Textbook: Fluid Mechanics Yunus A. Cengel, John M. Cimbala
Course coordinator: Sture Holmberg
Mathematical part: Armin Halilovic
armin@sth.kth.se
Tel 08 790 4810
Room 5046
Homepage: www.sth.kth.se/armin
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Fluid flow is often described by a velocity vector field V=(u(t,x,y,z),v(t,x,y,z), w (t,x,y,z)).
Fluid flow is steady is when the velocity vector field at a fixed point (x,y,z) does not change with time.Thus, in steady flow, V=(u(x,y,z),v(x,y,z), w (x,y,z)) depends only on the point position (x,y,z).
Pathlines, streamlines, streaklines and timelines are field lines that we use to describe the flow.
In steady flow the streamlines, pathlines, and streaklines coincide.
PATHLINES
Definition: A pathline is the trajectory (path, curve) traveled by an individual fluid particle over some time period.
In the example below we consider pathlines through the four points (0.5, 0.5),
(0.5, 2.5), (0.5, 4.5) and (0.5, 6.5) .
The given velocity field V= (0.7+0.5x(t), 2+2.4 cos(5t) -0.5y(t)) changes with time t.
To find the pathline for the given velocity field
V=(u(t,x,y,z),v(t,x,y,z), w t,x,y,z))
and given start point P=(a,b,c), we solve this system of differential equation:
dx/dt=u(t,x,y,z)
dy/dt=v(t,x,y,z)
dz/dt=w(t,x,y,z)
with initial conditions x(t0)=a, y(t0)=b, z(t0)=c
See the following examples:
Example1,
pathline, 3D
Example2,
pathline, 2D
Exercises, pathline
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STREAMLINES
A streamline is a curve that is instantaneously ( for the fixed time point) tangent to the velocity vector of the flow.
To find the stream line for the given velocity field
V=(u(t,x,y,z),v(t,x,y,z), w t,x,y,z))
given time point t=tc and given start point P=(a,b,c), we substitute t=tc and solve the following system of differential equation:
dx/dt= u(tc,x,y,z)
dy/dt= v(tc,x,y,z)
dz/dt= w(tc,x,y,z)
with initial conditions x(t0)=a, y(t0)=b, z(t0)=c .
In the example below, for the given velocity field V= (0.7+0.5x(t), 2+2.4 cos(5t) -0.5y(t)),
we choose, for instance, t=1.9 and consider streamlines through the points (0.5, 0.5), (0.5, 2.5), (0.5, 4.5) and (0.5, 6.5) .
Exercises,
streamlines
Exercises,
stream function
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STREAKLINES
Streaklines are the locus of points of all the fluid particles that have passed
continuously through a fixed point.
This can be visualized experimentally by releasing dye (or smoke) into the fluid
in a time period at a fixed point.
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TIMELINES
A timeline is the locus of all the fluid particles that were marked at a previous instant in time, creating a curve that is displaced in time as the particles move.
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VOLUMETRIC STRAIN RATE,
COMPRESSIBLE / INCOMPRESSIBLE FLOW
The rate of change of volume of a fluid element per unit volume is called its volumetric strain rate.
The volumetric strain rate (in Cartesian coordinates) =div(V) =∂u/∂x+∂v/∂y+∂w/∂z
If div(V)=0 ( in some region) then the flow defined by V=(u,v,w) is incompressible in that region.
In the example below we consider the two dimensional velocity field V= (0.7+0.5x(t), 2+2.4 cos(5t) -0.5y(t)).
Since div(V)=0 everywhere we conclude that this flow is incompressible.
The fluid element changes the shape but the area remains constant as we see
in this animation.
Incompressible flow
Consider a steady two dimensional flow given by V=(0.5+0.1x+0.5y,
0.5 - 0.9y).
Since div(V) =∂u/∂x+∂v/∂y+∂w/∂z=
0.1 - 0.9 + 0 = -
0.8 we conclude that the flow is compressible.
Compressible flow
Homework
1
Homework
2
Test
1 with solution
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NAVIER_STOKES EQUATION
Derivation
of the NAVIER-STOKES EQUATION (for compressible flow)
Incompressible ( rho= const) Navier Stokes Equations in cylindrical coordinates:
NAVIER-STOKES
EQUATION (incompressible flow)
EULER EQUATION
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Exercises:
Grad, Div, Curl, NAVIER-STOKES EQUATION
LabM (AssignmentM)
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Example P1. Poiseuille flow. Consider steady, incompressible, laminar stationary flow of a Newtonian fluid with μ=0.001 / kg/(ms), ρ=1000kg/m 3 in a long round pipe in the z-direction, with constant circular cross-section of radius R=2 m. A Constant pressure gradient ∂P/∂z = –1/250 Pa/m is applied in the horizontal axis ( z-axis in our notation). We assume that the flow is axisymmetric
a) Use the Navier-Stokes equations in cylindrical coordinates to find the velocity field and pressure field.
b)
At t =0 we consider the sphere
x2+y2+z2 = 1. Find the shape of the surface
at t=2 s and show the 3D animation of the flow and shape changing of the
sphere.
a) Solution
a, pdf-file
b) Solution b
with animation (Maple)