Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3,500 dynes/cm² and viscosity = 0.028. Find the velocity of the blood at radius r= 0.002. Round your answer to the nearest whole number. The velocity of the blood is ___ cm/s.
Answer (2)
Identify the Law of Laminar Flow formula: v = 4 η L P ( R 2 − r 2 ) .
Substitute given values: P = 3500 , η = 0.028 , L = 3 , R = 0.01 , r = 0.002 .
Calculate the velocity: v = 4 × 0.028 × 3 3500 (( 0.01 ) 2 − ( 0.002 ) 2 ) = 1 .
Round the result to the nearest whole number: 1 .
Explanation
Problem Analysis We are given the Law of Laminar Flow scenario and asked to find the velocity of blood in a vessel at a specific radius. We have all the necessary parameters: the radius of the blood vessel, its length, the pressure difference, the viscosity of the blood, and the radius at which we need to calculate the velocity.
Formula Introduction The Law of Laminar Flow is given by the formula: v = 4 η L P ( R 2 − r 2 ) where:
v is the velocity of the blood,
P is the pressure difference (3,500 dynes/cm²),
η is the viscosity (0.028),
L is the length of the vessel (3 cm),
R is the radius of the vessel (0.01 cm),
r is the radius at which the velocity is to be calculated (0.002 cm).
Substitution and Simplification Now, we substitute the given values into the formula: v = 4 × 0.028 × 3 3500 (( 0.01 ) 2 − ( 0.002 ) 2 ) Let's simplify the expression step by step. First, calculate the denominator: 4 × 0.028 × 3 = 0.336 Then, calculate the difference of squares: ( 0.01 ) 2 − ( 0.002 ) 2 = 0.0001 − 0.000004 = 0.000096 Now, substitute these values back into the formula: v = 0.336 3500 × 0.000096
Velocity Calculation Now, we perform the calculation: v = 0.336 3500 × 0.000096 = 0.336 0.336 = 1 So, the velocity v is 1 cm/s.
Rounding the Result Finally, we round the calculated velocity to the nearest whole number. Since the calculated velocity is exactly 1, rounding it to the nearest whole number gives us 1.
Final Answer Therefore, the velocity of the blood at radius r = 0.002 cm is 1 cm/s.
Examples
Understanding laminar flow is crucial in various real-world applications. For instance, in designing pipelines for oil or gas, engineers use the principles of laminar flow to optimize the flow rate and minimize energy losses due to friction. Similarly, in microfluidics, controlling laminar flow is essential for precise manipulation of fluids in lab-on-a-chip devices, enabling applications such as drug delivery and chemical analysis. In the human body, laminar flow is vital for efficient blood circulation, and deviations from it can indicate vascular diseases.
The calculated velocity of blood at a radius of 0.002 cm within a vessel is 1 cm/s, based on the application of the Law of Laminar Flow. The relevant variables such as pressure difference, viscosity, and vessel dimensions were substituted into the formula to derive this figure. The result was confirmed through step-by-step computation and rounding to the nearest whole number.
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