PHY 115 Fluid Flow Lecture Notes PDF

1 PHY 115 LECTURE NOTE (S. Olatunji) (CONTENT: Young Modulus, Fluid Flow, Turbulence, Streamline) FLUID FLOW Fluid refers to liquid and gas. Hydrostatics is the study of fluid at rest and hydrodynamics is the study of fluid in motion. In studying fluid flow, parameters such as velocity, p...

1 PHY 115 LECTURE NOTE (S. Olatunji) (CONTENT: Young Modulus, Fluid Flow, Turbulence, Streamline) FLUID FLOW Fluid refers to liquid and gas. Hydrostatics is the study of fluid at rest and hydrodynamics is the study of fluid in motion. In studying fluid flow, parameters such as velocity, pressure, density and temperature are considered. When parameters such as electric current and magnetism are studied in addition, such endeavour is termed magnetohydronomics. If we assume that density and pressure determine the temperature in a magnetohydronomics fluid then, PV = nRT and density ρ = nM/V; where M is molar mass. An incompressible fluid is one with constant density. Some characteristics of fluid flow include: steady/laminar flow, turbulent/non steady flow, viscous flow, non viscous flow, rotational and non rotational flows. LAMINAR/STEADY AND TURBULENT FLOWS Consider a moving liquid in a stationary pipe C Central axis B Liquid flow in A out Pipe Let the liquid be moving in layers A, B and C. Liquid in layer A moves faster than B and B is faster than C. This is due to high frictional force effects at the pipe’s edges (caused by adhesion between the particles of the liquid and pipe), which is stationary around the moving liquid layers. Downloaded from Samporhelp.000webhostapp.com  2 The flow of liquid in which the layers are distinct and the layers are not crossing such that there is frictional force within the layers is called laminar flow. In a laminar flow, velocity of particles at a given equidistance from the central axis of motion is constant. Further, if pressure, density and velocity of a moving fluid are independent of time then flow is laminar /steady. This is achievable at low flow speeds. If the layers of a flowing fluid intermix, it is a turbulent flow. In this case, velocity could be a complicated function of time e.g. Tides, water-falls etc. VISCOUS AND NON VISCOUS FLOWS Viscosity in fluid is analogous to friction in solid motion. A flow of fluid in which no energy is dissipated through viscous forces is called a non viscous flow; otherwise, it is called a viscous flow. ROTATIONAL AND NON ROTATIONAL FLOWS If an element of the fluid does not rotate about and axis through the centre of mass of the element, the flow is said to be non rotational, otherwise, it is rotational. Rotating element Non-rotational element Flow direction Flow direction Rotating element along the fluid flow direction Rotational flow Non-rotational flow Downloaded from Samporhelp.000webhostapp.com  3 STREAMLINES IN A MOVING FLUID Streamline is a curve whose tangent always lies along the direction of motion/velocity of the fluid at that point. Streamlined fluid Turbulent flow Line of flow is the path followed by an element of a moving fluid. In the steady/laminar flow, the streamlines coincide with the line of flow. No two streamlines can cross each other, this is because if they did, and oncoming fluid particle could go either one way or the other, and hence the flow could not be steady. VOLUME FLOW RATE V1Δt V2Δt P Flow direction Q y x a2 a1 Suppose a fluid flows in at x, and moves out at y. Let a1 and a2 be the cross-sectional areas of the pipe at x and y. Let V1 and V2 be the speeds within sections P and Q after the passage of a very small time Δt. Volume across a1 within time Δt = a1V1 Δt, Downloaded from Samporhelp.000webhostapp.com  4 and volume a2 within the same = a2V2 Δt If ρ1 and ρ2 are the densities at P and Q then the mass of fluid crossing a1 (i.e. Δm1) = ρ1a1v1Δt This is equal to that around a2 a1 (i.e. Δm2) = ρ2a2v2Δt. By conservation of mass rule Δm1=Δm2 That is ρ1a1v1Δt = ρ2a2v2Δt =……………. In general, ρav = constant----- (1) flow continuity equation. For an incompressible fluid, density is constant thus (1) becomes av = constant = R i.e., av = R -------- (2) ie, R = volume flow rate (ie. volume flux) Equation (2) implies that in an incompressible fluid flow the speed of flow is inversely proportional to the cross sectional area. v = R (1/a) -------- (3) POISEUILLE’S EQUATION Consider a stationary liquid in a moving pipe. The velocity will be much, near the edges of the pipe and decrease towards the centre as the liquid drifts along the pipe’s direction. This is a reverse of a moving liquid in a stationary pipe. Central axis r Stationary Liquid v dr Moving pipe Downloaded from Samporhelp.000webhostapp.com  5 At a small distance dr away from the central axis, there is a small change in the velocity dv. Thus, velocity gradient = dv/dr Viscosity is found to be proportional to the velocity gradient and cross-sectional area of the pipe ie ∝ ie = ɳ ∙ , where ɳ = coefficient of viscosity / ie, ɳ= = / its SI unit = Pas Also, for fluid flowing in a pipe, volume per second (flow rate) Ѵ depends on radius of the pipe; pressure gradient (ΔP/L) and on ɳ i.e. Ѵ = = ɳr -------- (4) By dimension method, = ɳ r ( ) -------- (5) If k (constant of the proportionality) = π/8 and solving for the values of x, y and z, we have ∆ Ѵ = Poiseuille’s equation ɳ Exercise1: Prove (5) from (4) above. Exercise 2: A 1km length pipe is to convey oil with coefficient of viscosity 0.3 Pas up to a tank. Calculate the pump pressure needed to maintain a flow rate of 0.05m3 /s if the diameter of the pipe is 15 cm. Downloaded from Samporhelp.000webhostapp.com  6 STOKE’S LAW Viscosity F Upthrust a A Gravity pull (weight) W F B Fg Consider a sphere falling through a viscous fluid, Its velocity increases due to gravity pull until a constant velocity called TERMINAL VELOCITY Vt is attained. At this velocity, viscosity is balanced by the gravitational force. Viscosity force F depends on terminal velocity, radius of the sphere and coefficient of viscosity ɳ Hence F = KVtr ɳ By dimension method, F = KVtxr y ɳz ----------(6) And if k = 6π, then F = 6πrɳVt ---------- (7) stoke’s law (equation) Exercise 3: Prove (7) from (6) Downloaded from Samporhelp.000webhostapp.com  7 The resultant downward force F at A is F = W - U - F which causes the sphere to accelerate to terminal velocity. If ρ is the density of the fluid and δ is that of the sphere then = , and = ; So from F = W – U = ( − ) ---------- (8) Equate (7) and (8) ( ) We have = YOUNG’S MODULUS (Y) Y is defined as the relation: Stress = F/A = Strain = e/L = Graph of stress/Strain gives Y by its slope. Exercise 4: Sketch the graph to obtain Young’s Modulus Y. Exercise 5: Incompressible liquid flows in a horizontal tube AB from A to B. If the areas of cross- sections are 30cm2 and 15 cm2 and velocity at end A is 12 m/s, calculate the velocity at the end B. Exercise 6: YOU ARE ADVISED TO ATTEMPT MORE RELATED QUESTIONS FROM SUITABLE TEXT BOOKS ON THE TOPICS TREATED; I.E. DO NOT LIMIT YOUR STUDY EFFORTS TO THIS NOTE. Downloaded from Samporhelp.000webhostapp.com