Fluid Dynamics | Part IB, 2002

Use Euler's equation to derive Bernoulli's theorem for the steady flow of an inviscid fluid of uniform density ρ\rho in the absence of body forces.

Such a fluid flows steadily through a long cylindrical elastic tube having circular cross-section. The variable zz measures distance downstream along the axis of the tube. The tube wall has thickness h(z)h(z), so that if the external radius of the tube is r(z)r(z), its internal radius is r(z)h(z)r(z)-h(z), where h(z)0h(z) \geqslant 0 is a given slowly-varying function that tends to zero as z±z \rightarrow \pm \infty. The elastic tube wall exerts a pressure p(z)p(z) on the fluid given as


where p0,kp_{0}, k and RR are positive constants. Far upstream, rr has the constant value RR, the fluid pressure has the constant value p0p_{0}, and the fluid velocity uu has the constant value VV. Assume that gravity is negligible and that h(z)h(z) varies sufficiently slowly that the velocity may be taken as uniform across the tube at each value of zz. Use mass conservation and Bernoulli's theorem to show that u(z)u(z) satisfies

hR=1(Vu)1/2+14λ[1(uV)2], where λ=2ρV2kR\frac{h}{R}=1-\left(\frac{V}{u}\right)^{1 / 2}+\frac{1}{4} \lambda\left[1-\left(\frac{u}{V}\right)^{2}\right], \quad \text { where } \quad \lambda=\frac{2 \rho V^{2}}{k R}

Sketch a graph of h/Rh / R against u/Vu / V. Show that if h(z)h(z) exceeds a critical value hc(λ)h_{c}(\lambda), no such flow is possible and find hc(λ)h_{c}(\lambda).

Show that if h<hc(λ)h<h_{c}(\lambda) everywhere, then for given hh the equation has two positive solutions for uu. Explain how the given value of λ\lambda determines which solution should be chosen.

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