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Fundamental standing modes and their overtones play an important role incoronal seismology. We examine how a significant field-aligned flow affectsstanding modes supported by coronal loops, modeled here as cold magnetic slabs.Of particular interest are the period ratios of the fundamental to its$(n-1)$-th overtone ($P_1/nP_n$) for both kink and sausage modes, and thethreshold half-width-to-length ratio for sausage modes. For standing kinkmodes, the flow significantly reduces $P_1/nP_n$ in general, the effect beingparticularly strong for larger $n$ and when the density contrast$\rho_0/\rho_e$ between loops and their surroundings is weak. That said, evenwhen $\rho_0/\rho_e$ approaches infinity, this effect is still substantial,reducing the minimal $P_1/nP_n$ by up to 13.7% (24.5%) for $n=2$ ($n=4$)relative to the static case, when the Alfv\'en Mach number $M_A$ reaches 0.8where $M_A$ measures the loop flow speed in units of the internal Alfv\'enspeed. For standing sausage modes, though not negligible, the flow effect inreducing $P_1/nP_n$ is not as strong. However, the thresholdhalf-width-to-length ratio is considerably larger in the flowing case than itsstatic counterpart. For $\rho_0/\rho_e$ in the range $[9, 1024]$ and $M_A$ inthe range $[0, 0.5]$, an exhaustive parameter study yields that this thresholdis well fitted by our Equation (23) which involves the two parameters in asimple way. This allows one to analytically constrain the combination$(\rho_0/\rho_e, M_A)$ for a loop with known width-to-length ratio when astanding sausage oscillation is identified therein. It also allows one tofurther examine the idea of partial sausage modes, and the flow is found toreduce significantly the spatial extent where partial modes are allowed.

Effects of field-aligned flows on standing kink and sausage modes supported by coronal loops