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Computing Prandtl number, Reynolds number, and Nusselt number for blood flow in nerves

To compute the Prandtl number, Reynolds number, and Nusselt number for blood flow in nerves, we need to first define some important quantities and make necessary assumptions for scale analysis.

Blood vessel

Let’s assume typical values for the properties of blood and the scale of nerve capillaries.

Assumed Values and Dimensions

• Dynamic viscosity of blood (μ): 3×10−3Pa⋅s
• Density of blood (ρ): 1060kg/m3
• Thermal conductivity of blood (k): 0.5W/mK
• Specific heat capacity of blood (cp​): 3600J/kgK
• Typical diameter of capillary (D): 10μm=10×10−6m
• Velocity of blood in capillary (U): 1mm/s=1×10−3m/s
• Characteristic temperature difference (ΔT): Assume 1K (small temperature gradient).

Reynolds Number (Re):

The Reynolds number gives an idea of the flow regime, whether it is laminar or turbulent. It is defined as:

Re = ρUD​/μ

Substituting the values:

Re = (1060kg/m3)*(1×10−3m/s)*(10×10−6m)​ / 3×10−3Pa⋅s

Re ≈ 3.53×10−3

Thus, the Reynolds number for blood flow in nerves is approximately 0.00353, indicating that the flow is highly laminar (since it is much less than 2300).

Prandtl Number (Pr):

The Prandtl number relates the momentum diffusivity (viscosity) to thermal diffusivity and is defined as:

Pr = cp​μ​/k

Substituting the assumed values:

Pr = (3600J/kgK)*(3×10−3Pa⋅s)/0.5W/mK

Pr = 21.6

The Prandtl number for blood flow in nerves is approximately 21.6, which indicates that momentum diffusion dominates over thermal diffusion.

Nusselt Number (Nu):

The Nusselt number is a dimensionless number representing the ratio of convective to conductive heat transfer across a boundary and is defined as:

Nu=hD​/k

Where:

• h is the convective heat transfer coefficient.
• D is the characteristic length (capillary diameter).
• k is the thermal conductivity.

For laminar flow in a circular tube (with Re<2300), the Nusselt number can be approximated by the correlation for fully developed laminar flow:

Nu ≈ 3.66 (for constant wall temperature)

Thus, assuming fully developed flow, the Nusselt number is approximately 3.66 for blood flow in capillaries.

Scale Analysis Conclusion:

• The Reynolds number (Re=0.00353) indicates laminar flow in capillaries.
• The Prandtl number (Pr=21.6) suggests that viscous effects dominate over thermal diffusivity.
• The Nusselt number (Nu≈3.66) implies the convection in this flow is moderate, and conductive heat transfer is more significant than convective.

Article Prepared by:

This Article is jointly prepared by the following students of IIT BHU (Varanasi).

• Mukesh Kumar Verma (21134018)
• Sanjay Yadav (21135161)
• Rohit Yadav (21135112)
• P.Abid Singh Rajput(21134021)
• Satish Chandra(21135157)