Near a corner of a laboratory water channel the flow is modelled by the velocity…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Potential flow, complex potentials and lift

Problem

Near a corner of a laboratory water channel the flow is modelled by the velocity potential ϕ=C(x3−3xy2),C=0.50 m−1 s−1,\phi = C\left(x^{3} - 3xy^{2}\right),\qquad C = 0.50\ \mathrm{m^{-1}\,s^{-1}} , with xx and yy in metres and ϕ\phi in m2 s−1\mathrm{m^{2}\,s^{-1}}. The flow is two-dimensional, steady, incompressible and inviscid, the water has density ρ=1000 kg m−3\rho = 1000\ \mathrm{kg\,m^{-3}}, and all the points considered lie in the same horizontal plane, so the gravity term is the same at each. ((a)) Evaluate ∇2ϕ\nabla^{2}\phi and the vorticity component ωz=∂v/∂x−∂u/∂y\omega_z = \partial v/\partial x - \partial u/\partial y for this field, and say what each result licenses. ((b)) Find the flow speed at A=(1.00,0)A = (1.00, 0) m and at B=(0.50,0.50)B = (0.50, 0.50) m. ((c)) Find pB−pAp_B - p_A.

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