Which of the following equations represents the velocity profile of a fluid flowing through a tube according to Poiseuille's law?

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:

$$v(r) = \frac{\Delta P}{4 \eta L} (R^2 - r^2)$$

where:

• $\Delta P$ is the pressure difference across the length of the tube.
• $\eta$ 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 ($0 \le r \le R$).

Now, let's compare this standard velocity profile equation with the given options:
A. $\frac{P x^2}{4L}$
B. $\frac{P(a^2 - x^2)}{4L}$
C. $\frac{P(x^2 + a^2)}{4L}$
D. $\frac{P(x^2 - a^2)}{4L}$

If we interpret the variables in option B as follows:

• $P$ represents $\frac{\Delta P}{\eta}$ (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:

$$v(x) = \frac{(\Delta P / \eta)}{4L} (R^2 - x^2) = \frac{\Delta P}{4 \eta L} (R^2 - x^2)$$

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 $\boxed{\text{B}}$.