Flow Through Constricted or Dilated Pipes and Channels: Part 1

Magnetohydrodynamic channel flows with weak transverse magnetic fields

“…The viscous wall layers are unchanged from the hydrodynamic solution of Smith [3,4]. Including a Prandtl transposition to create a body fitted structured grid, the lower wall layer has x = Re X, t = Re 2/3 T and y = 1/3 (Y + hF L ), with expansions given by…”

“…In addition, for each value of H in figure 3, the hump height is increased from h = 0 to h = 4 in 20 increments in order to ensure stability of the solutions at larger hump height. A comparison between the wall layer solution with H = 0 and the solution of Smith [3] is shown in figure 2 for the hump F U = F L = X exp(−X 2 /32) when X > 0, with h = 2.5. Typical streamfunction solutions for the steady MHD wall layer problem are shown in figure 3 for the Gaussian hump F U = F L = exp(−3X 2 ), with h = 4.…”

“…When Ha ∼ O(1), equation (2.4) may be used as the upstream flow in a slowly varying channel, a situation that allows a direct interaction between the applied magnetic field and the viscous flow throughout the channel. In particular, when Re 1, (x, y, t) = (Re X, y, Re T) and (u, v, p) ∼ (U, Re −1 V, P) + · · · , then the flow is known to be governed by the following boundary layer problem (see Genin & Podshibyakin [5], which is also documented in Blūms et al [3] for zero magnetic field (Ha = 0).…”

“…In fact, a strong near-wall response lies at the heart of the high Reynolds number theory of viscous flow through a slowly varying channel. In the hydrodynamic flow of Smith [3,4], the velocities where Re = ρV 0 L/μ is the Reynolds number, Rm = μ 0 σ V 0 L is the magnetic Reynolds number and Ha = B 0 L √ σ μ is the Hartmann number. In all cases considered here, the magnetic Reynolds number is sufficiently close to O(1) values.…”

“…On streamwise scales shorter than O(Re) and longer than O(Re 1/7 ), the hydrodynamic solution has been given by Smith [3,4]. In this study, we assume that 1.…”