Suppose the average speed of blood in the aorta is $0.26\text{ m/s}$, and the diameter is 2.0 cm. What is the pressure gradient in this aorta due to the viscosity of blood (assume a human body temperature)?

• A. $9.1\text{ Pa/m}$
• B. $18\text{ Pa/m}$
• C. $21\text{ Pa/m}$
• D. $23\text{ Pa/m}$
• E. $230\text{ Pa/m}$
• F. $73\text{ Pa/m}$

Answer: (F)

Verified step by step guidance

1. First, understand that the problem involves calculating the pressure gradient in the aorta due to the viscosity of blood. This can be approached using the Hagen-Poiseuille equation, which relates the pressure drop to the flow rate, viscosity, and dimensions of the tube.
2. The Hagen-Poiseuille equation is given by: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>&#x0394;p</mi></mrow><mrow><mi>L</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn><mi>&#x03BC;</mi><mi>Q</mi></mrow><mrow><mi>&#x03C0;</mi><msup><mi>r</mi><mn>4</mn></msup></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x0394;p</mi></math> is the pressure drop, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the length of the tube, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math> is the dynamic viscosity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> is the flow rate, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is the radius of the tube.
3. Calculate the radius of the aorta from the given diameter. The diameter is 2.0 cm, so the radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is half of that, which is 1.0 cm or 0.01 m.
4. Next, determine the flow rate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math>. The flow rate can be calculated using the formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>v</mi><mi>A</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> is the average speed of blood and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is the cross-sectional area of the aorta. The area <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> can be calculated using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>&#x03C0;</mi><msup><mi>r</mi><mn>2</mn></msup></math>.
5. Finally, substitute the values into the Hagen-Poiseuille equation to find the pressure gradient. Use the known viscosity of blood at human body temperature, which is approximately 0.004 Pa·s, and solve for the pressure gradient <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>&#x0394;p</mi></mrow><mrow><mi>L</mi></mrow></mfrac></math>.