Pharos UniversityME 253 Fluid Mechanics 2

Revision for Mid-Term ExamDr. A. Shibl

Streamlines• A Streamline is a curve that is

everywhere tangent to the instantaneous local velocity vector.

• Consider an arc length

• must be parallel to the local velocity vector

• Geometric arguments results in the equation for a streamline

dr dxi dyj dzk

dr

V ui vj wk

dr dx dy dz

V u v w

Kinematics of Fluid Flow

1 1 1

2 2 2

w v u w v ui j k

y z z x x y

1xx yy zz

DV u v w

V Dt x y z

1 1 1, ,

2 2 2xy zx yz

u v w u v w

y x x z z y

2

Stream Function forTwo-Dimensional

Incompressible Flow

• Two-Dimensional Flow

Stream Function

Stream Function forTwo-Dimensional

Incompressible Flow

• Cylindrical Coordinates

Stream Function (r,)

Is this a possible flow field

Given the y-component Find the X- Component of the velocity,

Determine the vorticity of flow field described byIs this flow irrotational?

Momentum Equation

• Newtonian Fluid: Navier–Stokes Equations

Example exact solutionPoiseuille Flow

Example exact solution Fully Developed Couette Flow

• For the given geometry and BC’s, calculate the velocity and pressure fields, and estimate the shear force per unit area acting on the bottom plate

• Step 1: Geometry, dimensions, and properties

Fully Developed Couette Flow

• Step 2: Assumptions and BC’s– Assumptions

1. Plates are infinite in x and z2. Flow is steady, /t = 03. Parallel flow, V=04. Incompressible, Newtonian, laminar, constant properties5. No pressure gradient6. 2D, W=0, /z = 07. Gravity acts in the -z direction,

– Boundary conditions1. Bottom plate (y=0) : u=0, v=0, w=02. Top plate (y=h) : u=V, v=0, w=0

Fully Developed Couette Flow

• Step 3: Simplify 3 6

Note: these numbers referto the assumptions on the previous slide

This means the flow is “fully developed”or not changing in the direction of flow

Continuity

X-momentum

2 Cont. 3 6 5 7 Cont. 6

Fully Developed Couette Flow

• Step 3: Simplify, cont.Y-momentum

2,3 3 3 3,6 7 3 33

Z-momentum

2,6 6 6 6 7 6 66

Fully Developed Couette Flow

• Step 4: Integrate

Z-momentum

X-momentum

integrate integrate

integrate

Fully Developed Couette Flow

• Step 5: Apply BC’s– y=0, u=0=C1(0) + C2 C2 = 0

– y=h, u=V=C1h C1 = V/h

– This gives

– For pressure, no explicit BC, therefore C3 can remain an arbitrary constant (recall only P appears in NSE).• Let p = p0 at z = 0 (C3 renamed p0)

1. Hydrostatic pressure2. Pressure acts independently of flow

Fully Developed Couette Flow

• Step 6: Verify solution by back-substituting into differential equations– Given the solution (u,v,w)=(Vy/h, 0, 0)

– Continuity is satisfied0 + 0 + 0 = 0

– X-momentum is satisfied

Fully Developed Couette Flow

• Finally, calculate shear force on bottom plate

Shear force per unit area acting on the wall

Note that w is equal and opposite to the shear stress acting on the fluid yx (Newton’s third law).

Momentum Equation

• Special Case: Euler’s Equation

20

Inviscid Flow for Steady incompressibleInviscid Flow for Steady incompressible

• For steady incompressible flow, the equation reduces to

where = constant.

• Integrate from a reference at along any streamline =C :

gvv p)(0 v

constant22

v 22

gzvp

gzp

21

Two-Dimensional Potential FlowsTwo-Dimensional Potential Flows

• Therefore, there exists a stream function such that in the Cartesian coordinate and

in the cylindrical coordinate

r

,r

v,ur

xy

vu

,,

Potential Flow

23

Two-Dimensional Potential Flows• The potential function and the stream function are conjugate pair

of an analytical function in complex variable analysis.

• The constant potential line and the constant streamline are orthogonal, i.e.,

and

to imply that .

xyyx

and

u,v- v,u

0

Stream and Potential Functions

• If a stream function exists for the velocity field u = a(x2 -- y2) & v = - 2axy & w = 0

Find it, plot it, and interpret it.• If a velocity potential exists for this velocity field.

Find it, and plot it.

yv

xu

xy

vu

,,

Summary• Elementary Potential Flow Solutions

26

Uniform Stream U∞y U∞x

Source/Sink m mln(r)

Vortex -Kln(r) K