Parabolic Inlet Flow in a Piecewise-Radius Pipe

A steady, incompressible viscous flow is modeled in a cylindrical pipe whose radius changes along the axis in three segments. A parabolic axial inflow is prescribed at the inlet, no-slip conditions are applied on the pipe walls, and a reference pressure is set at the outlet. The computed radial and axial velocity components and the pressure field are visualized in both a 2D axisymmetric cross-section and a 3D reconstruction.

Solve the steady incompressible Navier–Stokes equations in a cylindrical pipe whose radius changes along the axis. The model is formulated in axisymmetric (r, z) coordinates, so the solution represents a rotationally symmetric 3D flow. The unknown fields are the radial velocity u(r, z), axial velocity w(r, z), and pressure p(r, z):

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{rVel,zVel,pressure}=NDSolveValue{FluidFlowPDEComponent[{{u[r,z],0,w[r,z],p[r,z]},{r,θ,z}},"DynamicViscosity"0.01,"MassDensity"1,"RegionSymmetry""Axisymmetric"]{0,0,0},{DirichletCondition[{u[r,z]0,w[r,z]0.1(1.5^2-r^2)},z0],DirichletCondition[{u[r,z]0,w[r,z]0},r>0&&z≠0&&z≠9],DirichletCondition[u[r,z]0,z9],DirichletCondition[u[r,z]0,r0],DirichletCondition[p[r,z]0,r0&&z9]}},{u[r,z],w[r,z],p[r,z]},{r,z}∈

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Plot the velocity field in the axisymmetric cross-section:

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VectorPlot{rVel,zVel},{r,z}∈

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Visualize the flow in 3D:

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StreamPlot3DEvaluate[{Cos[ArcTan[x,y]]*rVel/.{rSqrt[x^2+y^2]},Sin[ArcTan[x,y]]*rVel/.{rSqrt[x^2+y^2]},zVel/.{rSqrt[x^2+y^2]}}],{x,y,z}∈

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Parabolic Inlet Flow in a Piecewise-Radius Pipe

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