Pharos University ME 253 Fluid Mechanics 2 Revision for Mid-Term Exam Dr. 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

Kinematics of Fluid Flow

Stream Function for Two-Dimensional Incompressible Flow Two-Dimensional Flow Stream Function y

Stream Function for Two-Dimensional Incompressible Flow Cylindrical Coordinates Stream Function y(r,q)

Is this a possible flow field

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

Determine the vorticity of flow field described by Is this flow irrotational?

Momentum Equation Newtonian Fluid: Navier–Stokes Equations

Example exact solution Poiseuille 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 Plates are infinite in x and z Flow is steady, /t = 0 Parallel flow, V=0 Incompressible, Newtonian, laminar, constant properties No pressure gradient 2D, W=0, /z = 0 Gravity acts in the -z direction, Boundary conditions Bottom plate (y=0) : u=0, v=0, w=0 Top plate (y=h) : u=V, v=0, w=0

Fully Developed Couette Flow Note: these numbers refer to the assumptions on the previous slide Step 3: Simplify 3 6 Continuity This means the flow is “fully developed” or not changing in the direction of flow 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 3 3 Z-momentum 2,6 6 6 6 7 6 6 6

Fully Developed Couette Flow Step 4: Integrate X-momentum integrate integrate Z-momentum 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) Hydrostatic pressure 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 satisfied 0 + 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

Inviscid Flow for Steady incompressible For steady incompressible flow, the equation reduces to where  = constant. Integrate from a reference at  along any streamline =C :

Two-Dimensional Potential Flows Therefore, there exists a stream function such that in the Cartesian coordinate and in the cylindrical coordinate

Potential Flow

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 .

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.

Summary Elementary Potential Flow Solutions y f Uniform Stream U∞y U∞x Source/Sink mq mln(r) Vortex -Kln(r) Kq 26 26

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