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Why we are able to suck more drink through a larger diameter straw than a smaller diameter straw if $p_1 v_1 = p_2 v_2 = Q$ as per Bernoulli's Principle. The pressure difference I create in mouth while sucking will be almost same, but I get more liquid in my mouth with a larger diameter than a smaller one.

Is there any correlation between diameter of the pipe & flow rate, even when the pressure difference is the same?

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If we assume the flow is laminar, the flow in the straw is best described by Poiseuille flow. The linked website will tell you, that the pressures drop scales with the flow-rate $Q$ and diameter $d$ as

$$ \Delta p \propto \frac{Q}{d^4} $$

The proportionality further depends on the length of the straw and the viscosity of the fluid. You can directly derive it from the Navier-Stokes equations.

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