Two-Dimensional And Axisymmetric Motion
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PYQs
Two-Dimensional And Axisymmetric Motion
1) In an axisymmetric motion, show that stream function exists due to equation of continuity. Express the velocity components of the stream function. Find the equation satisfied by the stream function if the flow is irrotational.
[2015, 20M]
Consider the motion of fluid in cylindrical coordinates. The equation of continuity is given by:
∇⋅(ρ→q)+∂ρ∂t=0For incompressible fluid and steady flow,
∇⋅(→q)=0
⟹1r∂∂r(rqr)+1r∂∂θ(qθ)+∂∂z(qz)=0
For axi-symmetric flow, ∂∂θ=0
⇒1r∂∂r(rqr)+∂∂z(qz)=0⇒∂∂r(rqr)+r∂∂z(qz)=0
Now the condition that rqrdz−rqzdr may be an exact differential. Let it be equal to dψ.
⟹rqrdz−rqzdr=dψ=∂ψ∂rdr+∂ψ∂zdz
⟹rqr=∂ψ∂z and −rqz=∂ψ∂r
⟹qr=1r∂ψ∂z and qz=−1r∂ψ∂r
which satisfy continuity equation. Also, the streamlines are given by:
drqr=dzqz
⟹rqrdz−rqzdr=0
⟹dψ=0
⟹ψ=constant
⟹ψ exists due to equation of continuity.
If flow is irrotational, then ϕ (potential) exists, such that
→q=−∇ϕ
qr=−∂ϕ∂r, qz=−∂ϕ∂z
Also,
qr=1r∂ψ∂z, qz=−1r∂ψ∂r
Also,
∂∂z(∂ϕ∂r)=∂∂r(∂ϕ∂z)
∂∂z(−1r∂ψ∂z)=∂∂r(1r∂ψ∂r)
−1r∂2ψ∂z2=1r∂2ψ∂r2−1∂ψr2∂r
∂2ψ∂z2−1∂ψr∂r+∂2ψ∂r2=θ
2) Consider a uniform flow U0 in the positive x−direction. A cylinder of radius a is located at the origin. Find the stream function and the velocity potential. Find also the stagnation points.
[2015, 10M]
Since w=ϕ+iψ,
⟹ϕ=U0[rcosθ+a2rcosθ] and ψ=u0[rsinθ−a2rsinθ]
q=|dwdz|=u0|1−a2/z2|=u0|−a2e−i2θ/r2|⟹q=u0|1−e−i2θ|At stagnation point, q=0,
⟹1−e−i2θ=0 ⟹cos2θ−isin2θ=1 ⟹cos2θ=1 and sin2θ=0 ⟹2θ=2nπ and 2θ=nπ, n∈Z ⟹θ=mπ, m∈Z
3) Show that the velocity distribution in axial flow of viscous incompressible fluid along a pipe of annular cross-section radii r1<r2, is given by ω(r)=14μdpdz{r2−r21+r21−r21log(r2r1)log(rr2)}.
[2001, 30M]