A pressure difference of 6.00 * 104 Pa is required to maintain a volume flow rate of 0.800m3/s for a viscous fluid flowing through a section of cylindrical pipe that has radius 0.210 m. What pressure difference is required to maintain the same volume flow rate if the radius of the pipe is decreased to 0.0700 m?

Verified step by step guidance

1. Understand that the problem involves fluid dynamics, specifically the flow of a viscous fluid through a cylindrical pipe. The relationship between pressure difference, volume flow rate, and pipe radius can be described by the Hagen-Poiseuille equation.
2. Recall the Hagen-Poiseuille equation: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Q</mi><mo>=</mo><frac><mrow><mn>&#x3C0;</mn><msup><mi>r</mi><mn>4</mn></msup><mi>&#x394;p</mi></mrow><mrow><mn>8</mn><mi>&#x3BC;</mi><mi>L</mi></mrow></frac></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> is the volume flow rate, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is the radius of the pipe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x394;p</mi></math> is the pressure difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3BC;</mi></math> is the dynamic viscosity, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the length of the pipe.
3. Since the volume flow rate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> and the dynamic viscosity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3BC;</mi></math> are constant, and assuming the length of the pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is unchanged, the pressure difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x394;p</mi></math> is proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>-4</mn></msup></math>. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x394;p</mi><mi>&#x394;p&#x2032;</mi></mfrac></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>r&#x2032;</mi><mn>4</mn></msup><msup><mi>r</mi><mn>4</mn></msup></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r&#x2032;</mi></math> is the new radius.
4. Substitute the given values into the proportional relationship: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x394;p</mi><mi>&#x394;p&#x2032;</mi></mfrac></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mn>0.0700</mn><mn>4</mn></msup><msup><mn>0.210</mn><mn>4</mn></msup></mfrac></math>. Calculate the ratio of the fourth powers of the radii.
5. Use the calculated ratio to find the new pressure difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x394;p&#x2032;</mi></math> by multiplying the original pressure difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6.00</mn><mo>&#xD7;</mo><msup><mn>10</mn><mn>4</mn></msup></math> Pa by the ratio obtained in the previous step.

Key Concepts

Poiseuille's Law

Poiseuille's Law describes the flow of a viscous fluid through a cylindrical pipe. It states that the volume flow rate is proportional to the pressure difference and the fourth power of the pipe's radius, and inversely proportional to the fluid's viscosity and the pipe's length. This law is crucial for understanding how changes in pipe radius affect flow rate and pressure.

Viscosity

Viscosity is a measure of a fluid's resistance to flow. It affects how easily a fluid moves through a pipe, with higher viscosity indicating greater resistance. In the context of this problem, viscosity remains constant, but it plays a key role in determining the relationship between pressure difference and flow rate according to Poiseuille's Law.

Pressure Difference

Pressure difference is the driving force that causes fluid to flow through a pipe. It is the difference in pressure between two points along the pipe. In this problem, understanding how the pressure difference must change when the pipe's radius is altered is essential for maintaining the same flow rate, as dictated by Poiseuille's Law.