Consider blood flow in an artery. Blood is nonNewtonian; the...

Consider blood flow in an artery. Blood is nonNewtonian; the shear stress versus shear rate is described by the Casson relationship: $$\left\{\begin{array}{ll} \sqrt{\tau}=\sqrt{\tau_{c}}+\sqrt{\mu \frac{d u}{d r}} & \text { for } \tau \geq \tau_{c} \\ \tau=0 & \text { for } \tau<\tau_{c} \end{array}\right.$$ where $\tau_{e}$ is the critical shear stress, and $\mu$ is a constant having the same dimensions as dynamic viscosity. The Casson relationship shows a linear relationship between $\sqrt{\tau}$ and $\sqrt{d u / d r},$ with intercept $\sqrt{\tau_{c}}$ and slope $\sqrt{\mu} .$ The Casson relationship approaches Newtonian behavior at high values of deformation rate. Derive the velocity profile of steady fully developed blood flow in an artery of radius $R$. Determine the flow rate in the blood vessel. Calculate the flow rate due to a pressure gradient $d p / d x=-100 \mathrm{Pa} / \mathrm{m},$ in an artery of radius $R=1 \mathrm{mm},$ using the following blood data: $\mu=3.5 \mathrm{cP}$ $\tau_{c}=0.05$ dynes $/ \mathrm{cm}^{2}$

Consider blood flow in an artery. Blood is nonNewtonian; the shear stress versus shear rate is described by the Casson relationship:
$$\left\{\begin{array}{ll}
\sqrt{\tau}=\sqrt{\tau_{c}}+\sqrt{\mu \frac{d u}{d r}} & \text { for } \tau \geq \tau_{c} \\
\tau=0 & \text { for } \tau<\tau_{c}
\end{array}\right.$$
where $\tau_{e}$ is the critical shear stress, and $\mu$ is a constant having the same dimensions as dynamic viscosity. The Casson relationship shows a linear relationship between $\sqrt{\tau}$ and $\sqrt{d u / d r},$ with intercept $\sqrt{\tau_{c}}$ and slope $\sqrt{\mu} .$ The Casson relationship approaches Newtonian behavior at high values of deformation rate. Derive the velocity profile of steady fully developed blood flow in an artery of radius $R$. Determine the flow rate in the blood vessel. Calculate the flow rate due to a pressure gradient $d p / d x=-100 \mathrm{Pa} / \mathrm{m},$ in an artery of radius $R=1 \mathrm{mm},$ using the following blood data: $\mu=3.5 \mathrm{cP}$ $\tau_{c}=0.05$ dynes $/ \mathrm{cm}^{2}$

Key Concepts

Non-Newtonian Fluids

Non-Newtonian fluids exhibit a shear stress–shear rate relationship that deviates from the simple linear behavior of Newtonian fluids. Their viscosity can change with the rate of deformation, and many biological fluids, including blood, display such behavior. Understanding these fluids is crucial when modeling real-world applications where fluid properties vary with applied stress or strain rate.

Casson Fluid Model

The Casson fluid model is a constitutive equation used to describe the flow of certain non-Newtonian fluids, particularly those with a yield stress, like blood. It relates the square root of the shear stress to the square root of the shear rate, featuring a yield stress term as the intercept and a parameter related to the fluid's viscosity as the slope. This model simplifies the behavior of complex fluids by providing a linear relationship between the square roots of these variables.

Yield Stress

Yield stress is the minimum stress required to initiate flow in a material. In the context of the Casson model, it represents the threshold below which the fluid behaves as a solid and does not flow. When the applied stress exceeds this yield stress, the fluid begins to deform and flow, leading to the characteristic plug flow region observed in many non-Newtonian fluids.

Shear Stress and Shear Rate Relationship

The relationship between shear stress and shear rate is fundamental in rheology. For non-Newtonian fluids such as those described by the Casson model, this relationship is non-linear and typically involves a yield stress. This relationship determines how forces applied to the fluid translate into flow, influencing the velocity profile and overall dynamics of the system.

Steady Fully Developed Flow

Steady fully developed flow refers to a flow regime where the velocity profile does not change with time or along the direction of flow. In cylindrical coordinates, this concept allows for the simplification of the momentum balance equations, assuming that variations occur only in the radial direction and that the flow is driven by a constant pressure gradient. This concept is key in deriving analytical expressions for the velocity profile in a tube or artery.

Velocity Profile Derivation

The derivation of the velocity profile in a flow system involves integrating the momentum balance equations under appropriate boundary conditions, such as no-slip at the wall and symmetry at the centerline. In non-Newtonian flow models like the Casson fluid, the integration must account for the modified relationship between shear stress and shear rate, leading to distinct regions in the profile if a yield stress is present.

Pressure Driven Flow

Pressure driven flow is a common flow mechanism in which a pressure gradient causes the fluid to move through a conduit. This concept is pivotal in analyzing flows in pipes and arteries, as the imposed pressure difference directly dictates the shear stress distribution and, consequently, the flow characteristics of the fluid.

Flow Rate Calculation

Calculation of the flow rate involves integrating the velocity profile across the cross-sectional area of the conduit. This integration provides a measure of the volume of fluid transported per unit time. For non-Newtonian fluids, especially those with a yield stress, the integration must consider regions of plug flow and regions where shear occurs, making the analysis more complex than in Newtonian fluids.

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