Two-Fluid Classical and Momentumless Laminar Far Wakes

Two-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation laws for the system of partial differential equations (PDEs) in the upper and lower parts of the wake are derived. Lie point symmetries associated with the conserved vectors for the classical and momentumless wakes are obtained. The conserved quantity for the two-fluid classical wake is the total drag on the obstacle, which is rederived. A new conserved quantity for the two-fluid momentumless wake is obtained, which satisfies the condition that the total drag on the obstacle is zero. Using the conserved quantities, it is shown that the equation of the interface is y=kx12, where k is a constant and x and y are Cartesian coordinates with origin at the trailing edge of the obstacle. New invariant solutions for the two-fluid classical and momentumless wakes with k=0 are found. Both solutions depend on the dimensionless parameter χ=(ρ1μ1)/(ρ2μ2) where suffices 1 and 2 refer to the upper and lower parts of the wake. For the special case in which the kinematic viscosity ratio ν2/ν1=1, two further solutions for the two-fluid momentumless wake are derived with k=±6.