Skip to main content

Impact of thermodynamical rotational flow of cerebrospinal fluid in the presence of elasticity



To explore the experimental justification of cerebrospinal fluid (CSF) amplitude and elastic fluctuations of ventricles, we extend our previous computational study to models with rotational flow and suitable boundary conditions. In the present study, we include an elastic effect due to the interaction with the thermal solutal model which accounts for CSF motion which flows rotationally due to hydrocephalus flows within the spinal canal.


Using an analytical pertubation method, we have attempted a new model to justify CSF flow movement using the influences of wall temperature difference.


This paper presents results from a computational study of the biomechanics of hydrocephalus, with special emphasis on a reassessment of the parenchymal elastic module. CSF amplitude in hydrocephalus patients is 2.7 times greater than that of normal subjects.


This finding suggests a non-linear mechanical system to present the hydrocephalic condition using a numerical model. The results can be useful to relieve the complexities in the mechanism of hydrocephalus and can shed light to support clinically for a convincing simulation.

Peer Review reports


Hydrocephalus is the imbalance between the production and absorption of cerebrospinal fluid (CSF). The prevalence of hydrocephalus is 0.3% to 0.5% for every 1000 patients per year [1]. There are many complexities in the brain pathology and mechanism of hydrocephalus. Due to the limitations of in vivo studies, it is not possible to study this disorder in various clinical conditions since it leads to damage to their health condition. it is not possible to take accurate undistorted measurements while using a clinical method. Therefore, reducing the complex mathematical modeling becomes an important method for studying CSF biofluid with various parameters. The simulation of the temperature effects on hydrocephalus CSF in oscillating flow that affects high pressure is of importance. We tried to prove these using previous studies to simulate the hydrocephalic condition using computational fluid dynamic and fluid-structure interaction methods [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. However, some of them calculate the effect of temperature in their simulations [3, 27,28,29,30,31,32,33].

Previous thermal analyses of hydrocephalus patients have not considered the effect of rotational flow and elasticity of the ventricular walls. Recently, we have studied the CSF flow with elastic boundary moving in a laminar path bounded by a porous layer for hydrocephalus patients. In the present study, we include an elastic effect due to the interaction with the thermal solutal model which accounts for CSF motion which flows rotationally due to hydrocephalus flows within the spinal canal. Using an analytical method, we have attempted a new model to justify CSF flow movement using the influences of wall temperature difference. We used hydroceplaus patients refered in [4, 17, 33] for this simulation.

Mathematical formulation

Due to the geometric complexity of the flow in the cranial system, many of them attempt to show CSF flow and have analyzed the system with multiple compartments, developed as a simple geometry as a cylindrical tube. Considering these compartments in our mind to qualify the physical configuration Fig. 1 is a closed cylindrical model through which the flow has been developed. In this approach, governing equations are solved analytically [2]. We consider the CSF oscillating flow passes through porous parenchyma. We take \(z\) as the vertical axis and \(y\) chosen as perpendicular to it.

Fig. 1
figure 1

Schematic representation of CSF flow in the ventricular region

We consider the hydrodynamic incompressible, unsteady viscous fluid moving vertically in a finite medium embedded by brain parenchyma. Let \(\partial {\Omega }_{b}\) represents the boundary of \({\Omega }_{b}\) which represents the CSF flow surface. Also, we describe the coordinate system in such a way that the horizontal and vertical directions on the y-axis and z-axis respectively. Fluid particle rotating in a uniform velocity \(\Omega\) with respect to y-axis. Consequently, \(u\) and \(w\) are denoted as the velocity of fluid flow in \(y\) and \(z\) respectively. Considering the fluid initially at rest portion with uniform temperature and concentration and all the physical quantities depend only on \(y\) and \(t\). Then the fluid particle starts moving at a time \(t\) and it is maintained uniformly throughout the fluid moment. Without loss of generality, we neglect pressure gradient and gravitational forces in the fluid flow for our convenience.

The physical principles of rigorous CSF fluid flow physics are used to develop the flow and thermal diffusivity defined in the analytical construction from the previous research papers. we tried to show the acceleration of the Soret parameter as there is an increase in the temperature of fluid flow. Hence this conservation stabilizes that lead to a system of non-linear partial differential equations (PDE) as the conservation of mass and Navier Stokes with heat and diffusivity equation are given by

Equation of Mass [2]

$$\frac{\partial u}{\partial y}+\frac{\partial w}{\partial z}=0$$

Equation of motion [11] with rotational effect

$$\frac{\partial u}{\partial t}+G{w}_{0}\frac{\partial u}{\partial y}+2{{\Omega^\prime }}u=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left(\frac{{\partial }^{2}u}{{\partial y}^{2}}\right) -\frac{\nu }{K} u-\frac{RN}{\rho }u$$
$$\frac{\partial w}{\partial t}+{Gw}_{0}\frac{\partial w}{\partial y}-2{{\Omega^\prime }}w=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left(\frac{{\partial }^{2}w}{{\partial y}^{2}}\right)-\frac{\nu }{K} w -\frac{RN}{\rho }w$$

Energy Eq. [28] with a heat conduction parameter

$$\left(\frac{\partial T}{\partial t}+{w}_{0}\frac{\partial T}{\partial y}\right) =\frac{k}{\rho {C}_{p}}\left(\frac{{\partial }^{2}T}{{\partial y}^{2}}\right)+\frac{J}{\rho {C}_{p}}(T-{T}_{c})$$

Concentration Eq. [16] with Soret effect

$$\frac{\partial C}{\partial t}+{w}_{0}\frac{\partial C}{\partial y}=D\frac{{\partial }^{2}C}{{\partial y}^{2}}+\frac{D {K}_{T}}{{T}_{m}}\left(\frac{{\partial }^{2}T}{{\partial y}^{2}}\right)$$
$$u=0,w=0,T={T}_{0},C={C}_{0}\mathrm \ at \,y\,=\,0 \,and\, t\le 0$$
$$u={u}_{0}\left(1 + cos\lambda t\right), w=0,T={T}_{c},C={C}_{c} \,at \,y\,=\,1 ,t\,>\,0$$

Non-dimensional quantities relevant to the problem and neglecting the dash symbol for our convenience

$${y}{\prime}=\frac{y}{l}, {t^\prime}=\frac{t{w}_{0}}{l}, {w^\prime}=\frac{wl}{{w}_{0}}, {u^\prime}=\frac{u}{{u}_{0}}$$
$${\theta^\prime}=\frac{T-{T}_{c}}{{T}_{0}-{T}_{c}}, {\phi^\prime}=\frac{C-{C}_{c}}{{C}_{0}-{C}_{c}}, Sc=\frac{\nu }{D}, Re=\frac{{w}_{0}l}{\nu }, Sr=\frac{D {K}_{T}}{{T}_{m}\nu }\left(\frac{{T}_{0}-{T}_{m}}{{C}_{0}-{C}_{m}}\right)$$
$${G}_{pm}=\frac{RN{l}^{2}}{\mu }, Pe=\frac{\rho {c}_{p}l{w}_{0}}{k}, J=\frac{kl}{\rho {c}_{p}{w}_{0}}, \Omega =\frac{\Omega {l}^{2}}{\nu }$$

\(u\) and \(w\) refer the velocity of fluid in \(y\) and z direction respectively.\({w}_{0}\) is represented as characteristic velocity. Fluid density is referred as \(\rho\). Also, \(\theta\) and \(\phi\) refers as temperature and diffusivity of fluid. Da and \(\Omega\) expressed as darcy number and rotational parameter, \({G}_{pm}\ and\ Pe\) referred as particle mass parameter (resistance parameter) and Peclet number in mass and heat transfer. Soret number is represented as \(Sr\), heat conduction parameter is represented as \(J\), \({T}_{c} \ and \ {C}_{c}\) referred as boundary temperature and concentration of CSF. \({T}_{0} \ and\ {C}_{0}\) refers free steam temperature and concentration of CSF. The following were the governing Eq. (1) to (5) in dimensionless form.

$$Re\left(\frac{\partial u}{\partial t}+G\frac{\partial u}{\partial y}\right)+2\Omega w=\frac{{\partial }^{2}u}{{\partial y}^{2}}+\frac{1}{Da}u-{G}_{pm}u$$
$$Re \left(\frac{\partial w}{\partial t}+G\frac{\partial w}{\partial y}\right)-2\Omega w=\frac{{\partial }^{2}w}{{\partial y}^{2}}+\frac{1}{Da}w-{G}_{pm}w$$
$$\frac{\partial \theta }{\partial t}+\frac{\partial \theta }{\partial y}=\frac{1}{Pe}\frac{{\partial }^{2}\theta }{{\partial y}^{2}}+J\theta$$
$$\frac{\partial \phi }{\partial t}+\frac{\partial \phi }{\partial y}=\frac{1}{Re Sc}\frac{{\partial }^{2}\phi }{{\partial y}^{2}}+\frac{Sr}{Re}\frac{{\partial }^{2}\theta }{{\partial y}^{2}}$$

corresponding initial and boundary conditions are,

$$u=0,w\,=\,0,\theta =\phi =1 \,at \,y\,=\,0$$
$$u=1,w\,=\,0,\theta =\phi =0 \,at\, y\,=\,1$$

\(u\) and \(w\) is inspected as a function of y since the fluid flows rotationally with respect to y axis.

To solve the above momentum equation, we introduce the complex velocity, \(\mathrm{F }= u +iw\)

The momentum equation in an explicit form of complex form is

$$Re\left(\frac{\partial F}{\partial t}+G\frac{\partial F}{\partial y}\right)+2\Omega IF=-\frac{2}{\rho }\frac{\partial p}{\partial x}+\frac{{\partial }^{2}F}{{\partial y}^{2}}-\frac{1}{Da}F-{G}_{pm}F$$

We assume a suitable boundary condition \(F=0 , y=0,t\le 0\)

$$F=1+ \frac{\varepsilon }{2}({e }^{i\lambda t} +{e }^{-i\lambda t}) , y=1,t>1$$

Method of solution

We solve the above governing equation using the perturbation technique as the method is quite an error-less, assuming the trail solution for velocity, diffusivity, and heat transfer as

$$u\left( y, t\right) ={ u}_{0} \left(y\right)+\frac{\epsilon }{2}{e}^{i\lambda t}{u}_{1}\left(y\right) +\frac{\epsilon }{2}{e}^{-i\lambda t} {u}_{2}\left(y\right)$$
$$\theta ( y, t) ={ \theta }_{0} (y) +\frac{\epsilon }{2}{e}^{i\lambda t}{\theta }_{1} (y) +\frac{\epsilon }{2}{e}^{-i\lambda t} {\theta }_{2} (y)$$
$$\varphi ( y, t) ={ \varphi }_{0} (y) +\frac{\epsilon }{2}{e}^{i\lambda t}{\varphi }_{1} (y) +\frac{\epsilon }{2}{e}^{-i\lambda t} {\varphi }_{2} (y)$$

\(\lambda\) refers to oscillation frequency and \(\epsilon\) is an arbitrary constant parameter defined in such a way that\(\epsilon \ll 1\). Let us consider \({u}_{0},{ u}_{1,}{ u}_{2}\), \({\theta }_{0}, { \theta }_{1,}{ \theta }_{2,} { \varphi }_{0},{ \varphi }_{1},{ \varphi }_{2}\) refers base part, first and second orders of momentum equation, energy equation, and concentration equation respectively.

Zero (\({F}_{0})\), First \({(F}_{1})\), Second (\({F}_{2})\) order of complex form of the equation of motion:

$$\frac{1}{Re}\left(\frac{{\partial }^{2}{ F}_{0}}{{\partial y}^{2}}\right)- \frac{\partial {F}_{0}}{\partial y}-\frac{{R}_{m}}{Re}{ F}_{0}=0$$
$$\left(\frac{{\partial }^{2}{ F}_{1}}{{\partial y}^{2}}\right)-Re \frac{\partial {F}_{1}}{\partial y}-(Re+{R}_{m}){F}_{1}=0$$
$$\left(\frac{{\partial }^{2}{ F}_{2}}{{\partial y}^{2}}\right)-Re \frac{\partial {F}_{2}}{\partial y}-(Re+{R}_{m}){F}_{2}=0$$

were \({R}_{m}=\) \(\frac{1}{Da}-{G}_{pm}-2\Omega I\) boundary conditions are,

$${F}_{0}{= F}_{1}={ F}_{2}=0 \,at \,y=0$$
$${F}_{0}{= F}_{1}={ F}_{2}=1\, at \, y=1$$

Zero (\({\theta }_{0})\), First (\({\theta }_{1})\), Second (\({\theta }_{2})\) order of energy equations:

$$\left(\frac{{\partial }^{2}{ \theta }_{0}}{{\partial y}^{2}}\right)-Pe \frac{\partial {\theta }_{0}}{\partial y}+J Pe \ { \theta }_{0}=0$$
$$\left(\frac{{\partial }^{2}{ \theta }_{1}}{{\partial y}^{2}}\right)-Pe \frac{\partial {\theta }_{1}}{\partial y}+(J+i\lambda )Pe \ { \theta }_{1}=0$$
$$\left(\frac{{\partial }^{2}{ \theta }_{2}}{{\partial y}^{2}}\right)-Pe \frac{\partial {\theta }_{2}}{\partial y}+(J-i\lambda )Pe \ { \theta }_{2}=0$$

Then the corresponding boundary conditions are,

$${\theta }_{0}{= \theta }_{1}={ \theta }_{2}=1 \,at\, y=0$$
$${\theta }_{0}{= \theta }_{1}={ \theta }_{2}=0 \,at\, y=1$$

Zero (\({\varphi }_{0})\), First (\({\varphi }_{1})\), Second (\({\varphi }_{2})\) order of concentration equation

$$\left(\frac{{\partial }^{2}{ \varphi }_{0}}{{\partial y}^{2}}\right)-Re \,Sc\frac{\partial {\varphi }_{0}}{\partial y}=-Sr \,Sc\left(\frac{{\partial }^{2}{ \theta }_{0}}{{\partial y}^{2}}\right)$$
$$\left(\frac{{\partial }^{2}{ \varphi }_{1}}{{\partial y}^{2}}\right)-Re \,Sc\frac{\partial {\varphi }_{1}}{\partial y}-(i\lambda\, Sc\, Re){\varphi }_{1}=-Sr\, Sc\left(\frac{{\partial }^{2}{ \theta }_{1}}{{\partial y}^{2}}\right)$$
$$\left(\frac{{\partial }^{2}{ \varphi }_{2}}{{\partial y}^{2}}\right)-Re \,Sc\frac{\partial {\varphi }_{2}}{\partial y}+\left(i\lambda\, Sc\, Re\right){\varphi }_{2}=-Sr \,Sc\left(\frac{{\partial }^{2}{ \theta }_{2}}{{\partial y}^{2}}\right)$$

\(i\) represents the complex form of variable \({{\theta }_{1}{ ,\theta }_{2} , \varphi }_{1 },{ \varphi }_{2}\) of the above equations respectively.

Hence corresponding boundary conditions are,

$${\varphi }_{0}{= \phi }_{1}={ \phi }_{2}=1 \,at\, y=0$$
$${\varphi }_{0}{= \phi }_{1}={ \phi }_{2}=0 \,at\, y=1$$

The above equation is solved by the analytical way of perturbation method (Additional file 1: Appendix), hence, resultant equations can be expressed as

$${\theta }_{0}={A}_{7}{e}^{{m}_{7}y}+{A}_{8}{e}^{{m}_{8}y}$$
$${\theta }_{1}={A}_{9}{e}^{{m}_{9}y}+{A}_{10}{e}^{{m}_{10}y}$$
$${\theta }_{2}={A}_{11}{e}^{{m}_{11}y}+{A}_{12}{e}^{{m}_{12}y}$$
$${\phi }_{0}={A}_{13}{e}^{{m}_{13}y}+{A}_{14}{e}^{{m}_{14}y }+{A}_{19}{e}^{{m}_{7}y }+{A}_{20}{e}^{{m}_{8}y}$$
$${\phi }_{1}={A}_{15}{e}^{{m}_{15}y}+{A}_{16}{e}^{{m}_{16}y }+{A}_{21}{e}^{{m}_{9}y }+{A}_{22}{e}^{{m}_{10}y}$$
$${\phi }_{2}={A}_{17}{e}^{{m}_{17}y}+{A}_{18}{e}^{{m}_{18}y }+{A}_{23}{e}^{{m}_{11}y }+{A}_{24}{e}^{{m}_{12}y}$$

Result and discussion

In the present study, we adopt some default parameters using Additional file 1: Table 1. There are no relevant literature relating a few parameters like Soret number, and heat conduction parameter to compare with the following finding of our research article. In the following graphs, we summarized the system of governing Equations in the region, with the boundary conditions are solved analytically. To understand the behaviour of the oscillating flow characteristics, velocity \((F)\), temperature \(\left(\theta \right)\) and Concentration \((\varphi )\) are calculated by varying the emerging flow parameters like Reynolds number, Peclet number, Darcy number, Soret number, Schmidt number, particle mass parameter, heat conduction parameter, etc. The velocity of CSF for patients varies from -15 m/s to 15 m/s for various parameters like Reynolds number, Darcy Number, Elasticity parameters etc.,

\(t=0.01\), \(\rho =998.2\), \(\begin {aligned}\lambda =0.3, \epsilon =0.01, J = 0.4, \vartheta =0.8, k=0.67 \times {10}^{-16} ,\vartheta = 0.8,{G}_{pv}=0.167, \\ k=0.67\times {10}^{-16}, 150\le Re \le 420, Sr=0.2,Da=0.37 \end{aligned}\) were the values assigned for the graphical representation of various parameters (Figs. 2 and 3).

Fig. 2
figure 2

Analysis of CSF fluid in case of real and imaginary velocity with varying parameters

Fig. 3
figure 3

Analysis of fluid velocity, temperature, and diffusivity by varying parameters

The permeability of the fluid increased as there is a feasible signal in filaments present in the epidermal cells. Hence, Darcy's number increases as the velocity of the fluid increases gradually when the velocity of the fluid increases. This reflects that there is an increase in fluid in the porous medium say pia mater as the pressure increases. (Fig. 2a, b)

The resistance (particle mass parameter) increases at a peak level as the velocity increase, as the drop in fluid flow the resistance in the flow regions will be less (Fig. 2c, d)

The Reynolds number (Re) is used to predict the flow patterns for different fluids. It is the ratio of inertial forces to viscous forces within a fluid which is subjected to relative internal movement due to different fluid velocities. The inertial force increases predominantly to the viscosity of the fluid flow. It reflects that the hydrocephalus CSF fluid flow increases in SSS as there are drop-out fluid molecules in the third ventricle (Fig. 2e, f)

Elastic deformation plays a major role in CSF circulation. When the elasticity increases due to the bulk flow of hydrocephalus, for a certain time period as the flow is in a closed medium (Fig. 3a, b). As a result, this closed medium produces more enamor pressure than the normal subject when compared with hydrocephalic patients.

Peclet number is the measure of advective transport of fluid particles to mass diffusion rate. Here there is a decrease in the concentration of the fluid along its length due to the increase of Peclet number. Hence there is a transfer of heat when the diffusivity conduction with convection. As there is an excess secretion of CSF due to weak fluid particle movement which results in increases in the fluid temperature due to an increase in fluid velocity (Fig. 3c).

The heat conduction parameter is also increasing significantly showing that there is a decrease in thermal heat transfer when the fluid is in excess (Fig. 3d).

Schmidt number (Sc) is a change of momentum diffusivity with mass diffusivity and is used to characterize fluid flows with simultaneous momentum and mass diffusion convection processes. Here, the transport diffusivity decreases predominantly as the mass diffusivity decreases. As a result, the validity fluid shows that CSF concentration decreases as the Schmidt number increases (Fig. 3e) Soret number is the rate of change in temperature difference to the fluid concentration. In Figure 3f, an increase in Soret number shows larger temperature variations. Soret number plays a major role in this paper as there is temperature variation in the subarachnoid space when there is an increase in thermodynamical level enormously then it should be noted by the neurologist.


The following were the conclusions that are made from the present investigation. The velocity of a fluid flow increases for increasing Darcy number, resistance parameter and Reynolds number. An increase in Elasticity with respect to dimensional change results in an increase in CSF velocity. There is a significant difference in temperature fall due to the enhancement of heat conduction parameter and Peclet number. The impact of the increase in Schmidt number, and Soret number considerably reduces the CSF fluid concentration. An increase in fluid velocity reflects significant changes in high intracranial pressure in the flow regime. It is observed that the comparison of the results produced using the revised elastic modulus with those of an existing value used in similar to simulations generated in [2, 4, 28]. Future works may generalize the present approach to consider a three-dimensional computational CSF pulsatile flow model for Non-Newtonian characteristics with pressure differences in the viscoelastic nature. There are a few limitations in this model, that is results were made to analyse CSF using the mathematical model with few parameters, but according to neurologists there a certain other parameters used in the three-dimensional model that have been simulated in various perpontine region in brain which is quite smaller visible only through CINE-MRI scan. We attribute this criterial result in the future to include more pulsatility model that the outcome is more appropriate one.

Availability of data and materials

All data used for the present study are available and could be requested from the authors.



Cerebrospinal fluid


Velocity of CSF flow in y and z direction


Cental nervous system


Intracranial pressure


Subarachnoid space


Fluid density


Kinematic viscosity


Permeabilty of porous layer (pia mater)

w0 :

Characteristic velocity


Electric conductivity


Velocity of real and imaginary part


Imaginary part


Resistance parameter (Stokes resistance)

cp :

Specific heat capaity at Constant pressure

Tc :

Wall temperature of CSF flow

Cc :

Wall concentration of CSF flow

T0 :

Steam temperature of CSF flow

C0 :

Steam concentration of CSF flow


Rotation parameter


Dimensionless rotation parameter


Reynolds number


Heat conduction parameter


Pertubation parameter ε  0


Pertubation postive real constant


Thermal conductivity

KT :

Thermal diffusion ratio


Number density


Dimensionless time taken


Darcy number

Gpm :

Particle mass parameter


Peclet number


Soret number


Schmidt number for mass transfer


Temperature of fluid flow in brain


Transport diffusivity of the fluid

y and z:

Co-ordinate system


  1. Mulugeta B, Seyoum G, Mekonnen A, et al. Assessment of the prevalence and associated risk factors of pediatric hydrocephalus in diagnostic centers in Addis Ababa. Ethiopia BMC Pediatr. 2022;22:145.

    Article  PubMed  Google Scholar 

  2. Andreas AL, et al. Cerebrospinal fluid flow in the normal and hydrocephalic human brain. IEEE Trans Biomed Eng. 2007;54(2):291.

    Article  Google Scholar 

  3. Donnelly J, Czosnyka M. The thermodynamic brain. Crit Care. 2014;18(6):1–2.

    Article  Google Scholar 

  4. Gholampour S, et al. Numerical simulation of cerebrospinal fluid hydrodynamics in the healing process of hydrocephalus patients. J Appl Mech Tech Phys. 2017;58(3):386–91.

    Article  CAS  Google Scholar 

  5. Keong NC, et al. Diffusion tensor imaging profiles reveal specific neural tract distortion in normal pressure hydrocephalus. PLoS ONE. 2017;12(8):e0181624.

    Article  PubMed  PubMed Central  Google Scholar 

  6. Smillie A, Sobey I, Molnar Z. A hydroelastic model of hydrocephalus. J Fluid Mech. 2005;539:417–43.

    Article  Google Scholar 

  7. Zhu DC, et al. Dynamics of lateral ventricle and cerebrospinal fluid in normal and hydrocephalic brains. J Magnetic Reson Imaging. 2006;24(4):756–70.

    Article  Google Scholar 

  8. Tangen KM, et al. CNS wide simulation of flow resistance and drug transport due to spinal microanatomy. J Biomech. 2015;48(10):2144–54.

    Article  PubMed  Google Scholar 

  9. Neff S. Measurement of flow of cerebrospinal fluid in shunts by transcutaneous thermal convection. J Neurosurg Pediatr. 2005;103(4):366–73.

    Article  Google Scholar 

  10. Hemalatha B, Selvi S. Effect of cerebrospinal fluid dynamics with hydrocephalus in porous medium. Turkish J Computer Math Educ. 2021;12(10):5666–71.

    Google Scholar 

  11. Hemalatha B, Selvi S. Effect of suction on an unsteady congenital hydrocephalus in cerebrospinal fluid flow between porous plates with thermal diffusion. Int J Res Educ Scientific Methods. 2021;9:5.

    Google Scholar 

  12. Raptist A, Singh AK. Rotation effects on MHD free-convection flow past an accelerated vertical plate. Mech Res Commun. 1985;12(1):31–40.

    Article  Google Scholar 

  13. Balasundaram H, et al. Effect of Ventricular Elasticity Due to Congenital Hydrocephalus. Symmetry. 2021;13(11):2087.

    Article  CAS  Google Scholar 

  14. Toro EF, et al. Cerebrospinal fluid dynamics coupled to the global circulation in holistic setting: mathematical models, numerical methods and applications. Int J Numer Methods Biomed Eng. 2022;38(1):e3532.

    Article  Google Scholar 

  15. Keith Sharp M, Carare RO, Bryn AM. Dispersion in porous media in oscillatory flow between flat plates: applications to intrathecal, periarterial and paraarterial solute transport in the central nervous system. Fluids Barriers CNS. 2019;16(1):1–17.

    Article  Google Scholar 

  16. Prema K, Muthuracku A, Muthucumaraswamy R. Rotation Effects on Flow Past an Exponentially Accelerated Vertical Plate with Variable Temperature and Uniform Mass Diffusion in the Presence of Thermal Radiation. 2015.

  17. Gholampour S. FSI simulation of CSF hydrodynamic changes in a large population of non-communicating hydrocephalus patients during treatment process with regard to their clinical symptoms. PLoS ONE. 2018;13(4): e0196216.

    Article  PubMed  PubMed Central  Google Scholar 

  18. Gholampour S, et al. Evaluating the effect of hydrocephalus cause on the manner of changes in the effective parameters and clinical symptoms of the disease. J Clin Neurosci. 2017;35:50–5.

    Article  PubMed  Google Scholar 

  19. Ursino M. A mathematical study of human intracranial hydrodynamics part 1—the cerebrospinal fluid pulse pressure. Ann Biomed Eng. 1988;16(4):379–401.

    Article  CAS  PubMed  Google Scholar 

  20. Momjian S, Bichsel D. Nonlinear poroplastic model of ventricular dilation in hydrocephalus. J Neurosurg. 2008;109(1):100–7.

    Article  PubMed  Google Scholar 

  21. Kaczmarek M, Subramaniam RP, Neff SR. The hydromechanics of hydrocephalus: steady-state solutions for cylindrical geometry. Bull Math Biol. 1997;59(2):295–323.

    Article  CAS  PubMed  Google Scholar 

  22. Linninger AA, Sweetman B, Penn R. Normal and hydrocephalic brain dynamics: the role of reduced cerebrospinal fluid reabsorption in ventricular enlargement. Ann Biomed Eng. 2009;37(7):1434–47.

    Article  PubMed  Google Scholar 

  23. Sincomb S, et al. A model for the oscillatory flow in the cerebral aqueduct. J Fluid Mech. 2020;899:89.

    Article  Google Scholar 

  24. Gholampour S, Fatouraee N. Boundary conditions investigation to improve computer simulation of cerebrospinal fluid dynamics in hydrocephalus patients. Commun Biol. 2021;4(1):1–15.

    Article  Google Scholar 

  25. Gholampour S, et al. A mathematical framework for the dynamic interaction of pulsatile blood, brain, and cerebrospinal fluid. Computer Methods Prog Biomed. 2022;23:107209.

    Google Scholar 

  26. Balasundaram H, Sathiamoorthy S, Anwar BO. Mathematical modeling of thermo-solutal transport in pulsating flow in the hydrocephalus. J Mech Med Biol. 2022;22(10):2250071.

    Article  Google Scholar 

  27. Hirashima Y, et al. Intracerebral temperature in patients with hydrocephalus of varying aetiology. J Neurol Neurosurg Psychiatry. 1998;646:792–4.

    Article  Google Scholar 

  28. Rajasekaran S, Qu H, Zakalik H. Thermal measurement of cerebrospinal fluid flow rate in hydrocephalus shunt. 2015 IEEE SENSORS. IEEE, 2015.

  29. Madsen JR, et al. Evaluation of the ShuntCheck noninvasive thermal technique for shunt flow detection in hydrocephalic patients. Neurosurgery. 2011;68(1):198–205.

    Article  PubMed  Google Scholar 

  30. Déli E, Kisvárday Z. The thermodynamic brain and the evolution of intellect: the role of mental energy. Cogn Neurodyn. 2020;14(6):743–56.

    Article  PubMed  PubMed Central  Google Scholar 

  31. Herbowski L, Gurgul H. Thermodynamic approach to cerebrospinal fluid circulation. J Neurol Res. 2011;1(5):215–8.

    Google Scholar 

  32. Zakharov M, Sadovsky M. The role of blood circulatory system in thermal regulation of animals explained by entropy production analysis. arXiv preprint arXiv:1308.3663. 2013.

  33. McAllister JP, et al. An update on research priorities in hydrocephalus: overview of the third National Institutes of Health-sponsored symposium “Opportunities for Hydrocephalus Research: Pathways to Better Outcomes.” J Neurosurg. 2015;123(6):1427–38.

    Article  CAS  PubMed  Google Scholar 

Download references


Not applicable.


Not applicable.

Author information

Authors and Affiliations



HB performed the Conceptualization, Design of the study, Data curation, Formal analysis, Investigation, Methodology, and Writing the manuscript, Project administration, Resources, Software, and Validation. The Author read and approved the final manuscript.

Corresponding author

Correspondence to Hemalatha Balasundaram.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors of the manuscript titled “Impact of Thermodynamical Rotational Flow of Cerebrospinal Fluid in the Presence of Elasticity” report no competing interests concerning the materials or methods used in the study or the findings specified in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Additional file 1:

Appendix. Obtained from solving the governing equations Eqns. (6)-(9) from manuscript. Table S1. Properties involved in CSF flow used to validate various parameters.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit The Creative Commons Public Domain Dedication waiver ( applies to the data made available in this article, unless otherwise stated in a credit line to the data.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balasundaram, H. Impact of thermodynamical rotational flow of cerebrospinal fluid in the presence of elasticity. BMC Res Notes 16, 355 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: