I am available to act as that host. If you are applying to a scheme that funds a summer research placement and need a supervisor at Cambridge, write to me.

Postdoctoral fellowships I can host

Sat, 22 Aug 2026 09:01:00 +0100

Every postdoctoral fellowship below requires a host institution and a named supervisor who commits to the application before it can be submitted. I am available to act as host and supervisor for applications in my areas. If you are planning to apply to one of these schemes and need a Cambridge host, write to me.

PhD scholarships I can supervise

Sat, 22 Aug 2026 09:00:00 +0100

The scholarships below fund PhD study at Cambridge. Most of them you apply for yourself, alongside your application for admission — they are not awarded on a supervisor’s nomination, and nobody can put you forward for them. What a supervisor does is agree to take you, and an application backed by a supervisor who has already discussed the research with you is a materially stronger one. I am available to be that supervisor.

Fluid-structure instability

Sat, 22 Aug 2026 00:00:00 +0000

A university course on the instabilities that arise when a fluid flow and a deformable structure interact. (Taught at Cambridge as 4A10, Flow Instability and Fluid-Structure Interaction.) The HTML and PDF lecture notes may be found at this link. Here is the learning material:

EPSRC Doctoral Prize

Fri, 01 Mar 2024 14:36:56 +0100

EPSRC funds this prize for PhD students currently funded through EPSRC, who are near completion. Visit the link to see if you meet the eligibility criteria and to decide whether you want to pursue this opportunity. If you do, then the first step to take immediately is to identify a supervisor. (If my research interests you, write to me right-away. I have numerous research interests, so check my publications to determine if there is an overlap.)

Germ granule transport during early zebrafish embryogenesis

Mon, 04 Sep 2023 14:36:56 +0100

Apply here.

In some living organisms (e.g. zebrafish), the gametes carry a substance called the germplasm made of RNA and other proteins. This germplasm is present in the organism since its unicellular embryonic stage, and is transported through development to the adult reproductive cells. In the unicellular embryo, the germplasm phase-separates into granules called germ granules, and when the cell divides, the granules concentrate along the cell-division furrows. In this project, we investigate the first few steps of this transport starting from the unicellular stage through a few cell-division cycles.

History of Wind Tunnels by [Prof. Geoffrey Spedding](https://viterbi.usc.edu/directory/faculty/Spedding/Geoff)

Sat, 27 May 2023 04:17:47 +0100

A brief history of wind tunnels, the challenge they faced in their design and the science they enabled and inspired. Highly recommended for all fluid dynamics and aerodynamics enthusiasts.

Systematic control of wind turbine wakes and interaction between wind turbines

Thu, 05 Jan 2023 15:20:38 +0000

This project is supported by the EPSRC DTP awarded to Cambridge Department of Engineering.

A wind turbine converts the kinetic energy of incident wind to electricity, and thus by design causes a wake with a deficit of energy (see Figure 1). Any turbines operating in the wake thus underperform severely, causing the windfarm to generate between 30-60% of installed capacity. Naturally, mechanisms for redirecting wakes are sought but no effective candidates are known.

In this project, we pursue a systematic approach to understand the possibility and nature of the control the windfarm has on the collective wake of turbines.

Elastohydrodynamics of contact in adherent sheets

Mon, 22 Aug 2022 00:00:00 +0000

Adhesive contact between a thin elastic sheet and a substrate arises in a range of biological, physical and technological applications. By considering the dynamics of this process that naturally couples fluid flow, long-wavelength elastic deformations and microscopic adhesion, we analyse a sixth-order thin-film equation for the short-time dynamics of the onset of adhesion and the long-time dynamics of a steadily propagating adhesion front. Numerical solutions corroborate scaling laws and asymptotic analyses for the characteristic waiting time for adhesive contact and for the speed of the adhesion front.

Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid

Wed, 30 Mar 2022 00:00:00 +0000

We determine the globally minimum time 𝑇 needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients.

Problems From Another Time

Mon, 28 Mar 2022 09:42:21 +0100

Prof. Steven Strogatz, Cornell University, tweeted recently about Convegerence, the publication by the Mathematical Association of America. Great resource for mathematics aficionados. Their collection of Problems from another time, which is also conveniently indexed by subject and geography, is highly addictive. So handle with care.

Modest advice by [Stephen C. Stearns](https://stearnslab.yale.edu/)

Wed, 26 Jan 2022 16:47:38 +0000

Are you thinking of pursuing a PhD? You think you know what you are getting into? Have you already been inducted into a PhD program? Stop what you are doing and first read this advice timely and timeless advice. The perspective Prof. Stearns offers will help you throughout your life.

Fluid Dynamics

Thu, 13 Jan 2022 07:33:50 +0000

A university course in fluid dynamics for undergraduates in Maths and Physics. (Some version of this is taught at Warwick as MA3D1 and at Brown as ENGN0810.) The HTML and PDF lecture notes may be found at this link. Here is the learning material, including recordings of videos made in the unusual year of 2020:

Mechanics of fluid interfaces

Thu, 13 Jan 2022 06:20:01 +0000

This material treats the mechanics that arises from the presence of surface tension of liquid interfaces. Liquid interface dynamics, especially due to the action of surface tension, has always been and remains one of the most interesting topics in mechanics. Even the discovery of surface tension or the first measurement of surface tension was not trivial. Today, as certain aspects of our world are miniaturized due to the emergence of “nanotechnology”, effects of surface tension are more imminently noticeable.

Nonlinear dynamic 1 & 2 by [Prof. Predrag Cvitanovic](https://physics.gatech.edu/user/predrag-cvitanovic)

Sun, 09 Jan 2022 09:27:43 +0000

Offered by Prof. Predrag Cvitanovic, Georgia Institute of Technology. Part 1 starts Jan 11, 2022. More information about how to sign up, or loosely participate, can be found here. A course on chaos through which " … you might inadvertently learn something that your professor does not know." The rationale behind designing such a course and delivering it to the greater online community is well thought out. Highly recommended if you have or had a professor!

Feet and fins

Fri, 24 Dec 2021 02:23:21 +0000

Why do our feet look the way they do? Believe it or not, some of us do ask ourselves this question.

Fluid mechanical kinematic optimization

Sun, 03 Jan 2021 16:32:40 +0000

In 1696, Johann Bernoulli posed a challenge called the brachistochrone problem, which kickstarted the field of calculus of variations. Here is simplest fluid mechanical version of the brachistochcrone problem.

Work-minimizing kinematics for small displacement of an infinitely long cylinder

Thu, 25 Jun 2020 00:00:00 +0000

We consider the time-dependent speed of an infinitely long cylinder that minimizes the net work done on the surrounding fluid to travel a given distance perpendicular to its axis in a fixed amount of time. The flow that develops is two-dimensional. An analytical solution is possible using calculus of variations for the case that the distance travelled and the viscous boundary layer thickness that develops are much smaller than the circle radius.

Stiffness of the human foot and evolution of the transverse arch

Wed, 26 Feb 2020 00:00:00 +0000

The stiff human foot enables an efficient push-off when walking or running, and was critical for the evolution of bipedalism. The uniquely arched morphology of the human midfoot is thought to stiffen it, whereas other primates have flat feet that bend severely in the midfoot. However, the relationship between midfoot geometry and stiffness remains debated in foot biomechanics, podiatry and palaeontology. These debates centre on the medial longitudinal arch and have not considered whether stiffness is affected by the second, transverse tarsal arch of the human foot.

Dynamics and stability of running on rough terrains

Wed, 13 Mar 2019 00:00:00 +0000

Stability of running on rough terrain depends on the propagation of perturbations due to the ground. We consider stability within the sagittal plane and model the dynamics of running as a two-dimensional body with alternating aerial and stance phases. Stance is modelled as a passive, impulsive collision followed by an active, impulsive push-off that compensates for collisional losses. Such a runner has infinitely many strategies to maintain periodic gaits on flat ground.

Confinement effects on energy harvesting by a heaving and pitching hydrofoil

Tue, 01 Jan 2019 00:00:00 +0000

Wall confinement effects on the energy harvesting performance by a flapping hydrofoil (aspect ratio 4.5) have been investigated in a circulating water flume at a Reynolds number of 50,000. Measurements of hydrodynamic forces are taken for three different confinement configurations (unconfined, one-wall and two-wall confinement) and a series of confinement levels for each configuration. Compared with the unconfined situation, a significant improvement of efficiency performance is obtained for strong two-wall confinement due to the enhancement of the hydrodynamic forces, while only a modest increase is observed in the one-wall confinement configuration.

Dynamics of a camphoric acid boat at the air–water interface

Thu, 03 May 2018 00:00:00 +0000

We report experiments on an agarose gel tablet loaded with camphoric acid (c-boat) spontaneously set into motion by surface tension gradients on the water surface. We observe three distinct modes of c-boat motion: harmonic mode where the c-boat speed oscillates sinusoidally in time, a steady mode where the c-boat maintains constant speed, and an intermittent mode where the c-boat maintains near-zero speed between sudden jumps in speed. Whereas all three modes have been separately reported before in different systems, controlled release of Camphoric Acid (CA) from the agarose gel matrix allowed the observation of all the three modes in the same system.

Surface tension of flowing soap films

Tue, 20 Feb 2018 00:00:00 +0000

The surface tension of flowing soap films is measured with respect to the film thickness and the concentration of soap solution. We perform this measurement by measuring the curvature of the nylon wires that bound the soap film channel and use the measured curvature to parametrize the relation between the surface tension and the tension of the wire. We find that the surface tension of our soap films increases when the film is relatively thin or is made of soap solution of low concentration; otherwise, it approaches an asymptotic value of 30 mN/m.

Flowing soap films

Thu, 04 Jan 2018 05:37:57 +0000

Ever wondered why you can blow bubbles from soap solution in water, but not from water itself or from a solution of sugar or salt?

Hydrodynamic signatures of stationary Marangoni-driven surfactant transport

Thu, 28 Dec 2017 00:00:00 +0000

We experimentally study steady Marangoni-driven surfactant transport on the interface of a deep water layer. Using hydrodynamic measurements, and without using any knowledge of the surfactant physicochemical properties, we show that sodium dodecyl sulphate and Tergitol 15-S-9 introduced in low concentrations result in a flow driven by adsorbed surfactant. At higher surfactant concentration, the flow is dominated by the dissolved surfactant. Using camphoric acid, whose properties are a priori unknown, we demonstrate this method’s efficacy by showing its spreading is adsorption dominated.

Axisymmetric spreading of surfactant from a point source

Sun, 10 Dec 2017 00:00:00 +0000

Guided by computation, we theoretically calculate the steady flow driven by the Marangoni stress due to a surfactant introduced on a fluid interface at a constant rate. Two separate extreme cases, where the surfactant dynamics is dominated by the adsorbed phase or the dissolved phase, are considered. We focus on the case where the size of the surfactant source is much smaller than the size of the fluid domain, and the resulting Marangoni stress overwhelms the viscous forces so that the flow is strongest in a boundary layer close to the interface.

Marangoni elasticity of flowing soap films

Mon, 28 Aug 2017 00:00:00 +0000

We measure the Marangoni elasticity of a flowing soap film to be 22 mN/m irrespective of its width, thickness, flow speed, or the bulk soap concentration. We perform this measurement by generating an oblique shock in the soap film and measuring the shock angle, flow speed, and thickness. We postulate that the elasticity is constant because the film surface is crowded with soap molecules. Our method allows nondestructive measurement of flowing soap film elasticity and the value 22 mN/m is likely applicable to other similarly constructed flowing soap films.

Controllable biomimetic birdsong

Wed, 02 Aug 2017 00:00:00 +0000

Birdsong is the product of the controlled generation of sound embodied in a neuromotor system. From a biophysical perspective, a natural question is that of the difficulty of producing birdsong. To address this, we built a biomimetic syrinx consisting of a stretched simple rubber tube through which air is blown, subject to localized mechanical squeezing with a linear actuator. A large static tension on the tube and small dynamic variations in the localized squeezing allow us to control transitions between three states: a quiescent state, a periodic state and a solitary wave state.

Curvature-induced stiffening of fish fin

Wed, 31 May 2017 00:00:00 +0000

How fish modulate their fin stiffness during locomotive manoeuvres remains unknown. We show that changing the fin’s curvature modulates its stiffness. Modelling the fin as bendable bony rays held together by a membrane, we deduce that fin curvature is manifested as a misalignment of the principal bending axes between neighbouring rays. An external force causes neighbouring rays to bend and splay apart, and thus stretches the membrane. This coupling between bending the rays and stretching the membrane underlies the increase in stiffness.

Energy harvesting performance and flow structure of an oscillating hydrofoil with finite span

Sat, 01 Apr 2017 00:00:00 +0000

The energy harvesting performance and resulting flow structures of a hydrofoil oscillating in pitch and heave are studied experimentally in a water flume. The shape of a hydrofoil cross-section is shown to have negligible influence on the power generation for the geometries tested. It is found that contribution to efficiency from heaving motion increases with reduced frequency at optimal pitching amplitude. However, contribution to efficiency from pitching motion decreases with reduced frequency because the development of a leading-edge vortex during the stroke is delayed at the high reduced frequency.

Linear stability analysis for monami in a submerged seagrass bed

Sun, 10 Jan 2016 00:00:00 +0000

The onset of monami – the synchronous waving of seagrass beds driven by a steady flow – is modelled as a linear instability of the flow. Unlike previous works, our model considers the drag exerted by the grass in establishing the steady flow profile, and in damping out perturbations to it. We find two distinct modes of instability, which we label modes 1 and 2. Mode 1 is closely related to Kelvin–Helmholtz instability modified by vegetation drag, whereas mode 2 is unrelated to Kelvin–Helmholtz instability and arises from an interaction between the flow in the vegetated and unvegetated layers.

Waving marine grass

Mon, 04 Jan 2016 06:08:23 +0000

I bet you have seen gusts of wind exciting synchronized waves on the surface of tall grass field just like observed on the beach.

Dynamics of evaporative colloidal patterning

Tue, 29 Sep 2015 00:00:00 +0000

Drying suspensions often leave behind complex patterns of particulates, as might be seen in the coffee stains on a table. Here, we consider the dynamics of periodic band or uniform solid film formation on a vertical plate suspended partially in a drying colloidal solution. Direct observations allow us to visualize the dynamics of band and film deposition, where both are made of multiple layers of close packed particles. We further see that there is a transition between banding and filming when the colloidal concentration is varied.

Cheerios effect

Sun, 04 Jan 2015 06:39:01 +0000

Pour milk into a bowl, drop in some Cheerios (the breakfast cereal), and notice how they clump together. This attraction is mediated by gravity and surface tension. Each cheerio makes a little depression in the milk-air interface, and the neighbouring one just falls into it.

Capillary interactions between nearby interfacial objects

Tue, 14 May 2013 00:00:00 +0000

We develop a general method to study the capillary interactions between objects of arbitrary shape which float close to each other on an interface, a regime in which multipole expansion is not useful. The force is represented as a power series in the small distance between the objects, of which the leading order is finite. For objects with size a much larger than the capillary length lc, the force scales as (a/lc)1/2 and the prefactor depends on the mean radius of curvature R at the closest points.

Skating on a film of air: Drops impacting a surface

Wed, 15 Feb 2012 00:00:00 +0000

The commonly accepted description of drops impacting on a surface typically ignores the essential role of the air that is trapped between the impacting drop and the surface. Here we describe a new imaging modality that is sensitive to the behavior right at the surface. We show that a very thin film of air, only a few tens of nanometers thick, remains trapped between the falling drop and the surface as the drop spreads.

The mechanism of a splash on a dry solid surface

Tue, 10 Jan 2012 00:00:00 +0000

From rain storms to ink jet printing, it is ubiquitous that a high-speed liquid droplet creates a splash when it impacts on a dry solid surface. Yet, the fluid mechanical mechanism causing this splash is unknown. About fifty years ago it was discovered that corona splashes are preceded by the ejection of a thin fluid sheet very near the vicinity of the contact point. Here we present a first-principles description of the mechanism for sheet formation, the initial stages of which occur before the droplet physically contacts the surface.

The branch with the furthest reach

Tue, 03 Jan 2012 00:00:00 +0000

How should a given amount of material be moulded into a cantilevered beam clamped at one end, so that it will have the furthest horizontal reach? Here, we formulate and solve this variational problem for the optimal variation of the cross-section area of a heavy cantilevered beam with a given volume V, Young’s modulus E, and density ρ, subject to gravity g. We find that the cross-sectional area should vary according a universal profile that is independent of material parameters, with both the length and maximum reach-out distance of the branch that scale as $(EV/ρg)^1/4$, with a universal self-similar shape at the tip with the area of cross-section $a∼s^3$, s being the distance from the tip, consistent with earlier observations of tree branches, but with a different local interpretation than given before.

Algorithm for microfluidic assembly line

Mon, 28 Feb 2011 00:00:00 +0000

Microfluidic technology has revolutionized the control of flows at small scales giving rise to new possibilities for assembling complex structures on the microscale. We analyze different possible algorithms for assembling arbitrary structures, and demonstrate that a sequential assembly algorithm can manufacture arbitrary 3D structures from identical constituents. We illustrate the algorithm by showing that a modified Hele-Shaw cell with 7 controlled flow rates can be designed to construct the entire English alphabet from particles that irreversibly stick to each other.

Events before droplet splashing on a solid surface

Thu, 25 Mar 2010 00:00:00 +0000

A high-velocity (≈1 m/s) impact between a liquid droplet (≈1 mm) and a solid surface produces a splash. Classical observations traced the origin of this splash to a thin sheet of fluid ejected near the impact point, though the fluid mechanical mechanism leading to the sheet is not known. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. In this paper, we theoretically and numerically study the events within the time scale of about 1 μs over which the coupled dynamics between the gas and the droplet becomes important.

A generalized theory of viscous and inviscid flutter

Wed, 07 Oct 2009 00:00:00 +0000

We present a unified theory of flutter in inviscid and viscous flows interacting with flexible structures based on the phenomenon of 1 : 1 resonance. We show this by treating four extreme cases corresponding to viscous and inviscid flows in confined and unconfined flows. To see the common mechanism clearly, we consider the limit when the frequencies of the first few elastic modes are closely clustered and small relative to the convective fluid time scale.

Precursors to splashing of a liquid droplet on a solid surface

Tue, 31 Mar 2009 00:00:00 +0000

A high velocity impact between a liquid droplet and a solid surface produces a splash. Classical work traced the origin of the splash to a thin sheet of fluid ejected near the impact point. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. We demonstrate that, neglecting intermolecular forces between the liquid and the solid, the liquid does not contact the solid, and instead spreads on a very thin air film.

Short-time dynamics of partial wetting

Wed, 11 Jun 2008 00:00:00 +0000

When a liquid drop contacts a wettable surface, the liquid spreads over the solid to minimize the total surface energy. The first moments of spreading tend to be rapid. For example, a millimeter-sized water droplet will wet an area having the same diameter as the drop within a millisecond. For perfectly wetting systems, this spreading is inertially dominated. Here we identify that even in the presence of a contact line, the initial wetting is dominated by inertia rather than viscosity.

The feasibility of generating low-frequency volcano seismicity by flow through a deformable channel

Tue, 01 Jan 2008 00:00:00 +0000

Oscillations generated by flow of magmatic or hydrothermal fluids through tabular channels in elastic rocks are a possible source of low-frequency seismicity. We assess the conditions required to generate oscillations of approximately 1 Hz via hydrodynamic flow instabilities (roll waves), flow-destabilized standing waves set up on the elastic channel walls (wall modes), and unstable normal modes ringing in an adjacent fluid reservoir (clarinet modes). Stability criteria are based on physical and dimensional arguments, and discussion of destabilized elastic modes is supplemented with laboratory experiments of gas flow through a channel in a block of gelatine, and between a rigid plate and a rubber membrane.

Bounds on double-diffusive convection

Mon, 25 Dec 2006 00:00:00 +0000

We consider double-diffusive convection between two parallel plates and compute bounds on the flux of the unstably stratified species using the background method. The bound on the heat flux for Rayleigh–Bénard convection also serves as a bound on the double-diffusive problem (with the thermal Rayleigh number equal to that of the unstably stratified component). In order to incorporate a dependence of the bound on the stably stratified component, an additional constraint must be included, like that used by Joseph (Stability of Fluid Motion, 1976, Springer) to improve the energy stability analysis of this system.

An experimental study of the coalescence between a drop and an interface in Newtonian and polymeric liquids

Tue, 12 Sep 2006 00:00:00 +0000

When a water drop falls onto an oil-water interface, the drop usually rests for some time before merging with the water underneath the interface. We report experiments on this process using water- and oil-based Newtonian liquids and polymer solutions, with an emphasis on the non-Newtonian effects. We deduce that the drop surface is immobilized by contaminants pre-existing in the fluids, and find that the rest time scales with the matrix viscosity for Newtonian fluids.

Partial coalescence between a drop and a liquid-liquid interface

Tue, 23 May 2006 00:00:00 +0000

This Letter reports experimental results for partial coalescence when a drop merges with an interface. We find an intermediate range of drop sizes in which the merger is not complete but a daughter drop is left behind. This phenomenon is governed primarily by inertia and interfacial tension, and three regimes can be further delineated depending on the roles of viscosity and gravity. Scaling relationships are developed for the drop size ratio and the coalescence time.

Dynamics of roll waves

Fri, 10 Sep 2004 00:00:00 +0000

Shallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow.

Mechanisms of liquid slip and solid surfaces

Fri, 20 Aug 2004 00:00:00 +0000

One of the oldest unresolved problems in fluid mechanics is the nature of liquid flow along solid surfaces. It is traditionally assumed that across the liquid-solid interface, liquid and solid speeds exactly match. However, recent observations document that on the molecular scale, liquids can slip relative to solids. We formulate a model in which the liquid dynamics are described by a stochastic differential-difference equation, related to the Frenkel-Kontorova equation. The model, in agreement with molecular dynamics simulations, reveals that slip occurs via two mechanisms: localized defect propagation and concurrent slip of large domains.

A simple model illustrating the role of turbulence on phytoplankton blooms

Tue, 01 Apr 2003 00:00:00 +0000

The problem of the vertical distribution of phytoplankton is considered in the presence of gravitational settling, turbulent mixing, population growth due to cell division and a constant rate of loss due to predation and natural death. Nutrients are assumed to be plentiful so that the production rate depends only on the light available for photosynthesis. The non-linear saturation of plankton growth is modeled by allowing the attenuation rate of light to be a linear function of the plankton density.