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The study investigated the effect of the angular position of the head on the blood flow in the jugular vein of giraffes. The vein considered is elastic and collapsible such that its cross-sectional area is not uniform. Transmural pressure causes the blood to move along the vein. Mathematical equations describing the flow were developed, and the vein was considered to be inclined at an angle φ to the horizontal. A finite-difference scheme was used to solve the equations of motion for the flow. The results are presented via relevant tables and plots. Our findings show that a change in the position of the head causes variation in the external pressure, which in turn causes variation in the cross-sectional area of the vein. Moreover, a drop (or increase) in the inertial pressure of the blood may cause the vein to collapse (or distend), which again triggers a change in the pressure.

Many fluid-conveying vessels in animal bodies are highly elastic, and in most cases, deform substantially as they respond to the traction (i.e., pressure and viscous stress) exerted by the fluid. Therefore, the study of flows in elastic vessels is of considerable interest and importance for many biomedical and bio-mechanical applications [

Veterinarians use mathematical modeling to explain how blood flows in and out of the brains of upright animals. The internal jugular veins are the primary venous drain for human brain; however, these veins tend to collapse because they are positioned above the heart level [

In this section, we present the continuity equation, which is also called the mass conservation equation. It is derived from the law of conservation of mass, which states that the mass remains constant in a steady-state flow (i.e., the stored mass in a controlled volume does not change). In a steady flow, the flow rate does not change with time, implying that the inflow into the controlled volume is equal to the outflow. The continuity equation can be written as

ρ ∂ ρ ∂ t + ∂ ρ u i ∂ u i = 0. (1)

where ρ is the fluid density, t is time, u is fluid velocity. For an incompressible flow,

∂ ρ ∂ t = 0, (2)

and thus the continuity equation reduces to

∂ ρ u i ∂ u i = 0, (3)

which represents the rate of change of volume of a moving fluid element per unit volume.

The momentum conservation equation is derived from the law of conservation of momentum, which states that the rate of change of momentum in a controlled volume is equal to the sum of the momentum flux into the control volume and any external forces acting on the control volume. This implies that the total momentum of a closed system is constant. Thus, the change in momentum of a small volume element of a fluid is equal to the sum of the dissipative viscous forces, pressure gradient, gravity, and other forces acting on the fluid. The general momentum conservation equation can be written in the tensor form as

ρ ( ∂ u i ∂ t ) + U i ( ∂ u i ∂ x i ) = ρ f i + ( ∂ σ i , j ∂ x i ) , (4)

where i = 1 , 2 , 3 and j = 1 , 2 , 3 are the summation variables along the x, y, and z directions, respectively. The term ρ f i represents the body forces acting on the fluid, which in this study are considered to be the pressure and gravitational forces. The first and second terms on the left side of Equation (4) represent the local and convective accelerations, respectively. For the purpose of this study, the momentum equation can be written in the form:

ρ ( ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ x ) = − ∂ p ∂ x + ρ g z + μ ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) , (5)

where z is the distance between the end points of the jugular vein, μ is blood viscosity, g is the gravitational forces and x, y are space variables. If the origin is fixed, then the muscles exert a force on the vein, which creates an upward pressure. To determine the pressure gradient, the momentum equation was evaluated at the edge of the boundary layer, where ρ → ρ ∞ . When the fluid is in equilibrium, the upward pressure gradient due to the vein muscle is balanced by the downward pressure gradient due to the variation in the fluid density. Thus, we can write

∇ p = − ∂ p ∂ x − g ρ z . (6)

The body force term in the momentum equation along the x-axis can be expressed as

∇ p = − ∂ p ∂ x − g ρ z (7)

By using Equation (6), Equation (7) can be expressed as

∇ p = ρ ∞ g − ρ g z . (8)

However,

ρ ∞ g − ρ g z = ρ β ( T 1 − T 2 ) , (9)

where β = 1 ρ ( ∂ p ∂ T ) p is the coefficient of thermal expansion.

By combining Equations (7), (8), and (9), we get

− ∇ p + ρ g z = ρ g ( 1 + z ) + β ( T 1 − T 2 ) . (10)

Equation (10) represents the total pressure gradient term in the momentum equation along the x axis, such that the momentum equation can now be written as

ρ ( ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y ) = ρ g ( 1 + z ) + β ( T 1 − T 2 ) + μ ( ∂ u ∂ x 2 + ∂ 2 u ∂ y 2 ) . (11)

The momentum equation governing the flow is nondimensionalized for convenience, such that

X = x U L , P = p ρ U , U = u U , T = t U L , Y = y U L , (12)

where U is the characteristic velocity and L is the characteristic length of the system. The derivatives of Equation (12) can thus be written as

∂ U ∂ T + U ∂ U ∂ X + V o ∂ U ∂ Y = 1 F r ( 1 + z ) + G r θ + 1 R e ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) . (13)

The shape of the vein is given by z = L sin ϕ , and hence Equation (22) becomes

∂ U ∂ T + U ∂ U ∂ X + V o ∂ U ∂ Y = 1 F r ( 1 + L sin ϕ ) + G r θ + 1 R e ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) . (14)

Equation ((14)) is used to analyze the effect of inertia on the steady flow in a collapsing vein.

The system of nonlinear equations obtained for the flow problem in this work, that is, Equation (14), was solved using the numerical method of finite differences, where the derivatives in the governing equations are replaced by their corresponding finite difference approximations.

Equation (14) was discretized using the central difference approximation for the partial derivatives with respect to space, while the forward difference approximation was used for the partial derivatives with respect to time. Term ∂ U ∂ X can be approximated at discrete node points using the Taylor series expansion of U i , j k . The Taylor series expansion of U i + 1, j k and U i − 1, j k can be expressed in terms of U i , j k and its higher-order derivatives as

U i + 1 , j k = U i , j k + Δ x U ′ i , j k + ( Δ x ) 2 2 ! U ″ i , j k + ( Δ x ) 3 3 ! U ‴ i , j k + ⋯ (15)

and

U i − 1 , j k = U i , j k − Δ x U ′ i , j k + ( Δ x ) 2 2 ! U ″ i , j k − ( Δ x ) 3 3 ! U ‴ i , j k + ⋯ (16)

By subtracting Equation (15) from Equation (14) we get

U ′ i , j k = U i + 1, j k − U i − 1, j k 2 Δ x + H . O . T . (17)

The addition of Equations (14) and (15) gives

U ″ i , j k = U i + 1, j k − 2 U i , j k + U i − 1, j k ( Δ x ) 2 + H . O . T . (18)

Note that the derivatives in Equations (15)-(17) are with respect to x. Similarly, if the derivatives are taken with respect to y, the following equations are obtained:

U ′ i , j k = U i , j + 1 k − U i , j − 1 k 2 Δ x + H . O . T (19)

and

U ″ i , j k = U i , j + 1 k − 2 U i , j k + U i , j − 1 k ( Δ y ) 2 + H . O . T . (20)

The corresponding time derivative can be written as

U ′ i + 1, j k = U i , j k + 1 − U i , j k Δ t . (21)

Finally, Equation (14) is discretized using a finite difference scheme on a uniform mesh, which consists of a plane divided into a network of uniform rectangles of width Δ x and height Δ y , as shown in

After neglecting the higher-order terms and assuming

K = 1 F r ( 1 + L sin ϕ ) + G r θ (22)

to obtain after letting let Δ x = Δ y ; then, the equation becomes

( r 1 = Δ t 2 Δ x r 2 = Δ t 2 Δ y r = Δ t R e ( Δ x ) 2 (23)

to obtain

U i , j k + 1 = − r 1 ( U i + 1, j k − U i − 1, j k ) − V 0 r 2 ( U i , j + 1 k − U i , j − 1 k ) + K Δ t + r ( U i + 1, j k − 4 U i , j k + U i − 1, j k + U i , j + 1 k + U i , j − 1 k ) + U i , j k (24)

The boundary conditions used to solve Equation (24) are as follows:

x 0 , y , t = 0 = U ( x , y , 0 ) , U ( x 0 , y , t ) = 0 , U ( x f , y , t ) = U x f = U x 0 .

U ( x , y 0 , T ) = 0 = U y 0 , U ( x , y f , t ) = sin ( π y ) = U y f .

Equation (24), along with its boundary conditions, was solved using MATLAB to obtain the numerical solutions of the flow variables. According to

According to

We observe that when the head is at an angle of 90˚, the velocity of blood changes slowly with distance along the vein. This implies that when the head is in the horizontal position, the neck muscles relax and the vein assumes an elliptical shape (i.e., the cross-sectional area of the vein is at its maximum), which consequently slows down the blood flow. As the angle decreases further (e.g., ϕ = − 45 ∘ ), the velocity starts to increase again see

In this work, we showed that the angular position of the head affects the blood flow in the jugular vein of the giraffe. The following conclusions can be drawn from our study.

1) Blood flow velocity is higher when the head is in an upright position than in the horizontal position.

2) The Reynolds number is inversely proportional to the viscosity of blood. Hence, the blood flow becomes more laminar with increasing viscosity; this in turn decreases the velocity.

3) Conversely, a more viscous and turbulent flow results from a decrease in the blood viscosity. Hemodynamic studies show that any change in the hematocrit value results in a change in the blood viscosity.

4) An increase in the angle of inclination of the head reduces the blood pressure in the vein and vice versa.

This study was partially funded by:

1) Higher Education Loans Board (HELB)-Kenya.

2) Dedan Kimathi University of Technology Staff Development fund.

The authors declare no conflicts of interest regarding the publication of this paper.

Amenya, R.O., Sigey, J.K., Maloiy, G.M.O. and Theuri, D.M. (2021) Effect of Head Position Angles on the Blood Flow in the Jugular Vein of Giraffes. World Journal of Mechanics, 11, 165-175. https://doi.org/10.4236/wjm.2021.118012