Fluid Mechanics-Frank M White Solution Ch8

소개글

Fluid Mechanics-Frank M White Solution Ch8에 대한 보고서 자료입니다.

본문내용

8.1 Prove that the streamlines ψ (r, θ ) in polar coordinates, from Eq. (8.10), are orthogonal to the potential lines φ (r, θ ).
Solution: The streamline slope is represented by



Since the ψ − slope = −1/( φ − slope), the two sets of lines are orthogonal. Ans.

8.2 The steady plane flow in the figure has the polar velocity components v θ = Ωr and vr = 0. Determine the circulation Γ around the path shown.
Solution: Start at the inside right corner, point A, and go around the complete path: Fig. P8.2
Γ = V⋅ds=0  ∫ (R2 −R1)+ΩR2( π R2)+0(R1−R2)+ΩR1(− π R1) or: Γ = π Ω R2 2 −R1 2( ) Ans.

8.3 Why is a problem of potential flow more difficult to solve when it involves a boundary on which the pressure is prescribed? Solution: If the pressure is prescribed on a boundary, the problem will involve the Bernoulli’s equation, which is nonlinear in the velocity. The problem then becomes nonlinear, and is more difficult to solve than a problem purely in terms of the velocity potential.

8.4 Using cartesian coordinates, show that each velocity component (u, v, w) of a potential flow satisfies Laplace’s equation separately if ∇2 φ = 0. Solution: This is true because the order of integration may be changed in each case:


2
8.5 Is the function 1/r a legitimate velocity potential in plane polar coordinates? If so, what is the associated stream function ψ (r, θ )? Solution: Evaluation of the laplacian of (1/r) shows that it is not legitimate:


P8.6 A proposed harmonic function F(x, y, z) is given by

(a) If possible, find a function f(y) for which the Laplacian of F is zero. If you do indeed solve part (a), can your final function F serve as (b) a velocity potential, or (c) a stream function? Solution: Evaluate ∇2F and see if we can find a suitable f(y) to make it zero:

Solving for f(y) eliminated y3, which is not a harmonic function. (b) Since ∇2F = 0, it can indeed be a velocity potential, although the writer does not think it is a realistic flow pattern. (c) Since F is three-dimensional, it cannot be a stream function.

8.7 Given the plane polar coordinate velocity potential
φ = Br2cos(2 θ ), where B is a constant. (a) Show that a stream function also exists. (b) Find the algebraic form of ψ (r, θ ). (c) Find any stagnation points in this flow field.
Solution: (a) First find the velocities from φ and then check continuity:



Yes, continuity is satisfied, so ψ exists. Ans.(a) (b) Find ψ from its definition:


3
(c) By checking to see where vr and v θ = 0 from part (a), we find that the only stagnation point is at the origin of coordinates, r = 0. These functions define plane stagnation flow, Fig. 8.19b.

키워드

인하대,   유체역학,   솔루션