FUNDAMENTALS OF FLUID MECHANICSFLUID MECHANICS Chapter 6 Flow . - CAU

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FUNDAMENTALS OF FLUID MECHANICS Chapter 6 Flow Analysis Using Differential Methods 1

MAIN TOPICS I I. II. Fl id Element Fluid El M i Motion Conservation of Mass (Continuity equation) and Conservation Conservation of Linear Momentum (Navier (Navier--Stokes Equation) III. Inviscid Flow (Bernoulli equation) and Potential Flow (Stream function) Incompressible – Compressible Inviscid – Viscous Steady – Unsteady Rotational - Irrotational Mathematical equation? 2

Motion of a Fluid Element 1. Fluid Translation Translation:: The element moves from one point to another. 3. Fluid Rotation Rotation:: The element rotates about any or all of the x,y,z axes. axes Fluid Deformation Deformation:: 4. Angular Deformation:The element’s angles between the sides change. 2. Linear Deformation:The element’s sides stretch or contract. 3

1 Fluid Translation velocity and acceleration 1. The velocity of a fluid particle can be expressed V V ( x , y, z , t ) u i v j wk Velocity field The total acceleration of the fluid particle is given by DV V V dx V dy V dz a Dt t x dt y dt z dt dx dy dz u, v, w dt dt dt DV V V V V a u v w Dt t x y z Acceleration field DV is called the material, or substantial derivative. a Dt dV dt 4

Physical Significance DV V V V V a u v w Dt x y z t Total Acceleration of a particle Convective A l i Acceleration - Space Local Acceleration - Time V DV a (V )V Dt t 5

Scalar Component (외울필요 없음 없음)) u u u u ax i a u v w t x y z v v v v ay j a u v w z t x y w w w w az k a v w u t x y z Rectangular coordinates system Vr Vr V Vr V 2 V ar Vr Vz r t r r r z V V V V Vr V V a Vr Vz r t r r z Vz Vz V Vz Vz Vr Vz az t r r z C li d i l Cylindrical coordinates system 6

Translation All points in the element have the same velocity, velocity, then the element will simply translate from one position to another. 7

2 Linear Deformation 1/2 2. The shape of the fluid element, described by the angles at its vertices, remains unchanged, since all right angles continue to be right angles. angles. g in the x dimension requires q a nonzero value of A change u / x A y v / y A z w / z 8

Linear Deformation 2/2 The change in length of the sides may produce change in volume off the h element. l The change in V u x ( y z )( t ) u ( x y z )( t ) x x The rate at which the V is changing per unit volume d tto gradient due di t u// x 1 d V u V dt x If v/ y and w/ z are involved as in 22--D or 3D cases,, u v w v w u ( x y z )( t ) d V x ( y z )( t ) y ( x z )( t ) z ( x y )( t ) z x y z x y Volumetric dilatation rate 1 d V u v w V V dt x y z Di ergence of V Divergence For an incompressible fluid (constant density), the volumetric dilatation rate is zero zero. 9

3 Angular Rotation 1/4 3. The angular velocity (각속도) of line OA ωOA v x t v For small angles tan x t x x δα lim δt 0 δt v OA CCW x (시계반대방향 시계반대방향)) u OB CW y u OB 0 y CCW Positive direction 10

Angular Rotation 2/4 The rotation of the element about the zz-axis is defined as the average of the angular velocities of the two mutually perpendicular lines OA and OB about the zz-axis axis. y 1 1 v u z lim li 2 t 0 t 2 x y x CCW (시계반대방향 시계반대방향)) 1 w v x 2 y z In vector form 1 u w y 2 z x xi y j z k 11

Angular Rotation 3/4 1 w v 1 u w 1 v u y x z 2 x y 2 y z 2 z x 1 w v u w v u i k j 2 y z z x x y 1 1 w v 1 u w 1 v u 1 curlV V i j k 2 2 2 y z 2 z x 2 x y Defining vorticity 2 V - Angular rotation Definingg irrotation V 0 12

Angular Rotation 4/4 i 1 1 1 curlV V 2 2 2 x u j y v k z w 1 w v 1 u w 1 v u i j k 2 x y 2 y z 2 z x 13

Vorticity Defining Vorticity ζ which is a measurement of the rotation of a fluid element as it moves in the flow field: 2 curl V V 1 w v u w v u 1 i k V j 2 y z z x x y 2 In cylindrical coordinates system: system: 1 Vz V Vr Vz 1 rV 1 Vr V er e ez z r r z r r r 14

4 Angular Deformation 1/2 4. Angular deformation of a particle is given by the sum of the two angular deformation u u u y t u t y t y y / x / y v v v x t v t x t x x ξ(Xi)η(Eta) Rate of shear strain or the rate of angular deformation v u t t x y v u lim lim . t 0 t 0 t t x y 15

Angular Deformation 2/2 The rate of angular deformation in xy plane v u x y The rate of angular deformation in yz plane w v y z The rate of angular deformation in zx plane w u x z 16

Example 6 6.1 1 Vorticity For F a certain i twotwo-dimensional di i l flow fl field fi ld the h velocity l i is i given i by b 2 2 V 4 xyy i 2 ( x y ) j Is this flow irrotational? 17

Example 6 6.1 1 Solution u 4 xy v x2 y2 1 w v 0 2 y z 1 u w y 0 2 z x w 0 x This flow is irrotational 1 v u z 0 2 x y 18

Conservation Equations Continuity equation Conservation of Mass Momentum equation (Navier(Navier-Stokes Eq.) Conservation C ti off Linear Li Momentum M t Angular momentum equation Conservation of Angular Momentum Energy equation Conservation of Energy Representation Integral (control volume) representation Differential representation 19

Conservation of Mass 1/5 To derive the differential equation for conservation of mass in rectangular and in cylindrical coordinate system. The derivation is carried out by applying conservation of mass to a differential control volume. volume. With the control volume representation of the conservation of mass t CV dV CS V n dA 0 Th differential The diff i l form f off continuity i i equation? i ? 20

Conservation of Mass 2/5 dV V n dA 0 CV CS t The CV chosen is an infinitesimal cube with sides of length x, y, and z. differential control volume d V x y z CV t t u x u dx u x x 2 2 n V u u x x 2 u x x 2 Taylor series n V i Vi V on the right surface n V i Vi V on the left surface 21

Conservation of Mass 3/5 Net rate of mass Outflow in xx-direction u x u x u u y z u y z x y z x 2 x 2 x Net rate of mass Outflow in yy-direction N rate off mass Net Outflow in zz--direction v x y z y w x y z z 22

Conservation of Mass 4/5 Net rate of mass Outflow u v w x y z x y z The differential equation for Continuity equation u v w V 0 t x y z t d V V n dA 0 CV CS t d V x y z CV t t CS u v w x y z y z x V ndA 23

Conservation of Mass 5/5 Incompressible fluid (density is constant and uniform) u v w V 0 x y z Steady flow ( ) ( u ) ( v ) ( w) 0 V 0 z t x y 24

Example 6.2 6 2 Continuity Equation The Th velocity l i components for f a certain i incompressible, i ibl steady d flow fl field are u x 2 y2 z2 v xy yz z w ? Determine the form of the z component, p , w,, required q to satisfy y the continuity equation. 25

Example 6 6.2 2 Solution The continuity equation u v w 0 x y z u 2z x v x z y w 2 x ( x z ) 3x z z z2 w 3xz f (x, y) 2 26

Conservation of Linear Momentum Applying Newton’s second law to control volume DP F Psystem Vddm V dV M ( system ) V ( system ) Dt SYS D V m V V V V F m u v w Dt y x z t DV m m a Newton’s 2nd law Dt For a infinitesimal f system y off mass dm, dm, what’s the the differential form of linear momentum equation? equation? 27

Forces Acting on Element 1/2 The forces acting on a fluid element may be classified as body forces and surface forces; surface forces include normal forces and tangential (shear) forces forces. F FS FB Fsx i Fsy j Fsz k Fbx i Fby j Fbz k Surface forces acting on a fluid element can be described in terms of normal and shear stresses. stresses Fn F1 F2 n lim 1 lim 2 lim t 0 A t 0 A t 0 A 28

Forces Acting on Element 2/2 Fsx xx yx zx x y z y z x xy yy zy Fsy x y z y z x xz yz zz Fsz y z x Fbx g x x y z Fby g y x y z Fbz g z x y z xx p xx yy p yy zz p zz x y z Equation of Motion 29

Double Subscript Notation for Stresses xy The direction of the stress The direction of the normal to the plane on which the stress acts 30

Equation of Motion Fx ma x Fy ma y Fz ma z General equation of motion u xx yyx zx u u u g x u v w z x y z x y t xy yy zy v v v v g y u v w z x y z x y t w xz yz zz w w w g z u v w z x y z x y t These are the differential equations of motion for any fluid. How to solve u,v,w ? - These can’t be solved because of more variables than equations, which requires more equations called “constitutive equations” to solve the equations in the case of “Newtonian Newtonian fluids fluids” 31

Stress-Deformation Relationship StressRelationship:: constitutive equations 1/2 The stresses must be expressed in terms of the velocity and pressure field field. Cartesian coordinates in Newtonian and C Compressible ibl fluids fl id u xx p xx p 2 x v yy p yy p 2 y w zz p zz p 2 z v u xy yx x y 2 V 3 2 V 3 2 V 3 w u x z w v y z xz zx yz zy p 1 xx yy zz 3 32

Stress-Deformation Relationship StressRelationship:: constitutive equations 2/2 The stresses must be expressed in terms of the velocity and pressure field field. Cartesian coordinates in Newtonian and I Incompressible ibl fluids fl id u x v yy p yy p 2 y w zz p zz p 2 z v u xy yx x y xx p xx p 2 w u x z xz zx w v yz zy y z 1 p xx yy zz 3 33

The Navier Navier--Stokes Equations 1/2 These obtained equations q of motion are called the NavierNavierStokes Equations. Under incompressible Newtonian fl fluids uids, the Navieruids, NavierStokes equations are reduced to: 2u 2u 2u u u u u p u v w g x 2 2 2 z z x y x y t x 2v 2v 2v v v v v p u v w g y 2 2 2 z z x y y y t x 2w 2w 2w w w w w p u v w g z 2 2 2 z z x y z y t x 34

The Navier Navier--Stokes Equations 2/2 The Navier Navier--Stokes equations apply to both laminar and turbulent flow, flow, but for turbulent flow each velocity component fluctuates randomly with respect to time and this added complication makes an analytical solution intractable. The exact solutions referred to are for laminar flows in which the velocity is either independent of time (steady flow) or dependent on time (unsteady flow) in a well well-defined manner. 35

Laminar or Turbulent Flow 1/2 The flow of a fluid in a pipe may be Laminar ? Or Turbulent ? Osborne Reynolds, Reynolds, a British scientist and mathematician, was the first to distinguish g the difference between these classification of flow by using a simple apparatus as shown. 36

Laminar or Turbulent Flow 2/2 For “small enough flowrate” the dye streak will remain as a well--defined line as it flows along, with only slight blurring due well t molecular to l l diffusion diff i off the th dye d into i t the th surrounding di water. t For a somewhat larger “intermediate flowrate” the dye fl fluctuates in i time i andd space, andd intermittent i i bursts b off irregular i l behavior appear along the streak. For F “large l enough h flowrate fl t ” the h dye d streakk almost l immediately become blurred and spreads across the entire pipe in a random fashion. fashion 37

Time Dependence of Fluid Velocity at a Point 38

Indication of Laminar or Turbulent Flow The Th term flowrate fl t should h ld be b replaced l d by b Reynolds R ld number,, Re VL / ,where V is the average velocity in the pipe, number and L is the characteristic dimension of a flow. L is usuallyy D (diameter) in a pipe flow. - a measure of inertial force to the viscous force. It I iis not only l the h fluid fl id velocity l i that h ddetermines i the h character h off the h flow – its density, viscosity, and the pipe size are of equal importance. p For general engineering purpose, the flow in a round pipe Laminar R e 2100 Transitional Turbulent R e 4000 39

Some Simple Solutions for Viscous, Incompressible Fluids A principal difficulty in solving the NavierNavier-Stokes equations is because of their nonlinearity arising from the convective acceleration terms. terms. general analytical y schemes for solving g There are no g nonlinear partial differential equations. There are a few special cases for which the convective acceleration vanishes. In these cases exact solution are often possible. 40

Steady, Laminar Flow between Fixed 1/4 4 Parallel Plates 1/ 1. 2. 3. 4. Schematic: Assumptions: p Incompressible, p , Newtonian,, Steady, y, One dimensional flow u u v w V 0 0 0 u u y Continuity equation x y z x The Navier Navier--Stokes equations 2u 2u 2u u p u u u u v w g x 2 2 2 x y z x y z t x 2v 2v 2v v v v v p u v w g y 2 2 2 x y z y y z t x 2w 2w 2w w w w w p u v w g z 2 2 2 t x y z z y z x 2u p 2 0 x y p g 0 y p 0 z 41

Steady, Laminar Flow between Fixed 2/4 4 Parallel Plates 2/ 5 5. Boundary conditions (B.C.) (B C ) 6. Solve the equations with B.C. boundary condition) u 0 at y y -h p g y Integrating p 0 z 2u p Integrating 0 2 x y 0 1 c 2 0, c1 2 1 u 2 p gy f1 x u 1 2 p 2 y c1 y c 2 x c1 ? c2 ? p 2 h x p 2 2 y h x u 0 at y h (no (no--slip 42

Steady, Laminar Flow between Fixed Parallel Plates 3/4 Shear stress distribution yx u p y y x Volume flow rate per unit depth (z direction) 3 1 p 2 2 h p 2 q udy ( y h ) dy h h 2 3 x x p p1 p 2 p p 2 p1 constant 0 x 2 x1 x h h 2h 3 p q , where p1 is the inlet pressure and p 2 is the outlet pressure 3 2h 3 p 2h 3 1 q q p i V Flow resistance 3 3 3 R h 43

Steady, Laminar Flow between Fixed Parallel Plates 4/4 Average velocity per unit depth q h 2 p Vaverage 2h 3 Point of maximum velocity du 0 dy at yy 00 u u max h2 U 2 p 3 V average x 2 44

Couette Flow 1/3 (HW) Since only the boundary conditions have changed, changed, there is no need to repeat the entire analysis of the “both plates stationary” case. 45

Couette Flow 2/3 The boundary conditions for the moving plate case are u 0 at y 0 1 p 2 u c1 ? c2 ? y c1 y c2 2 x u U at y b U 1 p p c1 b b 2 x Velocity distribution p P 2 U x x b 2 c2 0 Uy 1 p 2 1 p u y by b 2 x 2 x u y b 2 p y b 1 U b 2 U x b y 46

Couette Flow 3/3 Simplest type of Couette flow U ri b ro ri ri /( ro ri ) p y 0 u U x b This flow can be approximated by the flow between closely spaced concentric cylinder is fixed and the other cylinder rotates with a constant angular velocity. velocity Flow in the narrow gap g of a jjournal bearing. 47

Steady, Laminar Flow (Hagen (Hagen--Poiseuille Flow) in Circular Tubes 1/5 1. 2. Schematic: Assumptions: Incompressible, Newtonian, Steady, Laminar, One dimensional flow vr 0, v 0, v z 0 v z 0 v z v z r z 3. Continuity equation 4 4. 5. The Th NavierNavier N i -Stokes S k equations i Boundary Conditions: At r 0, the velocity vz is finite. At r R, the velocity vz is zero. Solve the equation with B.C. 48 6.

From the NavierNavier-Stokes Equations in Cylindrical coordinates General motion of an incompressible Newtonian fluid is governed by the continuity equation and the momentum equation Mass conservation Navier-Stokes Equation in a cylindrical coordinate Acceleration 49

Steady Laminar Flow in Circular Tubes 2/5 Steady, Navier – Stokes equation reduced to p 0 g sin p g r sin f1 z , r Integrating p gy f1 z 1 p 0 g cos r 1 v z p Integrating 0 r z r r r 1 p 2 vz r c1 ln r c2 4 z c1 ? c2 ? 50

Steady Laminar Flow in Circular Tubes 3/5 Steady, At r 0, r 0 the velocity elocit vz is finite finite. At rr R, R the velocity elocit vz is zero. 1 p 2 c1 0, c2 R 4 z Velocity distribution 1 p 2 2 vz r R 4 z 51

Steady Laminar Flow in Circular Tubes 4/5 Steady, The shear stress distribution rz dv z r p dr 2 z Volume flow rate R 4 p Q u z 2 rdr . 0 8 z p p 2 p1 p / constant z R 4 p R 4 p D 4 Q p 8 z 8 128 R 52

Steady Laminar Flow in Circular Tubes 5/5 Steady, Average velocity V average Q Q R 2 p 2 A R 8 Point of maximum velocity dv z 0 dr at r 0 R p vz r 2V average 1 4 v max R 2 v max 2 53

Steady, Axial, Laminar Flow in an Annulus 1/2 (HW) For steady, laminar flow in annular tubes Boundary conditions vz 0 , at r ro vz 0 , at r ri 54

Steady Axial, Steady, Axial Laminar Flow in an Annulus 2/2 Th velocity The l i distribution di ib i 1 p 2 2 ri 2 ro2 r vz ln r ro 4 z ln(ro / ri ) ro Th volume The l rate off flow fl Q ro ri p 4 4 (ro2 ri 2 ) 2 v z (2 r )dr ro ri 8 z ln(ro / ri ) The maximum velocityy occurs at r rm vz 0 r r ri rm 2 ln( r / r ) o i 2 o 2 1/ 2 55

Inviscid Flow Shear stresses develop in a moving fluid because of the viscosity of the fluid. For F some common fl fluid, id suchh as air, i the th viscosity i it is i small, small ll, andd therefore it seems reasonable to assume that under some circumstances we may be able to simply neglect the effect of viscosity. viscosity Flow fields in which the shear stresses are assumed to be negligible g g are said to be inviscid, or frictionless. frictionless. D fi th Define the pressure, p, as the th negative ti off the th normall stress t p xx yy zz 56

Euler’s Euler s Equation of Motion Under inviscid flows: frictionless condition, condition, the equations of motion are reduced to Euler’s Equation: Equation: u u u u p w u v g x z x y x t v v v v p u g y v w Euler’s Equation x y z y t w w w w p g z u v w x y z z t DV g p Dt 57

Bernoulli Equation 1/3 Euler’s equation for steady flow along a streamline is g p ( V ) V Selecting the coordinate system with the zz-axis vertical so that the acceleration of gravity vector can be expressed as g g z 1 V V V V V V 2 g z p 2 Vector identity . (V V ) V V 58

Bernoulli Equation 2/3 1 2 V g z V V V 2 pp ds V perpendicular to V p 1 2 ds V ds g z ds V V ds 2 With ds dxi dyj dzk tangential vector on a streamline p p p p p p p ds i j k dxi dyj dzk dx dy dz dp y z x y z x 1 V 2 1 V 2 V 2 V 2 1 V 2 V 2 1 2 V ds i j k dx d i dy d j dz d k d dx d dy ddz dV 2 2 2 x y z 2 x y z 2 z z z z z z g z ds g i j k dxi dyyj dzk g dx dyy dz ggdz y z y z x x 59

Bernoulli Equation 3/3 p 1 2 ds V ds g z ds 0 2 1 d V 2 gdz 0 2 d dp Integrating V2 gz constant 2 dp For steady, steady, inviscid, incompressible fluid (commonly called ideal fluids) along a streamline Bernoulli equation is given by V2 ggz constant 2 p Bernoulli equation 60

1/2 2 Irrotational Flow 1/ Irrotation ? The irrotational condition is V 0 In rectangular coordinates system v u w v u w 0 x y y z z x In cylindrical coordinates system 1 Vz V Vr Vz 1 rV 1 Vr 0 z r z r r r r 61

2/2 2 Irrotational Flow 2/ A general flow field would not be irrotational flow. p uniform flow field is an example p of an A special irrotational irrotation al flow 62

Bernoulli Equation for Irrotational Flow 1/3 The Bernoulli equation for steady, incompressible, and inviscid flow is p V2 gz constant 2 The Th equation i can be b applied li d between b any two points i on the h same streamline. In general, the value of the constant will vary from streamline streamline to streamline. streamline. Under additional irrotational condition, condition, the Bernoulli equation ? Starting with Euler Euler’ss equation in vector form g p V (V )V 1 1 ( V ) V p gk V V V V 2 t ZERO Regardless of the direction of ds 63

Bernoulli Equation for Irrotational Flow 2/3 With irrotaional condition V 0 1 1 (V )V p gk V V V V 2 1 1 1 2 V V V p gk 2 2 dr dr dxi dyj dzk 1 1 2 Not a streamline V dr p dr gk dr 2 1 dp dp 1 2 d V gdz d V 2 gdz 0 2 2 64

Bernoulli Equation for Irrotational Flow 3/3 Integrating for incompressible flow 2 dp V gz contant 2 V2 gz constant 2 p Thi equation This i is i valid lid between b any two points i in i a steady, d incompressible, inviscid, and irrotational flow irrespective of streamlines. streamlines 2 1 2 p1 V p 2 V2 z1 z2 2g 2g 65

Stream Function 1/6 Streamlines Streamlines:: Lines tangent to the instantaneous velocity vectors at every point. Stream St function f ti Ψ(x,y) Ψ Ψ((x,y) ( ) [Psi] [P i] ? Used U d to t representt the th velocity l it component u(x,y,t) and v(x,y,t) of a “two two--dimensional incompressible”” flow. incompressible flow Define a function Ψ(x,y), Ψ(x,y), called the stream function, which relates the velocities shown byy the figure g in the margin g as u y v x 66

Stream Function 2/6 The stream function Ψ(x,y) Ψ(x,y) satisfies the twotwo-dimensional form of the incompressible continuity equation 2 2 u v 0 0 x y x y y x Ψ(x,y) is still unknown for a particular problem, but at least we have simplify i lif the th analysis l i by b having h i to t determine d t i only l one unknown, unknown k , Ψ(x,y) , rather than the two unknown function u(x,y) and v(x,y). 67

Stream Function 3/6 Another advantage of using stream function is related to the fact that line along which Ψ(x,y) constant are streamlines. How H tto prove ? From F th the definition d fi iti off the th streamline t li that th t the th slope l at any point along a streamline is given by dy v dx streamline u Velocity Ve oc ty and a d velocity ve oc ty component co po e t along a o g a streamline st ea e 68

Stream Function 4/6 The change of Ψ(x,y) Ψ(x,y) as we move from one point (x,y) to a nearly point (x dx,y dy) is given by d dx dy vdx udy x y d 0 vdx udy 0 Along a line of constant Ψ v dy d dx streamline u This is the definition for a streamline. Thus, if we know the stream function Ψ(x,y) we can plot lines of constant Ψto provide the family of streamlines that are helpful in visualizing the pattern of flow. flow. There are an infinite number of streamlines that make up a particular flow field, since for each constant value assigned to Ψa streamline can be drawn. 69

Stream Function 5/6 The actual numerical value associated with a particular streamline is not of particular significance, but the change in the value of Ψ is related to the volume rate of flow. flow dq : volume rate of flow passing between the two streamlines. streamlines. Flow never crosses streamlines by definition definition. dq udy vdx q 2 1 dy dx d y x d 2 1 If q 0, the flow is from left to right. right If q 0, the flow is from right to left. 70

Stream Function 6/6 Thus the volume flow rate between any two streamlines can be written as the difference between the constant values of Ψ defining two streamlines. streamlines The velocity will be relatively high wherever the streamlines are close together together, and relatively low wherever the streamlines are far apart. 71

Example 6.3 6 3 Stream Function The Th velocity l i component in i a steady, d incompressible, i ibl two dimensional flow field are u 2y v 4x Determine the corresponding stream function and show on a sketch several streamlines. Indicate the direction of glow along the streamlines. 72

Example 6 6.3 3 Solution From the definition of the stream function u 2y y v 4x x y 2 f1 ((x)) 2 x 2 f 2 (y) 2 x y 2 2 Ψ 0 2 x y C 2 2 For simplicity, we set C 0 2 2 y x 1 /2 Ψ 0 73

1/3 3 Velocity Potential Φ( Φ(x Φ(x,y,z,t) x,y,z,t) x y z t) 1/ The Th stream function f i for f two two--dimensional di i l incompressible i ibl flow is Ψ(x,y) For F an iirrotational i l flow, fl the h velocity l i components can be b expressed in terms of a scalar function Φ( Φ(x,y,z,t) x,y,z,t) as u x v y w z where Φ( Φ(x,y,z,t) x,y,z,t) is called the velocity potential potential. V 0 V 74

2/3 3 Velocity Potential Φ( Φ(x Φ(x,y,z,t) x,y,z,t) x y z t) 2/ In vector form V For an incompressible flow V 0 Also called a potential flow For incompressible, incompressible irrotational flow 2 2 2 2 V V 2 2 2 0 x y z Laplace’s equation Laplacian operator 75

3/3 3 Velocity Potential Φ( Φ(x Φ(x,y,z,t) x,y,z,t) x y z t) 3/ Inviscid, incompressible, irrotational fields are governed by Laplace’s equation. equation. This type flow is commonly called a potential flow. flow. To complete the mathematical formulation of a given problem, boundary conditions have to be specified. These are usually velocities specified on the boundaries of the flow field of interest. 76

Motion of a Fluid ElementMotion of a Fluid Element 1. 1. Fluid Fluid Translation: The element moves from one point to another. 3. 3. Fluid Fluid Rotation: The element rotates about any or all of the x,y,z axes. Fluid Deformation: 4. 4. Angular Deformation:The element's angles between the sides Angular Deformation:The element's angles between the sides

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