Question #42104

An incompressible, electrically conducting fluid of density $\rho$ and electrical conductivity $\sigma$ flows steadily along the $+x$-direction through a horizontal Venturi tube of length $L$. The cross-sectional area of the tube decreases exponentially as $A(x) = A_0 e^{-x/L}$ from $x=0$ to $x=L$. The fluid velocity at the inlet ($x=0$) is $v_0$. A non-uniform transverse magnetic field $\vec{B}(x) = B_0 e^{-x/2L} \hat{k}$ is applied across the tube. The side walls of the tube are electrically short-circuited such that the induced electric field is zero. Assuming one-dimensional inviscid flow, what is the pressure drop $\Delta P = P(0) - P(L)$ across the tube?