xSkill: notes, lectures and practice for CBSE and CSE

1.1 Pressure and its variation A fluid exerts pressure on every surface it touches. Pressure is a scalar: it has a value at a point, not a direction. At a depth h in a liquid of uniform density rho, the pressure is the surface pressure plus rho g h. The relation is the same for all points at the same depth in a connected liquid at rest, which is the idea behind Pascal's law and all hydraulic devices.

P = \frac{F}{A}, P = P_{0} + ρ g h, F_{b u oy an t} = ρ_{l i q u i d} V_{s u bm er g e d} g

Pressure definition, pressure at depth in a liquid at rest, and the buoyant force on a submerged body

1.2 Flow of an ideal fluid For an incompressible fluid in steady flow, the volume crossing any section per second is constant, so the speed rises where the tube narrows. Bernoulli's equation is the energy statement for ideal flow: the sum of pressure, kinetic energy per unit volume and gravitational potential energy per unit volume stays constant along a streamline. It is valid for steady, non-viscous, incompressible flow along a streamline.

A_{1} v_{1} = A_{2} v_{2}, P + \frac{1}{2} ρ v^{2} + ρ g h = constant

Equation of continuity and Bernoulli's principle for an ideal fluid

v_{e f f l ux} = 2 g h, v_{v e n t u r i} = A_{1} \frac{2 ( Δ P )}{ρ ( A _{1}^{2} - A _{2}^{2} )}

Torricelli's law for a small side opening and the Venturi meter relation

1.3 Viscous flow and terminal speed Viscosity is internal friction between layers of a moving fluid. Newton's expression for viscous force on a layer is F = eta A (dv/dx), where eta is the coefficient of viscosity, measured in Pa s or poise. A small sphere falling through a viscous liquid reaches a terminal speed when its weight equals the sum of the buoyant force and the viscous drag. For slow motion Stokes' law gives the drag as 6 pi eta r v, and the terminal speed follows from balancing the three forces.

F = η A \frac{d v}{d x}, F_{d r a g} = 6 π η r v, v_{t} = \frac{2 r ^{2} ( ρ - σ ) g}{9 η}

Newton's viscous force, Stokes' law and the terminal speed of a sphere in a viscous liquid

\frac{V}{t} = \frac{π P r ^{4}}{8 η l}, R e = \frac{ρ v d}{η}

Poiseuille's formula for laminar flow in a tube and the Reynolds number

1.4 Surface tension and capillarity The surface of a liquid behaves like a stretched sheet because surface molecules have extra energy. Surface tension T can be read as the force per unit length of a line drawn on the surface, or as the work needed to create one square metre of new surface, so T is quoted in N m^-1 or J m^-2. A curved liquid surface has an excess pressure across it: 2T/r for a drop or a cavity and 4T/r for a soap bubble, because the bubble has two surfaces. In a narrow tube, surface tension lifts a liquid to a height h that balances the weight of the column in the tube.

T = \frac{F}{l}, Δ P_{d r o p} = \frac{2 T}{r}, Δ P_{b u bb l e} = \frac{4 T}{r}, h = \frac{2 T cos θ}{ρ g r _{t u b e}}

Surface tension, excess pressure inside a drop or bubble, and capillary rise

Exam

Always check which formula matches the object: 2T/r for a drop or an air bubble inside a liquid, 4T/r for a soap bubble with two surfaces. Half the marks in surface tension numericals are lost on this one choice.