Unit 3: Movement of Solids Within Fluids PDF

Unit 3 MOVEMENT OF SOLIDS WITHIN FLUIDS CONTENTS INTRODUCTION. LOW AND HIGH SPEED REGIMES. DRAG FORCES. STOKES’ LAW. • Low Speed. • High Speed. LIMIT SPEED. • Sphere moving at low speed. • Speed moving at high speed. • Example: Bubbles rising in a drink. APLICATIONS. • Viscosity measurements....

Unit 3 MOVEMENT OF SOLIDS WITHIN FLUIDS CONTENTS INTRODUCTION. LOW AND HIGH SPEED REGIMES. DRAG FORCES. STOKES’ LAW. • Low Speed. • High Speed. LIMIT SPEED. • Sphere moving at low speed. • Speed moving at high speed. • Example: Bubbles rising in a drink. APLICATIONS. • Viscosity measurements. Stokes’ Viscometer. • Sedimentation and Centrifugation. INTRODUCTION According to Newton's second law, if a force 𝐹 is applied to a body of mass 𝑚, the body experiences an acceleration     𝑑𝑣 𝐹=𝑚 𝑎=𝑚 𝑑𝑡 If 𝐹 = 0 ⇒ 𝑣 = 𝑐𝑜𝑛𝑠𝑡 → Uniform movement If 𝐹 = 𝑐𝑜𝑛𝑠𝑡 ⇒ 𝑎 = 𝑐𝑜𝑛𝑠𝑡 ⇒ 𝑣 = 𝑣(𝑡) → Uniformly accelerated movement If 𝐹 = 𝐹 𝑡 ⇒ 𝑎 = 𝑎(𝑡) 𝑦 𝑣 = 𝑣 (𝑡) → Non-uniform and time-dependant movement When 𝐹 → 0 ⇒ 𝑎 = 0 y 𝑣 = 𝑐𝑜𝑛𝑠𝑡 → The movement tends to be uniform and is characterized by a constant and uniform speed called the LIMIT SPEED. KINEMATICS REMINDER Kind of Movement Definition/Equations URM 𝑣 = 𝑣 = 𝑐𝑜𝑛𝑠𝑡 𝑒 = 𝑒 + 𝑣𝑡 UARM 𝑣 = 𝑣 + 𝑎𝑡 𝑒 = 𝑒 + 𝑣 𝑡 + 𝑎𝑡 UCM UACM 𝜛 = 𝜛 = 𝑐𝑜𝑛𝑠𝑡 𝛼 = 𝛼 = 𝑐𝑜𝑛𝑠𝑡 𝜑 = 𝜑 + 𝜛𝑡 𝜛 = 𝜛 + 𝛼𝑡 𝜑 =𝜑 +𝜛 + 𝑎 = 𝑟𝛼 and 𝑎 =𝜛 𝑟 LOW AND HIGH SPEED REGIMES. ℜ  To distinguish between low and high speed, the dimensionless parameter that characterizes the movement regime of a solid body inside a fluid is used: the Reynolds number (ℜ) 𝜌 𝑣𝐿 ℜ= 𝜂 o o o o L, characteristic length of the moving body ρ0, density of the fluid η, viscosity of the fluid v, speed of the moving body  For the case of a spherical particle, 𝐿 is the radius of the particle. 𝜌 𝑣𝑅 ℜ= 𝜂  If ℜ < 1 → LOW SPEED REGIME (LSR)  If 1 < ℜ < 100 → HIGH SPEED REGIME (HSR)  Si ℜ > 100 → VERY HIGH SPEED REGIME (VHSR) DRAG FORCES. STOKES’ LAW.  Consider a population of particles subjected to the action of a force field.  When moving inside a fluid, the particle will experience a frictional force 𝐹 . This force is called the drag force.  The drag force is characterized by: o It is parallel to the direction of the particle's velocity and opposite to it. o Opposes the acting field of forces. o At low speeds (ℜ < 1) it is due to the viscosity of the fluid, 𝐹 ∝ 𝑣. o At high speeds (ℜ > 1) it is due to the acceleration that the body communicates to the fluid that is around it and that is dragged by it, 𝐹 ∝ 𝑣 . o At higher speeds (ℜ > 100), 𝐹 ∝ 𝑣 with ** being number higher than 2 a DRAG FORCES. STOKES’ LAW. LOW SPEEDS. STOKES’ LAW.  The drag force is due to the viscosity of the fluid. As the particle moves, the fluid layer adhering to it experiences the viscous force with respect to the immediately adjacent fluid layer.  Drag force is proportional to speed 𝐹 = 𝜙𝑣 = 𝑘𝐿𝜂𝑣.  𝜙 is the coefficient of molecular friction that depends on the shape of the particle and the characteristics of the fluid • 𝑘 is the shape factor of the particle 𝜙 = 𝑘𝐿𝜂 • 𝐿 is a characteristic length of the particle • 𝜂 is the viscosity of the fluid DIMENSIONAL ANALYSIS For a spherical particle  𝑘 = 6𝜋 and 𝐿 = 𝑅  The friction coefficient is: 𝜙 = 6𝜋𝑅𝜂  Drag force for an spherical particle 𝐹 = 6𝜋𝑅𝜂𝑣 STOKES’ LAW DRAG FORCES. STOKES’ LAW. HIGH SPEEDS, ℜ > 𝟏  The drag force is due to the acceleration of the fluid moving around the particle and not due to viscous effects.  The drag force is proportional to 𝑣 : 𝐹 = 𝜌 𝑣 • 𝐶 is the drag coefficient. Must be determined experimentally. • 𝑆 is the cross section of the body in the direction of motion ↑ S F • 𝜌 is the density of the fluid DIMENSIONAL ANALYSIS For a spherical particle =  𝑆 = 𝜋𝑟 is the cross-sectional area of ​the moving sphere  𝜌𝑣 ⁄2 is the kinetic energy per unit volume of the fluid with speed 𝑣  Viscosity does not appear 𝐹 = 𝐶 𝜋𝑅 𝜌 𝑣 LIMIT SPEED Si está en , reposo no hay fuerza de arrastre  When a body is subjected to the action of a variable force that tends to 0, the movement of the body tends to reach a uniform speed called the limit speed. If ∑ 𝐹 (𝑡) → 0 ⇒ 𝑎 = 𝑎 𝑡 → 0 ⇒ 𝑣 = 𝑐𝑜𝑛𝑠𝑡 → 𝑙𝑖𝑚𝑖𝑡 𝑠𝑝𝑒𝑒𝑑 • ∑𝐹 = 𝐹 − 𝐹 • When ∑ 𝐹 = 0 ⟹ 𝐹 − 𝐹 = 0 ⇒ 𝐹 = 𝐹 LOW SPEEDS 𝐹 = 𝐹 = 𝜙𝑣 ⟹ 𝐹 𝑣 = 𝜙 HIGH SPEEDS 𝐹 = 𝜌 𝑣 ⇒ 𝑣 = LIMITE SPEED  We are going to study the limit speed of a sphere that moves in a fluid under the effects of gravity. If ∑ 𝐹 (𝑡) → 0 ⇒ 𝑎 = 𝑎 𝑡 → 0 ⇒ 𝑣 = 𝑐𝑡𝑒 → 𝑙𝑖𝑚𝑖𝑡 𝑠𝑝𝑒𝑒𝑑  The force driving the motion is 𝐹 = 𝑃 − 𝐸  P is the weight of the sphere and E its buoyancy; expressing both forces as a function of the densities: 𝑃 = 𝑚𝑔 = 𝜌s 𝑉𝑔 and 𝐸 = 𝜌 𝑉𝑔  ∑𝐹 = 𝐹 − 𝐹 = 𝑃 − 𝐸 − 𝐹  When ∑ 𝐹 = 0 ⟹ 𝑃 − 𝐸 − 𝐹 = 0 ⇒ 𝐹 = 𝑃 − 𝐸 = 𝜌 − 𝜌 𝑉𝑔 P S LOW SPEEDS 𝐹 = 𝐹 = 𝜙𝑣 ⟹ >0 𝑣 = = ⇒ 𝑣 = Lo P ( HIGH SPEED 𝐹 = 𝜌𝑣 ⇒ 𝑣 = = ⇒ 𝑣 = APPLICATIONS VISCOSITY MEASUREMENT (STOKES VISCOMETER) 𝜂 = ( ) We use a small ball of lead (ρe =11.35 g/cm3) with a radius of 1.85 mm and drop it into an oil column with a density of ρf =11.35 g/cm3. The time it takes for the sphere to travel a distance of 50 cm is 4.57 s. Calculate the viscosity of the fluid. Sol: 7,13 poises APLICATIONS SEDIMENTATION  The speed limit is reached in a very short distance.  If the particles are spherical: 𝑣 =  If all the particles have the same radius, they will sediment with the same speed.  The larger the particle radius, the higher the sedimentation speed. CENTRIFUGATION  The particles are subjected to a centrifugal inertial force field. So the acceleration will be 𝑎 = 𝜛 𝑅  If the particles are spherical: 𝑣 = • r is the distance from the center of the centrifuge to the position  If 𝑎 ≫ 𝑔 very small radius particles can sediment. where the particle is located.  It is used to measure molecular masses. • 𝜛 is the angular rotation speed of the centrifuge.