Viscosity & Stokes' Law Viscosity & Stokes' Law Think of fluid flow like a fixed budget of energy. A fluid particle has two main ways to "spend" its energy: moving fast (kinetic energy) or pushing against things (pressure energy). Because the total energy along a smooth path is limited, if a fluid speeds up it "spends" more on motion and has less to spare as static pressure. That trade-off is the heart of ideal-flow thinking. Real fluids, however, also spend part of their budget internally: neighboring layers rub against each other. This internal rubbing is viscosity. It converts ordered motion into heat and gradually eats away the mechanical energy that Bernoulli’s ideal world would preserve. You feel viscosity in everyday life when honey pours slowly, motor oil cushions engine parts, or blood squeezes through very narrow capillaries. At small speeds in smooth, orderly (laminar) flow, we can model viscous resistance precisely and even predict the maximum constant speed a falling particle will settle into. At higher speeds or with sharp changes in geometry, motion becomes erratic (turbulent), energy losses surge, and simple formulas fail. This lesson builds an intuition-first route to the key laws: Newton’s law of viscosity for shear between layers, Stokes’ law for the drag on a tiny sphere, terminal velocity from a clean force balance, and Reynolds number as the map that tells us which formula belongs to which regime. Along the way we will keep units consistent, call out boundaries where each law applies, and link back to the ideal tools (continuity and Bernoulli) so you know when they still offer quick estimates. remember Crowded hallway analogy: In a wide hallway, students amble and bump into walls (higher lateral "pressure"). At a narrow door they sprint in line (high speed, fewer sideways bumps), mirroring the speed–pressure trade-off in ideal flow. Real fluids also waste some of that energy to internal rubbing—viscosity. Internal friction in a fluid that resists relative motion between adjacent layers; felt as "thickness" of a liquid. Viscosity Coefficient of viscosity ( ) Proportionality constant in Newton’s law of viscosity; SI unit Pa ,s or kg ,m -1 ,s -1 . Laminar (streamline) flow Orderly flow where each fluid particle follows a smooth path and layers slide without mixing; velocity at a point is steady in time. Chaotic flow with eddies and mixing; velocity at a point fluctuates randomly with time and energy losses are large. Turbulent flow Velocity gradient Rate of change of layer speed with perpendicular distance, d v/ d x . Viscosity per unit density: = / ; SI unit m 2/s . Kinematic viscosity ( ) Reynolds number ( R e ) Dimensionless index of flow regime: R e = v D for speed v in a conduit of diameter D . Viscous drag on a small sphere in laminar flow: F d = 6 r v . Stokes' law Final constant speed of a body falling through a fluid when net force becomes zero due to balance of weight, buoyancy, and viscous drag. Terminal velocity Very low R e 1 flow where inertia is negligible and viscous forces dominate; Stokes’ law strictly applies here. Creeping flow Newton’s law of viscosity Shear force on area A is proportional to the velocity gradient; minus sign shows opposition to motion. It calculates the resistive force exerted by a fluid due to the shear rate and the fluid's inherent viscosity. tip Sign convention: the viscous force on a faster-moving layer acts opposite to its motion, so F carries a negative sign relative to the velocity gradient. Units and dimensions matter on NEET. For : SI units are Pa ,s = kg ,m -1 ,s -1 ; CGS unit is poise (P) where 1 , Pa ,s = 10 , P . Dimensions: [M ,L -1 ,T -1 ] . Kinematic viscosity has m 2/s . Always convert radii to meters and speeds to m/s before substituting. Fluid (at 20–25 °C) in Pa ,s Notes Air 1.8 10 -5 Gas; is large because is tiny. Water 1.0 10 -3 Low-viscosity benchmark. Blood 3.0 10 -3 to 4.0 10 -3 Non-Newtonian tendencies in small vessels. Glycerin 1.5 Very viscous; great for Stokes experiments. Honey 2 - 10 Strongly temperature dependent. Microscopic picture: In liquids, closely packed molecules exchange momentum when layers slide, producing shear stress proportional to the velocity gradient. In gases, momentum transport by random molecular motion creates an effective viscosity even though density is low. Many everyday liquids behave nearly Newtonian over a wide range of shear rates, which is why F d v/ d x works well. Temperature effect: For liquids, viscosity decreases sharply with temperature (warmer honey flows easily). For gases, viscosity increases mildly with temperature. Unless data are given, take standard 25 C values on NEET. Velocity m/s A falling sphere quickly accelerates and asymptotically approaches terminal velocity v t as drag grows with v. Release: v = 0 Long time: v → v t v t Time dependent control Approach to terminal speed as viscous drag rises linearly with velocity in Stokes regime. vt 2D PLOT Approach to terminal velocity v = vt (1 - exp(-t/tau)) vt Terminal velocity tau Time constant Valid for a rigid sphere of radius r in laminar flow with very small Reynolds number. Stokes’ drag on a sphere This formula is valid for calculating the viscous drag force on a small, spherical object moving slowly through a fluid under laminar flow c This drag force applies only when the sphere is perfectly shaped, moving at low velocity, and far from any walls or boundaries. tip Where Stokes’ law applies: R e 1 , perfectly spherical particle, no slip at the surface, no nearby walls, and steady uniform flow. It fails for rough shapes, large R e , or when the sphere is very close to a boundary. v t = 2 r 2 ( s - f ) g 9 Weight downward Buoyant force upward by Archimedes Viscous drag upward at speed v Net force zero Equate forces Solve for terminal speed Terminal velocity of a sphere in a viscous liquid Small sphere; laminar (creeping) flow so Stokes’ law holds Constant densities s (sphere) and f (fluid), uniform g No wall effects, steady vertical fall Higher radius and density difference increase v t ; higher viscosity reduces it. Terminal velocity (sphere) Calculates the constant terminal velocity reached by a sphere when the drag force balances the net gravitational force. neet-alert Trap: Use radius, not diameter. Insert the density difference ( s - f ) . If the object is lighter than the fluid (e.g., an air bubble in water), reverse the sign and motion direction. Scaling from the formula: v t r 2 , v t ( s - f ) , and v t 1/ . Doubling the radius makes terminal speed four times; halving viscosity doubles v t . These quick ratios save time on exam day. Where this model is useful Sedimentation of cells in diagnostic centrifugation (small R e ). Design of oil-drop and falling-sphere viscometers. Predicting raindrop settling speeds before coalescence makes them larger (and then R e grows). How do we know if laminar models are valid? Use Reynolds number. For a pipe of diameter D carrying an incompressible fluid of density at mean speed v , R e = v D . Small R e means viscous forces dominate and flow is orderly. Large R e means inertia dominates and eddies appear. Rough guide: laminar if R e 1000 , turbulent if R e 2000 , transitional in between (for smooth pipes). Reynolds number Helps determine if viscous forces dominate, resulting in orderly laminar flow, or if inertia dominates. Thresholds are not exact constants; they depend on entry conditions and surface roughness. For neat calculations at this level, use: laminar R e < 1000 , turbulent R e > 2000 . remember Feature Laminar Turbulent Velocity at a point Steady Fluctuating Energy loss per length Small Large Mixing Minimal Strong Applicability of Bernoulli Closer (with modest corrections) Poor (large viscous head loss) Example Glycerin in a capillary Fast water in a rough pipe Velocity profile in a long, narrow tube under steady pressure difference is parabolic in laminar flow, with zero velocity at the wall (no-slip) and maximum at the center. The volume flow rate then follows Poiseuille’s law. Valid for incompressible Newtonian liquids, long straight tubes, no-slip walls, and R e below critical. Poiseuille’s law (laminar tube flow) This equation determines the constant volume flow rate of a fluid moving in a steady, laminar regime through a cylindrical pipe. tip Poiseuille pitfalls: If temperature changes along the tube, is not constant. Entrance/exit effects and bends break the assumptions. The R 4 sensitivity is a classic exam lever. easy Use Stokes’ law in creeping flow. Viscous drag F d r = 0.50 , mm = 5.0 10 -4 , m = 1.5 , Pa ,s v = 5.0 10 -3 , m/s A steel bead of radius 0.50 , mm moves through glycerin ( = 1.5 , Pa ,s ) at v = 5.0 10 -3 , m/s . Find the viscous drag. medium m/s Find the terminal velocity of a steel ball ( s = 7.8 10 3 , kg/m 3 , radius 1.0 , mm ) falling in glycerin ( f = 1.26 10 3 , kg/m 3 , = 1.49 , Pa ,s ). Take g = 9.8 , m/s 2 . r = 1.0 , mm = 1.0 10 -3 , m s - f = 6.54 10 3 , kg/m 3 = 1.49 , Pa ,s g = 9.8 , m/s 2 Terminal velocity v t Use v t = 2 r 2 ( s - f ) g 9 . Viscosity Rearrange v t = 2 r 2 ( s - f ) g 9 to = 2 r 2 ( s - f ) g 9 v t . r = 0.75 , mm = 7.5 10 -4 , m s - f = 1.24 10 3 , kg/m 3 v t = 1.0 10 -3 , m/s g = 9.8 , m/s 2 A glass sphere of radius 0.75 , mm and density 2.5 10 3 , kg/m 3 falls slowly in glycerin ( f = 1.26 10 3 , kg/m 3 ). If the measured terminal speed is 1.0 10 -3 , m/s , estimate the viscosity of glycerin. Pa·s hard D = 0.020 , m v = 0.50 , m/s = 1000 , kg/m 3 = 1.0 10 -3 , Pa ,s Water ( =1000 , kg/m 3 , =1.0 10 -3 , Pa ,s ) flows in a smooth pipe of diameter 2.0 , cm at average speed 0.50 , m/s . Classify the flow. Use R e = v D . Reynolds number and regime medium Venturi meter glass pipe with three vertical manometer tubes showing pressure drop at the narrow throat. Venturi tube with manometers: higher levels at wide sections (higher pressure), lowest at the throat where streamlines pack and speed is highest. Ideal-flow tools still help your intuition: continuity predicts speed-up in the throat, and Bernoulli links that to lower static pressure. Real pipes also lose head to viscosity, so actual pressure drop is larger than ideal. This is why long, narrow catheters need higher driving pressures to maintain flow. Airflow over an aerofoil: faster stream on the curved upper surface gives lower pressure, producing lift; viscosity shapes the boundary layer and can trigger turbulence if R e is high. Schematic of upper fast flow (low pressure) and lower slow flow (high pressure) around a wing. Near solid walls a thin boundary layer forms where viscosity is crucial: speed rises from zero at the surface (no slip) to the outer-flow value. If adverse pressure gradients or high R e disturb it, the layer can separate and turbulence appears—destroying the neat Bernoulli picture. Close-up of a nozzle with high-speed water jet and arrows indicating kinetic energy rise and pressure fall. Garden nozzle: narrowing the exit speeds up the jet (continuity) and lowers static pressure (Bernoulli), while internal viscosity causes extra pressure drop along the hose. Along a streamline in ideal, inviscid, incompressible flow: P + 1 2 v 2 + g h = constant . In real flows, add viscous head loss. For incompressible steady flow, volume rate is constant: A 1 v 1 = A 2 v 2 . Narrower section → higher speed. A 1 v 1 = A 2 v 2 Steady state; no accumulation inside the control volume Incompressible fluid so density is constant Single inlet and outlet streamtubes Equation of continuity for incompressible steady flow Viscosity resists motion. For the same pressure difference, a higher gives a smaller flow rate (Poiseuille: Q 1/ ). More viscous liquid must flow faster if we push harder; viscosity is like "extra weight." Terminal velocity does not depend on the fluid—only on the object. It depends directly on s - f and inversely on . Denser fluid or fluid density closer to the object both reduce v t . Definition uses a characteristic length, conventionally the diameter D in pipes. Using radius halves R e and can flip your regime classification. Reynolds number uses radius; using diameter or radius is the same. Balance picture to recall 2 r 2 ( s - f ) g 9 . At terminal speed: W = B + D → Weight equals Buoyancy plus Drag. neet-alert Units trap: 1 , Pa ,s = 10 , P (poise). If is given in cP (centipoise), divide by 1000 to convert to Pa ,s . Boundary checks for Stokes’ law As r 0 , drag F d 0 and v t 0 for fixed densities. As 0 , v t (unphysical; the law is not meant for inviscid flow). As s f , v t 0 ; particle becomes neutrally buoyant and hardly settles. For R e 1 , drag is not linear in v ; quadratic drag models are needed. Always convert mm to m and cP to Pa ,s . Keep g at 9.8 , m/s 2 unless a rounded value is specified. Round final numerical answers to two significant figures for NEET. tip neet-alert Symbol confusion: In this lesson we use s for sphere density and f for fluid density. Avoid using for density since is common for surface tension in other topics. Link to biology: Blood is a non-Newtonian fluid in narrow vessels; apparent viscosity increases as diameter shrinks (Fåhræus–Lindqvist effects). NEET numericals, however, treat blood as Newtonian unless stated otherwise. Use given and check regimes with R e . Dimension check practice: Stokes’ drag [F d] = [ ][L][v] = (M L -1 T -1 )(L)(L T -1 ) = M L T -2 , which is the unit of force (newton). Quick dimension checks catch many algebra slips. Energy view with losses: In long pipe runs, Bernoulli’s sum drops along the direction of flow because viscous shear converts mechanical energy into heat. Engineers add a viscous head-loss term; for our scope, recognize that larger R e and roughness increase losses. Use carefully: it ignores viscosity, compressibility, and pumps/turbines between sections. Bernoulli along a streamline (ideal) Volume flow rate Q = A v is the same at any cross-section. Continuity (incompressible) Worked-ratio trick: If a glass bead’s radius triples while everything else stays the same, v t rises by 3 2 =9 times. For an exam shortcut, write proportionalities before touching a calculator. Wall effects: If the sphere falls in a narrow tube, the drag is higher than 6 r v due to disturbed streamlines near the wall, reducing the observed v t . NEET problems usually neglect this unless tube dimensions are provided to discuss corrections. Non-spherical particles: Stokes’ law is specific to spheres. Irregular grains or disks need drag coefficients measured experimentally; on the exam, the word "sphere" is a cue that Stokes applies. Quick Reynolds estimate in a pipe: R e 1000 ,v ,D/ mPa ,s if you enter v in m/s , D in m , and viscosity in mPa ,s . tip Experimental measurement of viscosity: In a falling-sphere viscometer, measure v t for a known r , s , and f , then compute . Repeat with different radii to verify the r 2 dependence and ensure R e stays small. When R e becomes moderate ( 100 to 10 5 depending on geometry), drag becomes roughly proportional to v 2 . Then the approach to terminal speed is much quicker and the terminal value scales as r rather than r 2 . Such regimes lie beyond Stokes’ law and are not tested with detailed numerics in NEET. Viscosity Internal friction in fluids that resists shear. Coefficient of viscosity Proportionality in Newton’s law; unit Pa ,s . Kinematic viscosity = / with unit m 2/s . Stokes’ law F d = 6 r v for a small sphere in laminar flow. Terminal velocity Final constant speed when net force is zero. Reynolds number R e = v D/ ; classifies flow regime. Poiseuille’s law Laminar tube-flow rate Q = R 4 P 8 L . Recap: Key terms