Engineering Simplification Axi-Symmetric Flow through Circular pipe 6 R r A=r2 δA=2r6r Elemental discharge through elemental area δQ=v.SA=v.2rr6r Total Q through pipe section given by integration Q=vdA= 27rvdr Mean velocity Vm=Q/A=Q/(TR2)
Engineering Simplification Axi-Symmetric Flow through Circular Pipe Elemental discharge through elemental area : Q = v . A = v . 2rr Total Q through pipe section given by integration : Mean velocity Vm = Q/A = Q/(R2 ) r r R A = 2rr v A =r 2 Vm Q = = A R Q vdA rvdr 0 2
Fluid kinematics Under thecontinuum' hypothesis, a fluid body is considered to be made up of infinitesimal fluid"particles tightly packed together and interact with each other Each fluid" contains numerous molecules Fluid motion is described in terms of velocity and acceleration of fluid'particles', and not individual molecules Fluid Kinematics Study of the motion of fluid (position, velocity and acceleration) without consideration offorces producing the motion Fluid dynamics Analyses of fluid motion in relation to forces producing the motion
Fluid kinematics • Under the ‘continuum’ hypothesis, a fluid body is considered to be made up of infinitesimal fluid ‘particles’ tightly packed together and interact with each other • Each fluid ‘particle’ contains numerous molecules • Fluid motion is described in terms of velocity and acceleration of fluid ‘particles’, and not individual molecules Fluid Kinematics Study of the motion of fluid (position, velocity and acceleration) without consideration of forces producing the motion Fluid Dynamics Analyses of fluid motion in relation to forces producing the motion
Fluid kinematics Inflow CV-I Fixed control surface and system System boundary at time r System boundary at time t+ or o A definite mass of matter which distinguishes it from its surrounding matter o Has constant mass System boundary moves Used in bernoulli's equation Control Volume(C v) Definite region in space enclosed by control surfaces, fixed relative to observer Control Volume boundary is fixed e Mass can flow in or out of c. v Used in Continuity and momentum equations
Fluid kinematics System • A definite mass of matter which distinguishes it from its surrounding matter • Has constant mass • System boundary moves • Used in Bernoulli’s equation Control Volume (C.V.) • Definite region in space enclosed by control surfaces, fixed relative to observer • Control Volume boundary is fixed • Mass can flow in or out of C.V. • Used in Continuity and Momentum equations
Flow Analyses Eulerian method Fluid motion and properties(pressure, density, velocity etc) are described as functions of space and time Lagrangian method Follows the motion of individual fluid particles determines how the fluid properties of the particles change with time Location o: T=7(xo, yo, r) Particle A I、=T(t)
Flow Analyses Eulerian Method Fluid motion and properties (pressure, density, velocity etc) are described as functions of space and time Lagrangian Method Follows the motion of individual fluid particles determines how the fluid properties of the particles change with time
Flow kinematics Streamline Imaginary line through fluid such that at an instant, velocity of ever particle on the line is tangent to it Stream-tube Imaginary tube formed by all streamlines passing through a closed curve no fluid can enter or leave a stream tube except through its ends Used in continuity equation
Flow kinematics Streamline : Imaginary line through fluid such that at an instant, velocity of ever particle on the line is tangent to it Stream-tube Imaginary tube formed by all streamlines passing through a closed curve. No fluid can enter or leave a stream tube except through its ends Used in continuity equation