Mechanics & Fluid Dynamics

UPSC Mathematics Optional · Paper II · Lagrangian Mechanics to Fluid Flow

1. Generalised Coordinates & Lagrangian 2. Hamilton's Equations 3. Impulsive Forces & Collision 4. Fluid Kinematics 5. Equation of Continuity 6. Euler's Equation 7. Bernoulli's Equation 8. Irrotational Flow & Potential 9. Sources, Sinks & Doublets 10. Vortex Motion 11. Viscous Flow — Navier-Stokes

§1 · Generalised Coordinates & Lagrangian Mechanics

Generalised Coordinates & Degrees of Freedom

For a system with \(n\) particles and \(k\) constraints: DOF \(= 3n-k\). Generalised coordinates \(q_1,\ldots,q_s\) parameterize the configuration space.

Lagrange's Equations of Motion PYQ★★★

\[L = T - V\] (Lagrangian = kinetic − potential energy)

\[\frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i}=Q_i\]

where \(Q_i\) = generalised force for non-conservative forces. For conservative systems: \(Q_i=0\).

Cyclic / ignorable coordinate: If \(\partial L/\partial q_i=0\), then \(\partial L/\partial\dot q_i=p_i=\text{const}\) (conserved generalised momentum).

Common Kinetic Energies in Generalised Coordinates PYQ★★

Plane polar: \(T=\tfrac{1}{2}m(\dot r^2+r^2\dot\theta^2)\)
Spherical: \(T=\tfrac{1}{2}m(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2)\)
Rigid body rotation: \(T=\tfrac{1}{2}I\omega^2\) or \(T=\tfrac{1}{2}\boldsymbol\omega^T\mathbf{I}\boldsymbol\omega\)

§2 · Hamiltonian Mechanics

Hamiltonian & Hamilton's Equations PYQ★★★

Generalised momentum: \(p_i=\dfrac{\partial L}{\partial\dot q_i}\).

Hamiltonian: \(H=\sum_i p_i\dot q_i - L\). For conservative natural systems: \(H=T+V\) (total energy).

Hamilton's equations:

\[\dot q_i=\frac{\partial H}{\partial p_i},\qquad\dot p_i=-\frac{\partial H}{\partial q_i}\]

If \(H\) has no explicit time dependence: \(H=E=\text{const}\) (energy conservation).

Poisson Brackets

\(\{f,g\}=\sum_i\left(\dfrac{\partial f}{\partial q_i}\dfrac{\partial g}{\partial p_i}-\dfrac{\partial f}{\partial p_i}\dfrac{\partial g}{\partial q_i}\right)\)

\(\dot f=\{f,H\}+\dfrac{\partial f}{\partial t}\). Conservation: \(\dot f=0\) iff \(\{f,H\}=0\) (and \(f\) not explicit in \(t\)).

Fundamental: \(\{q_i,p_j\}=\delta_{ij}\). \(\{q_i,q_j\}=0\). \(\{p_i,p_j\}=0\).

§3 · Impulsive Forces & Collision

Collision — Newton's Law of Restitution PYQ★★

Coefficient of restitution \(e\in[0,1]\): \(\dfrac{\text{velocity of separation}}{\text{velocity of approach}}=e\).

Conservation of linear momentum: \(m_1u_1+m_2u_2=m_1v_1+m_2v_2\).

Loss of KE \(=\dfrac{m_1m_2(1-e^2)}{2(m_1+m_2)}(u_1-u_2)^2\). For \(e=1\): elastic (no loss). \(e=0\): perfectly inelastic.

§4 · Fluid Kinematics

Eulerian & Lagrangian Descriptions

Eulerian: velocity field \(\mathbf{v}(x,y,z,t)\) at fixed points.

Lagrangian: track individual fluid particles.

Material (substantial) derivative: \(\dfrac{D}{Dt}=\dfrac{\partial}{\partial t}+(\mathbf{v}\cdot\nabla)\) = rate of change following a fluid particle.

Streamlines, Pathlines & Streaklines

Streamline: curve tangent to velocity at each point at a given time. Equation: \(\dfrac{dx}{u}=\dfrac{dy}{v}=\dfrac{dz}{w}\).
Pathline: trajectory of a single fluid particle.
Streakline: locus of particles that have passed through a given point.
For steady flow: all three coincide.

§5 · Equation of Continuity

Continuity Equation PYQ★★★

General (compressible): \(\dfrac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0\)

Incompressible (\(\rho=\text{const}\)): \(\nabla\cdot\mathbf{v}=0\), i.e. \(\dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}+\dfrac{\partial w}{\partial z}=0\).

2D incompressible: stream function \(\psi\) exists: \(u=\dfrac{\partial\psi}{\partial y}\), \(v=-\dfrac{\partial\psi}{\partial x}\).

§6 · Euler's Equation of Motion

Euler's Equation (Inviscid Flow) PYQ★★★

\[\rho\frac{D\mathbf{v}}{Dt}=\rho\left[\frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right]=-\nabla p+\rho\mathbf{F}\]

where \(\mathbf{F}\) = body force per unit mass (e.g. gravity \(\mathbf{g}\)).

§7 · Bernoulli's Equation

Bernoulli's Equation PYQ★★★

For steady, inviscid, incompressible flow along a streamline:

\[p+\frac{1}{2}\rho v^2+\rho gz=\text{constant along streamline}\]

For irrotational flow: constant throughout entire fluid.

Unsteady form: \(\rho\dfrac{\partial\phi}{\partial t}+\dfrac{1}{2}\rho v^2+p+\rho gz=\text{const}\) where \(\phi\) = velocity potential.

Applications of Bernoulli PYQ★★

Venturi meter: \(Q=A_1A_2\sqrt{\dfrac{2(p_1-p_2)}{\rho(A_1^2-A_2^2)}}\)
Pitot tube: \(v=\sqrt{2(p_{\text{stag}}-p)/\rho}\)
Torricelli: speed of efflux \(v=\sqrt{2gh}\)

§8 · Irrotational Flow & Velocity Potential

Velocity Potential \(\phi\) PYQ★★★

For irrotational flow (\(\nabla\times\mathbf{v}=\mathbf{0}\)): \(\mathbf{v}=\nabla\phi\).

Incompressible + irrotational: \(\nabla^2\phi=0\) (Laplace equation).

Complex potential (2D): \(w=\phi+i\psi\) where \(\phi\)=velocity potential, \(\psi\)=stream function. \(w\) is an analytic function of \(z=x+iy\).

Complex velocity: \(\dfrac{dw}{dz}=u-iv\).

Cauchy-Riemann in Fluid Context

\(u=\phi_x=\psi_y\), \(v=\phi_y=-\psi_x\). Streamlines \(\psi=\text{const}\), equipotentials \(\phi=\text{const}\) are orthogonal families.

§9 · Sources, Sinks & Doublets

Elementary 2D Flows PYQ★★★

Uniform flow (speed \(U\), angle \(\alpha\)): \(w=Ue^{-i\alpha}z\). \(\phi=Ux\), \(\psi=Uy\) (for \(\alpha=0\)).
Source (strength \(m\)) at origin: \(w=\dfrac{m}{2\pi}\ln z\). \(\phi=\dfrac{m}{2\pi}\ln r\), \(\psi=\dfrac{m}{2\pi}\theta\).
Sink: Source with \(-m\).
Vortex (circulation \(\kappa\)) at origin: \(w=-\dfrac{i\kappa}{2\pi}\ln z\). \(\phi=\dfrac{\kappa\theta}{2\pi}\), \(\psi=-\dfrac{\kappa}{2\pi}\ln r\).
Doublet (strength \(\mu\)) at origin: \(w=-\dfrac{\mu}{2\pi z}\) (source+sink combined in limit). \(\phi=-\dfrac{\mu\cos\theta}{2\pi r}\).

Flow Past a Cylinder PYQ★★★

Uniform flow \(U\) past cylinder radius \(a\): \(w=U\left(z+\dfrac{a^2}{z}\right)\).

With circulation \(\kappa\): \(w=U\left(z+\dfrac{a^2}{z}\right)-\dfrac{i\kappa}{2\pi}\ln z\).

Blasius theorem: Force on body \(= \dfrac{1}{2}i\rho\oint\left(\dfrac{dw}{dz}\right)^2dz\).

Kutta-Joukowski: Lift \(L=\rho U\kappa\) (lift per unit span = density × velocity × circulation).

§10 · Vortex Motion

Vorticity & Kelvin's Theorem PYQ★★

Vorticity: \(\boldsymbol\omega=\nabla\times\mathbf{v}\). Irrotational iff \(\boldsymbol\omega=0\).

Kelvin's Circulation Theorem: In inviscid, barotropic flow with conservative body forces, the circulation around any material loop is constant: \(\dfrac{D\Gamma}{Dt}=0\) where \(\Gamma=\oint_C\mathbf{v}\cdot d\mathbf{r}\).

Helmholtz's theorems: Vortex lines move with the fluid; circulation of a vortex tube is constant.

§11 · Viscous Flow — Navier-Stokes

Navier-Stokes Equations PYQ★★

\[\rho\frac{D\mathbf{v}}{Dt}=-\nabla p+\mu\nabla^2\mathbf{v}+\rho\mathbf{F}\]

\(\mu\) = dynamic viscosity. \(\nu=\mu/\rho\) = kinematic viscosity.

Reynolds number: \(\text{Re}=\dfrac{\rho UL}{\mu}=\dfrac{UL}{\nu}\). Low Re: viscous (laminar); High Re: inertia dominant (turbulent).

Plane Poiseuille & Couette Flow PYQ★★★

Poiseuille flow (flow between parallel plates driven by pressure gradient):

\(u(y)=\dfrac{1}{2\mu}\dfrac{dp}{dx}(y^2-h^2)\). Parabolic profile. Volume flux \(Q=\dfrac{2h^3}{3\mu}\left|\dfrac{dp}{dx}\right|\).

Circular Poiseuille (Hagen-Poiseuille): \(u(r)=\dfrac{1}{4\mu}\left(-\dfrac{dp}{dz}\right)(R^2-r^2)\). \(Q=\dfrac{\pi R^4}{8\mu}\left(-\dfrac{dp}{dz}\right)\).

Couette flow (between moving plates, no pressure gradient): \(u(y)=U\dfrac{y}{h}\). Linear profile.

Stokes' Drag on a Sphere PYQ★★

Drag force on sphere of radius \(a\) moving at velocity \(U\) in viscous fluid (low Re):

\(F=6\pi\mu aU\) (Stokes' drag formula).

Terminal velocity: \(F_{\text{drag}}+F_{\text{buoy}}=W\) → \(U_{\text{term}}=\dfrac{2a^2(\rho_s-\rho_f)g}{9\mu}\).