ANS:A - V/V_max = (x/r)^1/7

For turbulent fluid flow in a pipe, the expression for Prandtl's one-seventh power law relating the velocity VVV at distance xxx from the pipe wall to the maximum velocity VmaxV_{\text{max}}Vmax​ is: VVmax=(xr)1/7\frac{V}{V_{\text{max}}} = \left( \frac{x}{r} \right)^{1/7}Vmax​V​=(rx​)1/7

Explanation:

1. Prandtl's One-Seventh Power Law:
   • Prandtl's one-seventh power law describes the radial velocity profile in fully developed turbulent flow in a pipe.
   • According to this law, the velocity VVV at a distance xxx from the centerline (where rrr is the pipe radius) varies with the seventh root of the radial distance xxx.
   • This means that as you move away from the centerline towards the pipe wall, the velocity decreases following a power law relationship.
2. Options Analysis:
   • VVmax=(xr)1/7\frac{V}{V_{\text{max}}} = \left( \frac{x}{r} \right)^{1/7}Vmax​V​=(rx​)1/7: This correctly represents Prandtl's one-seventh power law, where xxx is the radial distance from the pipe centerline and rrr is the pipe radius.
   • VVmax=(rx)1/7\frac{V}{V_{\text{max}}} = \left( \frac{r}{x} \right)^{1/7}Vmax​V​=(xr​)1/7: This would describe an inverse relationship, which is not consistent with Prandtl's law.
   • VVmax=(x⋅r)1/7\frac{V}{V_{\text{max}}} = \left( x \cdot r \right)^{1/7}Vmax​V​=(x⋅r)1/7: This would imply a multiplication, which is not the form of Prandtl's law.
   • None of these: This option is incorrect because the correct form of Prandtl's one-seventh power law is represented by the first option.

Conclusion: