This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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The question asks for a suitable expression for Poiseuille's equation for the flow of liquid through a tube. Poiseuille's law describes the laminar flow of an incompressible Newtonian fluid through a cylindrical pipe. One of the key results of Poiseuille's law is the velocity profile of the fluid within the tube. The velocity v of the fluid at a radial distance r from the center of the tube is given by the equation: v(r) = ( P)/(4 L) (R^2 - r^2) where: P is the pressure difference across the length of the tube. is the dynamic viscosity of the fluid. L is the length of the tube. R is the radius of the tube. r is the radial distance from the center of the tube (0 r R). Now, let's compare this standard velocity profile equation with the given options: A. (P x^2)/(4L) B. (P(a^2 - x^2))/(4L) C. (P(x^2 + a^2))/(4L) D. (P(x^2 - a^2))/(4L) If we interpret the variables in option B as follows: P represents ( P)/() (the pressure difference divided by viscosity). a represents the radius of the tube R. x represents the radial distance r from the center. Then, option B becomes: v(x) = (( P / ))/(4L) (R^2 - x^2) = ( P)/(4 L) (R^2 - x^2) This exactly matches the velocity profile equation derived from Poiseuille's law. The velocity is maximum at the center (x=0) and zero at the walls (x=a=R). The other options do not correctly represent the parabolic velocity profile of laminar flow in a pipe. The final answer is B. That's 2 down. 3 left today — send the next one.