Fluid Dynamics - Equation Of
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Derivation of Fluid Flow Equations
TPG4150 Reservoir Recovery Techniques 2017 1 Fluid Flow Equations DERIVATION OF FLUID FLOW EQUATIONS Review of basic steps Generally speaking, flow equations for flow in porous materials are based on a set of mass, momentum and energy conservation equations, and constitutive equations for the fluids and the porous material involved. For simplicity, we will in the following assume isothermal conditions, so that we not have to involve an energy conservation equation. However, in cases of changing reservoir temperature, such as in the case of cold water injection into a warmer reservoir, this may be of importance. Below, equations are initially described for single phase flow in linear, one- dimensional, horizontal systems, but are later on extended to multi-phase flow in two and three dimensions, and to other coordinate systems. Conservation of mass Consider the following one dimensional rod of porous material: Mass conservation may be formulated across a control element of the slab, with one fluid of density ρ is flowing through it at a velocity u: u ρ Δx The mass balance for the control element is then written as: ⎧Mass into the⎫ ⎧Mass out of the ⎫ ⎧ Rate of change of mass⎫ ⎨ ⎬ − ⎨ ⎬ = ⎨ ⎬ , ⎩element at x ⎭ ⎩element at x + Δx⎭ ⎩ inside the element ⎭ or ∂ {uρA} − {uρA} = {φAΔxρ}. x x+ Δx ∂t Dividing by Δx, and taking the limit as Δx approaches zero, we get the conservation of mass, or continuity equation: ∂ ∂ − (Aρu) = (Aφρ). ∂x ∂t For constant cross sectional area, the continuity equation simplifies to: ∂ ∂ − (ρu) = (φρ) . ∂x ∂t Next, we need to replace the velocity term by an equation relating it to pressure gradient and fluid and rock properties, and the density and porosity terms by appropriate pressure dependent functions. -
Continuity Equation in Pressure Coordinates
Continuity Equation in Pressure Coordinates Here we will derive the continuity equation from the principle that mass is conserved for a parcel following the fluid motion (i.e., there is no flow across the boundaries of the parcel). This implies that δxδyδp δM = ρ δV = ρ δxδyδz = − g is conserved following the fluid motion: 1 d(δM ) = 0 δM dt 1 d()δM = 0 δM dt g d ⎛ δxδyδp ⎞ ⎜ ⎟ = 0 δxδyδp dt ⎝ g ⎠ 1 ⎛ d(δp) d(δy) d(δx)⎞ ⎜δxδy +δxδp +δyδp ⎟ = 0 δxδyδp ⎝ dt dt dt ⎠ 1 ⎛ dp ⎞ 1 ⎛ dy ⎞ 1 ⎛ dx ⎞ δ ⎜ ⎟ + δ ⎜ ⎟ + δ ⎜ ⎟ = 0 δp ⎝ dt ⎠ δy ⎝ dt ⎠ δx ⎝ dt ⎠ Taking the limit as δx, δy, δp → 0, ∂u ∂v ∂ω Continuity equation + + = 0 in pressure ∂x ∂y ∂p coordinates 1 Determining Vertical Velocities • Typical large-scale vertical motions in the atmosphere are of the order of 0. 01-01m/s0.1 m/s. • Such motions are very difficult, if not impossible, to measure directly. Typical observational errors for wind measurements are ~1 m/s. • Quantitative estimates of vertical velocity must be inferred from quantities that can be directly measured with sufficient accuracy. Vertical Velocity in P-Coordinates The equivalent of the vertical velocity in p-coordinates is: dp ∂p r ∂p ω = = +V ⋅∇p + w dt ∂t ∂z Based on a scaling of the three terms on the r.h.s., the last term is at least an order of magnitude larger than the other two. Making the hydrostatic approximation yields ∂p ω ≈ w = −ρgw ∂z Typical large-scale values: for w, 0.01 m/s = 1 cm/s for ω, 0.1 Pa/s = 1 μbar/s 2 The Kinematic Method By integrating the continuity equation in (x,y,p) coordinates, ω can be obtained from the mean divergence in a layer: ⎛ ∂u ∂v ⎞ ∂ω ⎜ + ⎟ + = 0 continuity equation in (x,y,p) coordinates ⎝ ∂x ∂y ⎠ p ∂p p2 p2 ⎛ ∂u ∂v ⎞ ∂ω = − ⎜ + ⎟ ∂p rearrange and integrate over the layer ∫p ∫ ⎜ ⎟ 1 ∂x ∂y p1⎝ ⎠ p ⎛ ∂u ∂v ⎞ ω(p )−ω(p ) = (p − p )⎜ + ⎟ overbar denotes pressure- 2 1 1 2 ⎜ ⎟ weighted vertical average ⎝ ∂x ∂y ⎠ p To determine vertical motion at a pressure level p2, assume that p1 = surface pressure and there is no vertical motion at the surface. -
Chapter 3 Newtonian Fluids
CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics Chapter 3: Newtonian Fluids CM4650 Polymer Rheology Michigan Tech Navier-Stokes Equation v vv p 2 v g t 1 © Faith A. Morrison, Michigan Tech U. Chapter 3: Newtonian Fluid Mechanics TWO GOALS •Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields QUICK START V W x First, before we get deep into 2 v (x ) H derivation, let’s do a Navier-Stokes 1 2 x1 problem to get you started in the x3 mechanics of this type of problem solving. 2 © Faith A. Morrison, Michigan Tech U. 1 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics EXAMPLE: Drag flow between infinite parallel plates •Newtonian •steady state •incompressible fluid •very wide, long V •uniform pressure W x2 v1(x2) H x1 x3 3 EXAMPLE: Poiseuille flow between infinite parallel plates •Newtonian •steady state •Incompressible fluid •infinitely wide, long W x2 2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL 4 2 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics Engineering Quantities of In more complex flows, we can use Interest general expressions that work in all cases. (any flow) volumetric ⋅ flow rate ∬ ⋅ | average 〈 〉 velocity ∬ Using the general formulas will Here, is the outwardly pointing unit normal help prevent errors. of ; it points in the direction “through” 5 © Faith A. Morrison, Michigan Tech U. The stress tensor was Total stress tensor, Π: invented to make the calculation of fluid stress easier. Π ≡ b (any flow, small surface) dS nˆ Force on the S ⋅ Π surface V (using the stress convention of Understanding Rheology) Here, is the outwardly pointing unit normal of ; it points in the direction “through” 6 © Faith A. -
Compressible Flows for Fluidic Control in Microdevices
COMPRESSIBLE FLOWS FOR FLUIDIC CONTROL IN MICRODEVICES by Dustin S. Chang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Chemical Engineering) in The University of Michigan 2010 Doctoral Committee: Professor Mark A. Burns, Chair Professor Robert M. Ziff Associate Professor Luis P. Bernal Associate Professor Michael Mayer © Dustin S. Chang 2010 To my mom and dad, Sweson and Chaohui, and my grandparents, Po-High and Peyuk … Thank you for all your sacrifices. ii ACKNOWLEDGEMENTS To the casual observer, a doctoral thesis may appear to be the product of a single dedicated—albeit misguided—individual after multiple years of toil. Anyone that has survived the experience will most assuredly tell you that he could not have done it without the many people who supported him along the way. Beginning long before graduate school, my family has been a constant source of encouragement, well-meaning criticism, and material support. I am hugely grateful for my aunts and uncles who never failed to go the extra mile in taking care of me, teaching me, and challenging me to improve. Aunt Hsueh-Rong, Aunt Nancy, Aunt Chao-Lin, Uncle Jeff, Aunt Tina, and Uncle Ming ... Thank you for your love. The only people potentially deserving greater appreciation than these are my mother and father and my grandparents. I have no doubt that none of my meager accomplishments would have been possible without your prayers and unconditional support, both moral and financial. I also have no doubt that all of the aforementioned people are infinitely more enamored with the idea of a Ph.D. -
Pressure and Piezometry (Pressure Measurement)
PRESSURE AND PIEZOMETRY (PRESSURE MEASUREMENT) What is pressure? .......................................................................................................................................... 1 Pressure unit: the pascal ............................................................................................................................ 3 Pressure measurement: piezometry ........................................................................................................... 4 Vacuum ......................................................................................................................................................... 5 Vacuum generation ................................................................................................................................... 6 Hydrostatic pressure ...................................................................................................................................... 8 Atmospheric pressure in meteorology ...................................................................................................... 8 Liquid level measurement ......................................................................................................................... 9 Archimedes' principle. Buoyancy ............................................................................................................. 9 Weighting objects in air and water ..................................................................................................... 10 Siphons ................................................................................................................................................... -
Aerodynamic Analysis of the Undertray of Formula 1
Treball de Fi de Grau Grau en Enginyeria en Tecnologies Industrials Aerodynamic analysis of the undertray of Formula 1 MEMORY Autor: Alberto Gómez Blázquez Director: Enric Trillas Gay Convocatòria: Juny 2016 Escola Tècnica Superior d’Enginyeria Industrial de Barcelona Aerodynamic analysis of the undertray of Formula 1 INDEX SUMMARY ................................................................................................................................... 2 GLOSSARY .................................................................................................................................... 3 1. INTRODUCTION .................................................................................................................... 4 1.1 PROJECT ORIGIN ................................................................................................................................ 4 1.2 PROJECT OBJECTIVES .......................................................................................................................... 4 1.3 SCOPE OF THE PROJECT ....................................................................................................................... 5 2. PREVIOUS HISTORY .............................................................................................................. 6 2.1 SINGLE-SEATER COMPONENTS OF FORMULA 1 ........................................................................................ 6 2.1.1 Front wing [3] ....................................................................................................................... -
CONTINUITY EQUATION Another Principle on Which We Can Derive a New Equation Is the Conservation of Mass
ESCI 342 – Atmospheric Dynamics I Lesson 7 – The Continuity and Additional Equations Suggested Reading: Martin, Chapter 3 THE SYSTEM OF EQUATIONS IS INCOMPLETE The momentum equations in component form comprise a system of three equations with 4 unknown quantities (u, v, p, and ). Du1 p fv (1) Dt x Dv1 p fu (2) Dt y p g (3) z They are not a closed set, because there are four dependent variables (u, v, p, and ), but only three equations. We need to come up with some more equations in order to close the set. DERIVATION OF THE CONTINUITY EQUATION Another principle on which we can derive a new equation is the conservation of mass. The equation derived from this principle is called the mass continuity equation, or simply the continuity equation. Imagine a cube at a fixed point in space. The net change in mass contained within the cube is found by adding up the mass fluxes entering and leaving through each face of the cube.1 The mass flux across a face of the cube normal to the x-axis is given by u. Referring to the picture below, these fluxes will lead to a rate of change in mass within the cube given by m u yz u yz (4) t x xx 1 A flux is a quantity per unit area per unit time. Mass flux is therefore the rate at which mass moves across a unit area, and would have units of kg s1 m2. The mass in the cube can be written in terms of the density as m xyz so that m x y z . -
1 the Continuity Equation 2 the Heat Equation
1 The Continuity Equation Imagine a fluid flowing in a region R of the plane in a time dependent fashion. At each point (x; y) R2 it has a velocity v = v (x; y; t) at time t. Let ρ = ρ(x; y; t) be the density of the fluid 2 −! −! at (x; y) at time t. Let P be any point in the interior of R and let Dr be the closed disk of radius r > 0 and center P . The mass of fluid inside Dr at any time t is ρ dxdy: ZZDr If matter is neither created nor destroyed inside Dr, the rate of decrease of this quantity is equal to the flux of the vector field −!J = ρ−!v across Cr, the positively oriented boundary of Dr. We therefore have d ρ dxdy = −!J −!N ds; dt − · ZZDr ZCr where N~ is the outer normal and ds is the element of arc length. Notice that the minus sign is needed since positive flux at time t represents loss of total mass at that time. Also observe that the amount of fluid transported across a small piece ds of the boundary of Dr at time t is ρ−!v −!N ds. Differentiating under the integral sign on the left-hand side and using the flux form of Green’s· Theorem on the right-hand side, we get @ρ dxdy = −!J dxdy: @t − r · ZZDr ZZDr Gathering terms on the left-hand side, we get @ρ ( + −!J ) dxdy = 0: @t r · ZZDr If the integrand was not zero at P it would be different from zero on Dr for some sufficiently small r and hence the integral would not be zero which is not the case. -
Fluid Mechanics Policy Planning and Learning
Science and Reactor Fundamentals – Fluid Mechanics Policy Planning and Learning Fluid Mechanics Science and Reactor Fundamentals – Fluid Mechanics Policy Planning and Learning TABLE OF CONTENTS 1 OBJECTIVES ................................................................................... 1 1.1 BASIC DEFINITIONS .................................................................... 1 1.2 PRESSURE ................................................................................... 1 1.3 FLOW.......................................................................................... 1 1.4 ENERGY IN A FLOWING FLUID .................................................... 1 1.5 OTHER PHENOMENA................................................................... 2 1.6 TWO PHASE FLOW...................................................................... 2 1.7 FLOW INDUCED VIBRATION........................................................ 2 2 BASIC DEFINITIONS..................................................................... 3 2.1 INTRODUCTION ........................................................................... 3 2.2 PRESSURE ................................................................................... 3 2.3 DENSITY ..................................................................................... 4 2.4 VISCOSITY .................................................................................. 4 3 PRESSURE........................................................................................ 6 3.1 PRESSURE SCALES ..................................................................... -
Chapter 2 Wave Propagation in Viscous Fluid
Chapter 2 Wave Propagation in Viscous Fluid This chapter summarizes with the derivation of the mathematical form of the acoustic wave propagation in the fluid. Before we derive the final form of the wave propagation equation in viscous fluid, we first look at two conservation (mass and momentum) of equations and state equation in the fluid. Detailed derivations can be found in the literatures [4, 6, 27, 28]. We limit our discussion only on the lossy 1-dimensional plane wave. 2.1 One-dimensional Viscous Wave Equation 2.1.1 One-dimensional Continuity Equation (Conversation of Mass) Consider the flow of a compressible fluid through a duct of arbitrary cross section (area S) in one dimension. The control volume (CV), is the segment between x and x + ∆x . 5 We want to know the rate at which the inside mass changes. First, we made two assumptions: 1. The CV is fixed in space 2. The flow is one-dimensional, so the mass flow only depends on t and x. Mass flow in Mass flow out = ρ uS |x = ρ uS |x+∆x x x+∆x , where ρ, u are the average mass flow density and average mass flow speed, respectively. For ∆x →0 , ρ and u will become a true point function. The time rate of increase of mass inside the CV is equal to net mass flow into the CV through the CV surfaces or in mathematical terms, ∂ (Sρ ∆x)= ρ uS | −ρ uS | ( 1 ) ∂t x x+∆x Since S is a constant and ∆x is not a function of time, Eq. -
Fuel Injection
Fuel injection • Fuel injection is a system for mixing fuel with air in an internal combustion engine. It has become the primary fuel delivery system used in gasoline automotive engines, having almost comppyletely replaced carburetors in the late 1980s. • The carburetor was invented by Karl Benz (founder of Mercedes‐Benz) in 1885 and patented in 1886. • Carburetors were the usual fuel delivery method for almost all gasoline (petrol)‐ fuelled engines up until the late 1980s, when fuel injection became the preferred method of automotive fuel delivery. In the U.S. market, the last carbureted cars were the 1990 Oldsmo bile CtCustom CiCruiser, BikBuick EttEstate Wagon, and SbSubaru Justy, and the last carbureted light truck was the 1994 Isuzu. Elsewhere, Lada cars used carburetors until 1996. A majority of motorcycles still use carburetors due to lower cost and throttle response problems with early injection set ups, but as of 2005, many new models are now being introduced with fuel injection. Internal combustion Engines: Carburetor, Fuel injection Carburetors are still found in small engines and in older or specialized automobiles, such as those designed for stock car racing. Dr. Primal Fernando • A fuel injection system is designed and calibrated specifically for the type(s) of [email protected] fuel it will handle. Most fuel injection systems are for gasoline or diesel Ph: (081) 2393608 applications. 1 2 Gas Review November 1913 Well, lets see if we can figure it out…… Used on tractors, boats, and stationary engines, including the Waterloo -
ESCI 342 – Atmospheric Dynamics I Lesson 10 – Vertical Motion, Pressure Coordinates
ESCI 342 – Atmospheric Dynamics I Lesson 10 – Vertical Motion, Pressure Coordinates Reading: Martin, Section 4.1 PRESSURE COORDINATES Pressure is often a convenient vertical coordinate to use in place of altitude. If the hydrostatic approximation is used, the relationship between pressure and altitude is given by the hydrostatic equation, p g . (1) z In height coordinates the vertical velocity is defined as w Dz Dt . In pressure coordinates the vertical velocity is defined as Dp , (2) Dt and is commonly called simply omega. The units of are Pa/s (often microbars per second, b/s, is also used). Since pressure decreases upward, a negative omega means rising motion, while a positive omega means subsiding motion. w and are related as follows: Dp p p p p u v w . (3) Dt t x y z On the synoptic scale we can assume hydrostatic vertical balance, so that (3) becomes Dp p p p u v gw gw , (4) Dt t x y since the local pressure tendencies and horizontal pressure advection terms are much, much smaller in magnitude than the vertical pressure advection terms. The total derivative in pressure coordinates is D uv .1 (5) Dt t x y p The conversion of a height derivative to a pressure derivative is accomplished using the chain rule as follows z . (6) p p z g z In pressure coordinates, the directions of the unit vectors ( iˆ , ˆj , and kˆ ) are the same as in height coordinates. The x and y axes are still horizontal, and not oriented along the constant pressure surface.2 The vertical axis is still vertical (perpendicular to x and y.) 1 If you take the total derivative of pressure you end up with the seeming absurdity that p p p uv .