A steady two-dimensional incompressible flow has stream function $\psi=xy^3$ (SI units), density 1 kg/m³ and dynamic viscosity 0.1 kg/(m s). At (x,y)=(1,1) m, find the magnitude of the x pressure gradient in Pa/m (one decimal place).
Using $u=\psi_y=3xy^2$ and $v=-\psi_x=-y^3$, the x acceleration is $uu_x+vu_y=9xy^4-6xy^4=3xy^4$. Also ∇²u=6x. With no x body force, $p_x=\mu\nabla^2u-\rho(uu_x+vu_y)=0.6-3=-2.4$ Pa/m, giving magnitude 2.4. The opposite stream-function sign convention changes the viscous contribution relative to the acceleration and gives magnitude 3.6. The official grading accepts both 2.3–2.5 and 3.5–3.7; both ranges are preserved.
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