EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7138 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mathematical Analysis for Magnetohydrodynamic Peristaltic Flow of Cerebrospinal Fluid in the Glymphatic System M. A. El-Shorbagy1, Ahmed A. Elsawy2, Abdallah A. Henedy2,3, Hossam A. Nabwey1, B. Alreshidi1, Essam T. Abdelwahab2,3,∗ 1 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 2 Basic Engineering Sciences Department, Faculty of Engineering, Menoufia University, Shebin El-Kom, Egypt 3 Faculty of Engineering, Menoufia National University, Menoufia, Shebin El-Kom, Egypt Abstract. Cerebrospinal fluid (CSF) flow within perivascular routes (PVS) is essential for main- taining brain health and function. This flow is a part of the glymphatic system, which clears waste products and metabolic byproducts from the brain. The PVS boundaries consist of a non- permeable arterial vessel wall and neural tissue. The arterial vessel peristalsis and the elasticity of the neural tissue are responsible for generating the CSF flow. This article examines the ef- fect of a transverse magnetostatic field and pertinent physiological conditions on CSF flow. It specifically considers the impact of brain elasticity and porosity at the neuroglial layers, arterial wall curvature, and the peristalsis amplitude. The effective Navier–Stokes governing equations describe the peristalsis of CSF flow subjected to a magnetostatic field. The perturbation strat- egy is utilized to address the governing equations. This leads to a set of equations describing CSF stream-direction velocity, pressure gradient, and neural tissue deformation. The graphical outcomes highlight the significant influence of a range of physiological and flow factors on the be- havior of CSF. The key parameters encompass neural tissue porosity and stiffness, neuroglial layer deformation, magnetostatic field strength, and arterial wall curvature. The results reveal that enhancing the magnetostatic field intensity reduces CSF pressure distribution, stream-direction velocity, and arterial deformation. The numerical computations and graphical representations of the obtained analytical solutions are performed using the MATLAB software package. This study is highly significant due to its implications for the development and advancement of different neu- rological disorders for instance Alzheimer’s disease, meningeal inflammation, neurotrauma, and hemorrhagic stroke. 2020 Mathematics Subject Classifications: 76W05, 76Z05, 92C35, 74F10 Key Words and Phrases: Perturbation method, peristalsis, magnetostatic field, glymphatic network, CSF ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7138 Email addresses: essam.tharwat@sh-eng.menofia.edu.eg (E. T. Abdelwahab) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 2 of 28 Nomenclature r The radial coordinate axis z The stream-direction axis b The PVS thickness r1 The arterial vessel average radius r2 The neural tissue lumen radius h(z, t) The wall deformation h̄ The peristalsis amplitude λ The peristalsis wavelength t Time frame d The deformation of neuroglial layer c The peristalsis speed p The CSF pressure D The neuroglial elastic layer deformation with neural tissue Pe The neural tissue pressure Pa, Pb The Pressure values at the extreme points of the PVS (Pa > Pb) Po The zeroth-order pressure Me The neural tissue porosity Ee The neural tissue stiffness Ea The arterial vessel wall elastic modulus Eg The neural tissue Young’s modulus Kg The neuroglial layer porosity parameter ta The arterial vessel layer thickness Re Reynolds number µ The CSF dynamic viscosity ρ The CSF density ε PVS film thickness B Hartmann number σ Electrical conductivity βo Magnetic field strength Wo The zeroth order stream-direction flow velocity Uo The zeroth order radial flow velocity Ue The permeation velocity n⃗ The elastic neuroglial layer normal unit vector t⃗ The elastic neuroglial layer tangent unit vector V⃗ The vector of velocity field êr The radial unit vector êz The stream-direction unit vector U The radially oriented velocity W The stream-direction velocity R1 The PVS layer lumen radius R2 The PVS layer exoradius ω The peristalsis frequency in the arterial vessel L The PVS length M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 3 of 28 1. Introduction The glymphatic system is an indispensable brain-wide PVS network responsible for removing metabolic waste products and regulating water balance in the brain. This mech- anism function involves driving CSF function along the periarterial spaces and supporting the astrocyte-mediated fluid interchange between the interstitial fluid (ISF) and CSF. This intricate mechanism is essential for preserving brain activity and function especially during sleep and anesthesia [1]. Disruptions in its function is associated to a range of pathologi- cal and neurological conditions, including aging, Alzheimer’s disease, hemorrhagic stroke, and neurotrauma [2, 3]. The dynamics of CSF are regulated by various elements includ- ing arterial pulsation, respiration, posture, and sleep quality. This highlights the strong influence of maintaining normal sleep patterns on glymphatic function and overall brain activity [4, 5]. CSF is a vital, clear, and colorless liquid that envelops the central nervous system (CNS) and serves an essential role in providing nourishment, safeguarding the brain, and removing waste products from this essential organ [6]. This vital fluid also maintains brain stability by controlling chemical composition and temperature. Additionally, it serves a crucial function by protecting the brain from mechanical trauma and supporting fragile nervous tissue [7]. The dynamics of peristalsis exist due to the alternating contraction and expansion of the flexible walls of a distensible channel fluid. It occurs in hollow organs such as the esophagus, stomach, intestines, urine motion in the ureter, chyme movement through the gastrointestinal system, blood transport through cardiovascular network, as well as CSF in the glymphatic system. Several recent theoretical studies investigated the peristaltic transport of biofluids within different conduits and their interaction with var- ious physiological and flow factors [8–20]. Peristaltic motion plays a crucial role in the glymphatic system, facilitating the transport of CSF within the brain. In the context of the glymphatic system, peristalsis in combination with other mechanisms such as longi- tudinal pressure gradients and directional flow resistance contributes to driving fluid flow through the para-arterial spaces surrounding cerebral arteries [21–24]. The intricate and synchronized movement of neuroglial and arterial walls plays a pivotal role in enabling the glymphatic system to perform essential functions such as effectively clearing waste, facilitating the movement of ISF, and potentially contributing to the maintenance of brain immunity [25]. The influence of magnetostatic fields on CSF flow is a nuanced topic that bridges the disciplines of medical imaging, neurobiology, and biophysics with significant implications for both diagnostic and therapeutic practices. Magnetostatic resonance imaging (MRI), an essential method of modern neuroimaging, leverages strong magnetostatic fields to produce high-resolution images of the brain and spinal cord and is capable of visualizing CSF dynamics using techniques such as phase-contrast MRI. These imaging modalities are instrumental in diagnosing and monitoring conditions like hydrocephalus, where CSF flow is disrupted [26]. Theoretical frameworks such as magnetohydrodynamics (MHD) postulate that magnetostatic fields can influence the behavior of conductive fluids. While CSF is predominantly composed of water and exhibits low ionic conductivity, the presence M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 4 of 28 of ions and charged particles suggests that strong magnetostatic fields, such as those used in high-field MRI scanners, could potentially induce subtle alterations in CSF flow dynamics. Understanding these potential interactions is vital for optimizing MRI protocols and improving the accuracy of CSF flow measurements [27]. Therapeutic applications of magnetostatic fields are gaining attention for their poten- tial to influence CSF flow and mitigate hemorrhagic stroke-related complications. One of the most promising techniques is magnetostatic resonance-guided focused ultrasound. This noninvasive treatment leverages magnetostatic fields to guide focused ultrasound waves, creating precise openings in neural tissue to enhance CSF drainage. It is particularly beneficial in treating hydrocephalus, a condition often seen in hemorrhagic stroke patients where CSF accumulates due to impaired drainage pathways. By facilitating better CSF flow, magnetostatic resonance-guided focused ultrasound can reduce intracranial pressure and alleviate symptoms associated with fluid buildup. Additionally, electromagnetic fields are being explored for their neuromodulator ef- fects, which could indirectly benefit CSF dynamics. The modulation of electromagnetic fields might also help in normalizing disrupted CSF flow patterns caused by hemorrhagic stroke-induced brain damage and swelling [28]. Moreover, advances in technology have led to the development of devices that use electromagnetic fields to regulate intracranial pressure. These devices aim to stabilize CSF flow and reduce the complications arising from abnormal CSF dynamics, providing a noninvasive therapeutic option for hemor- rhagic stroke patients. In summary, therapeutic methods utilizing magnetostatic fields, such as magnetostatic resonance-guided focused ultrasound and electromagnetic neuro- modulation, offer promising avenues for managing CSF flow abnormalities in hemorrhagic stroke patients. These techniques not only enhance CSF drainage and reduce intracranial pressure but also hold potential for broader applications in improving brain function and promoting recovery after hemorrhagic stroke. The nonionizing constant magnetostatic fields produced by devices like smartphones and headphones, as well as mobile network towers, can have varying effects on the CSF in humans, including children. Lonappan et al. [29] demonstrated that exposure to these electromagnetic fields can induce distinct changes in the dielectric parameters of CSF, with both normal and abnormal samples exhibiting different behaviors in terms of dielectric constant and conductivity at specific frequencies. Additionally, Ali [30] highlighted concerns about the potential neurological effects associated with the emitted mobile phone electromagnetic radiation, including cog- nitive dysfunction and neurogenesis impairment. While there is ongoing debate about the safety of exposure to electromagnetic fields, especially from mobile devices, Preece [31] suggested that a short-term exposure within established guidelines may not pose a significant risk to humans. Moreover, Romanò et al. [22] presented a dominant order approximation of the momen- tum equations with lubrication approximation for investigating the flow of CSF through the glymphatic system. The effects of a peristaltic wave in the arterial vessel and the deformability of neural tissue were considered as a motive force for the flow. Addition- ally, they justified the negligible importance of convective transport compared to diffusion effects in the PVS using first principles and dimensional analysis considerations. Sub- M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 5 of 28 sequently, they focused on computing the leak velocity of CSF between the PVS space and brain interstation using their first-principles approach. Furthermore, Preece [31] con- centrated on the fluid dynamics of CSF along the PVS in the spinal cord. A detailed numerical analysis was conducted based on the lubrication approximation, utilizing crite- ria such as long wavelength and low Reynolds number. They indicated the strong effect of interaction between the peristalsis induced by the arterial vessel wall and the flow as- sociated with a convection-enhanced delivery infusion, especially at distances far from the infusion source. The model undertaken examined the axial transport in a circular area with non-permeable boundaries, but it did not investigate fluid transport across the tissue layer. On the other hand, Schley et al. [32] formulated a mathematical framework to examine the hypothesis that the PVS drainage of ISFs and solutes out of neural tissue is driven by pulsations of arterial walls. Prior empirical studies [33–35] demonstrated that the CSF transport is substantially influenced by the elastic features of blood vessel walls, the wall porosity of neural tissue, and the pulsation of the blood flow. However, the inherent difficulty of measuring these parameters in vivo necessitates the use of modeling- based approaches to enhance our understanding of fluid transport within the brain. To preserve neuro homeostasis, the glymphatic system promotes the dynamics of CSF for metabolic waste products to be removed from neural tissue. This mechanism is related to the onset of various neurological disorders involving traumatic brain injury, Parkinson’s, and Alzheimer’s diseases. Therefore, knowledge of the dynamics of CSF flow within the glymphatic network is necessary to identify the mechanisms underlying these conditions. Recent studies [36–43] indicate that magnetic fields can alter the movement of charged particles and ionic species in biological fluids, thereby altering transport phenomena at the microvascular and perivascular scales. Magnetic fields have a potential impact on the flow of ionic species and charged particles contained in biofluids through microvascular and perivascular mechanisms. Magnetostatic fields can affect flow resistance and enhance solute transport performance. The interaction between the induced electric currents and magnetic fields produces the Lorentz force. This force can alter CSF flow rates, pressure gradients, and velocity profiles. A helpful framework for investigating possible alterations to glymphatic transport pathways brought on by internal or external magnetic fields is provided using MHD models. The primary motivation of this study is to develop a mathematical formulation of PVS transport under a magnetostatic field to provide insights into how the factors of brain elasticity and porosity at the neuroglial layer boundaries, the arterial wall curvature, the strength of the magnetostatic field, and the peristalsis amplitude affect the direction and quantity of CSF flow. To achieve a dominant-order features of CSF flow, this model will neglect the influence of ciliated boundaries and non-Newtonian behavior [44–46]. The PVS is modeled to contain a Newtonian incompressible fluid of constant viscosity. The model described in this paper builds upon previous methods for calculating PVS fluid flow in simplified shapes under the influence of a transverse magnetostatic field. In this model, the Navier–Stokes equations are addressed using lubrication theory approximations following the steps of the semi-analytical perturbation technique. This study displays the promising influences of the magnetostatic field on CSF flow and brain health. This M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 6 of 28 investigation is essential as it explores the complex mechanisms involved in the peristaltic flow of CSF within the glymphatic system in the presence of a magnetostatic field and its direct relevance to the development of diverse neurological disorders, including Alzheimer’s disease, hemorrhagic stroke, and meningeal inflammation. It should be noticed that this study assumes constant physical properties and a steady- state flow regime. This assumption simplifies the actual physiological conditions of CSF motion within the glymphatic system. CSF flow is driven by pulsatile pressure variations associated with cardiac activity. Its rheological and transport properties can vary due to interactions with the surrounding neural tissues. This simplification results in a steady state approach which provides an essential theoretical framework for the MHD-induced CSF transport. It highlights the strong influence of magnetic, viscous, and porosity effects on PVS dynamics. Future studies could extend the present model to incorporate time de- pendent pulsatile effects, and temperature dependent variable viscosity of CSF to achieve a more comprehensive physiological representation. 2. Mathematical Modelling 2.1. Structural Geometry The model structure of the PVS space that transports the CSF is characterized as a thin annulus space surrounded by the walls of a non-permeable elastic arterial vessel and neural tissues (Figure 1). The neural tissue is padded from the inside by a porous, elastic neuroglial wall, which separates the PVS from the neural tissue [22]. This neu- roglial wall is modeled with the tube law. The elastic tube law describes how the elastic properties of pulsating walls in physiological systems can be modeled. Experimentally, it is simulated by finite length rubber tubes that demonstrate a diverse self-sustained oscillation, providing insight into the underlying dynamics system. It is used to define the interaction between the peristaltic deformation of the solid walls and the pressure difference of the CSF across them. The blood pulsation in the arterial vessel generates consequential sinusoidal waves, which represent the peristaltic deformation. This model is exposed to an external transverse magnetostatic field. It is applicable in cases of diag- nostic and therapeutic applications, such as MRI for visualizing the CSF dynamics in the brain and magnetostatic resonance-guided focused ultrasound as a therapeutic method for brain hemorrhagic stroke treatment. The ISF is used as a constant reference pressure. The cylindrical frame (R-Z) is chosen to represent the CSF flow in the PVS structure, The peristalsis propagates according to h(z, t) = h̄ sin [ 2π λ (z − ct) ] , and the permeable elastic neuroglial wall is deformed according to d(z, t). The PVS geometry is mapped to the glymphatic system with elastic annular structure to depict the anatomical arrangement where cerebrospinal fluid flows around cerebral arteries. This arrangement is surrounded on the inside by the arterial wall and on the M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 7 of 28 outside by the permeable astrocytic layer. This model approaches the real PVS system and refers to a good agreement with physiological structure that has been incorporated, see Figure 2. In this study, a uniform magnetostatic field of strength β0 is applied transversely to the direction of peristaltic motion of CSF within the tube. The magnetic field is oriented perpendicular to the streamwise direction (r-direction). This configuration is consistent with the approach used in low magnetic Reynolds number MHD flows in which the induced magnetic field is negligible compared to the applied field. Under these conditions, the Lorentz force FL = σβ2 0w acts as a resistive body force opposing the motion. The constancy of β0 emphasizes that magnetic effects remain spatially uniform across the domain. The PVS walls are assumed to be non-conducting, ensuring that the tangential component of the magnetic field remains continuous across the boundaries. Figure 1: Schematic graph for the PVS space between the elastic neural tissue and the arterial vessel, including the peristalsis and transversal magnetostatic field. 2.2. Mathematical Formulation The transformation of the cylindrical framework from the unsteady frame (r, z) to the steady one (R,Z) using the same wave speed c requires the following method: R = r, Z = z−ct, U(R,Z, T ) = u(r, z, t), W (R,Z, T ) = w(r, z, t), P (R,Z, T ) = p(r, z, t). The Navier–Stokes equations for the CSF flow with constant viscosity across the elastic PVS structure are represented as follows [22, 47]: Governing Equations: Continuity 1 r ∂(ru) ∂r + ∂w ∂z = 0 (1) M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 8 of 28 Figure 2: Real physiological structure of PVS system. r-momentum ∂u ∂t + u ∂u ∂r + w ∂u ∂z = −1 ρ ∂p ∂r + µ ρ [ 1 r ∂u ∂r + ∂2u ∂r2 + ∂2u ∂z2 ] (2) z-momentum ∂w ∂t + u ∂w ∂r + w ∂w ∂z = −1 ρ ∂p ∂z + µ ρ [ 1 r ∂w ∂r + ∂2w ∂r2 + ∂2w ∂z2 ] − σβ2 0w (3) Peristaltic propagation of the arterial vessel wall about the mean radii h(z, t) = h̄ sin [ 2π λ (z − ct) ] (4) Boundary conditions: Nonslip case at the elastic arterial vessel r = r1 + h w(r, z, t) = 0 (5) u(r, z, t) = ∂h(z, t) ∂t (6) Porosity case at the elastic neuroglial layer r = r1 + b+ d0 u⃗ · n⃗ = ∂d ∂t +Me(p− pe) (7) u⃗ · t⃗ = 0 (8) d = p− pe Ee (9) CSF pressure at the PVS inlet z = 0 and outlet z = l p = pa at z = 0 (10) M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 9 of 28 p = pb at z = l (11) Dimensionless parameters: r = bR, z = λZ, t = T/ω, u = bωU, w = λωW, p = µω ϵ2 P, B2 = σβ2 0b 2 µ , pa = µω ϵ2 Pa, pb = µω ϵ2 Pb (12) ω = c λ , ϵ = b λ , Re = ϵρωb2 µ , H = h b , R1 = r1 b , R2 = r2 b = R1 + 1, D0 = d b , l = λL (13) Applying the nondimensional parameters (12-13) to the governing equations (1-4) and their boundary conditions (5-11) yields the following nondimensional sets of equations. Dimensionless governing equations: 1 R ∂(RU) ∂R + ∂W ∂Z = 0 (14) ϵRe [ ∂U ∂T + U ∂U ∂R +W ∂U ∂Z ] = −∂P ∂R + ϵ2 [ 1 R ∂U ∂R + ∂2U ∂R2 + b2 λ2 ∂2U ∂Z2 ] (15) ϵRe [ ∂W ∂T + U ∂W ∂R +W ∂W ∂Z ] = −∂P ∂Z + [ 1 R ∂W ∂R + ∂2W ∂R2 + ϵ2 ∂2W ∂Z2 ] − ϵ2λ2ρσβ µ W (16) Dimensionless peristaltic deformation of the arterial vessel: H = H̄ sin[2π(Z − T )] (17) Nonslip wall conditions at R = R1 +H: W (R,Z, T ) = 0 (18) U(R,Z, T ) = ∂H(Z, T ) ∂T (19) Permeable wall conditions R2 = R1 + 1 +D: V⃗ · n⃗ = ∂D ∂T +Me(P − Pe) ⇒ U = ∂D ∂T +Me(P − Pe) (20) V⃗ · t⃗ = 0 ⇒ W = 0 (21) V⃗ = Uêr +Wêz, n⃗ = 1êr + 0êz, t⃗ = 0êr + 1êz (22) D = P − Pe Ee (23) M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 10 of 28 The pressure conditions at the extreme points of the PVS (inlet Z = 0 and outlet Z = L): P = Pa at Z = 0 (24) P = Pb at Z = L (25) Method of Solution: The semi-analytical perturbation strategy expands the flow quantities as a power series with a small wave number ϵ as follows: W = W0 + ϵW1 + ϵ2W2 + ϵ3W3 + · · · (26) U = U0 + ϵU1 + ϵ2U2 + ϵ3U3 + · · · (27) P = P0 + ϵP1 + ϵ2P2 + ϵ3P3 + · · · (28) D = D0 + ϵD1 + ϵ2D2 + ϵ3D3 + · · · (29) Applying the perturbation expansions (26-29) to the governing equations (14-16) with the corresponding boundary conditions (18-25) yields the following sets of differential equations: The zeroth order set: 1 R ∂(RU0) ∂R + ∂W0 ∂Z = 0 (30) −∂P0 ∂R = 0 (31) −∂P0 ∂Z + 1 R ∂W0 ∂R + ∂2W0 ∂R2 − b2ρσβ2 µ W0 = 0 (32) Nonslip case R = R1 +H: W0(R,Z, T ) = 0 (33) U0(R,Z, T ) = ∂H(Z, T ) ∂T (34) Permeable case R2 = R1 + 1 +D0: W0 = 0 (35) U0 = ∂D0 ∂T +Me(P0 − Pe) (36) The first order set: 1 R ∂(RU1) ∂R + ∂W1 ∂Z = 0 (37) −∂P1 ∂R = 0 (38) M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 11 of 28 Re [ ∂W0 ∂T + U0 ∂W0 ∂R +W0 ∂W0 ∂Z ] = −∂P1 ∂Z + [ 1 R ∂W1 ∂R + ∂2W1 ∂R2 ] (39) Nonslip case R = R1 +H: W1(R,Z, T ) = 0 (40) U1(R,Z, T ) = 0 (41) Permeable case R2 = R1 + 1 +D1: W1 = 0 (42) U1 = ∂D1 ∂T +MeP1 (43) The second order set: 1 R ∂(RU2) ∂R + ∂W2 ∂Z = 0 (44) Re [ ∂U0 ∂T + U0 ∂U0 ∂R +W0 ∂U0 ∂Z ] = −∂P2 ∂R + [ 1 R ∂U0 ∂R + ∂2U0 ∂R2 ] (45) Re [ ∂W1 ∂T + U0 ∂W1 ∂R + U1 ∂W0 ∂R +W0 ∂W1 ∂Z +W1 ∂W0 ∂Z ] = −∂P2 ∂Z + [ 1 R ∂W2 ∂R + ∂2W2 ∂R2 + ∂2W0 ∂Z2 ] (46) Nonslip case R = R1 +H: W2(R,Z, T ) = 0 (47) U2(R,Z, T ) = 0 (48) Permeable case R2 = R1 + 1 +D2: W2 = 0 (49) U2 = ∂D2 ∂T +Me(P2) (50) For simplifications, the zeroth-order solution is adequate and guarantee solution con- vergence, and the subsequent sets are complicated with little significance to the outcomes. The CSF velocity and the pressure solutions are expressed as follows: W (r) = C1I0( √ Br) + C2K0( √ Br)− 1 B dP0 dZ (51) M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 12 of 28 U = 1√ B ∂C2 ∂Z K1( √ Br)− 1√ B ∂C1 ∂Z I1( √ Br) + R 2B d2P0 dZ2 + C3 R (52) P0 = C4e √ A1/A0Z + C5e − √ A1/A0Z − A2 A1 (53) ∂P0 ∂Z = √ A1 A0 { Pa + A2 A1 − (Pe +A2/A1 − [Pa +A2/A1]e √ A1/A0L) [e− √ A1/A0L − e √ A1/A0L] } e √ A1/A0Z − √ A1 A0 (Pe +A2/A1 − [Pa +A2/A1]e √ A1/A0L) [e− √ A1/A0L − e √ A1/A0L] e− √ A1/A0Z (54) where C1 = 1 I0( √ Br1) [ 1 B dP0 dZ − C2K0( √ Br1) ] (55) C2 = 1 B dP0 dZ I0( √ Br1)− I0( √ Br2) K0( √ Br2)I0( √ Br1)− I0( √ Br2)K0( √ Br1) (56) C3 = r2Me(P0 − Pe) + r2√ B ∂C1 ∂Z I1( √ Br2)− r2√ B ∂C2 ∂Z K1( √ Br2)− r22 2B d2P0 dZ2 (57) C4 = Pa + A2 A1 − (Pe +A2/A1 − [Pa +A2/A1]e √ A1/A0L) [e− √ A1/A0L − e √ A1/A0L] (58) C5 = Pe +A2/A1 − [Pa +A2/A1]e √ A1/A0L e− √ A1/A0L − e √ A1/A0L (59) 2.3. Physiological parameters Physiological Parameters The real-life physiological parameters related to CSF flow in the glymphatic system for humans have been discussed in the literature. Xie et al. [48] presented an approximation for the value of intracranial pressure to be about Pe ≈ 2000 N/m−2. This is in line with the results of Sudres [49], who indicated that the ranges of intracranial pressure are 1300 ≤ Pe ≤ 2000 N/m−2 and the ranges of inlet pressure Pa at entrance of PVS are 5 × 10−7 ≤ Pa ≤ 5 × 102 N/m−2. In this study, Pb is assumed with a very small value Pb ≈ 0 compared with the inlet pressure Pa. Wang and Olbricht [50] presented that the measured values for the inner and outer radii of PVS layer are r1 ≈ 10−5 m for the peristaltic wave and r2 ≈ 1.1 × 10−5 m for the neural tissue. Recent studies [51–55] reported the different values of the relevant physiological parameters as follows: the PVS M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 13 of 28 film thickness, b ≈ 10−6 m; the amplitude of the peristaltic wave, h̄ ∈ [1.25, 5]× 10−7 m; the speed of the peristaltic wave, c ≈ 1 m/s (which cause a peristaltic wavelength of λ ≈ 0.2 m); the dynamic viscosity, µ = 9× 10−4 Pa·s; and the fluid density compared to the one of water, ρ ≈ ρwater = 103 kg/m3. Furthermore, Baumbach et al. [56] reported the wall thickness ta of the arterial vessel to be 10 to 100 times smaller than r1; i.e., 10−6 ≤ ta ≤ 10−5 m. Additionally, the arterial vessel wall elastic modulus Ea has a range of 105 ≤ Ea ≤ 106 N/m−2. Finally, the neural tissue elasticity and the porosity factors were Eg ≈ 0.01 Pa and kg ≈ 10−11 m·Pa−1s−1. Based on these factors, the ranges of the nondimensional groups of interest for the present study are listed in Table 1. Notably, the lubrication theory is a strong approximation for the thin film cerebrospinal fluid flow for this study. Thus, this results in assumptions of long wavelength and low Reynolds number and small amplitude ratio [ε ≪ 1 and Re ≪ 1]. Thus, higher orders of ε and inertial effects due to Re are negligible. The low Reynolds number indicated the high viscosity dominance created by viscous damping and the low inertial effect. The small CSF velocities vary between 10−5 and 10−6 m/s. Then, the estimated range for Re was [5 × 10−12 : 2 × 10−4]. The long wavelength approach was involved due to the short length scale for PVS width l. Then, the range for long wavelength λ was estimated as [0.2 : 0.6], while the PVS length (L = l/λ) was [2 : 20]. This analysis concentrates on conditions where the Reynolds number is low, and inertial effects are negligible. Table 1: Range of parameters. Parameter Characterization Assessed range Re Reynolds number (ερωb2/µ) [5× 10−12 : 2× 10−4] ε Film thickness of PVS (b/λ) [10−6 : 3.75× 10−2] R1 The PVS layer lumen radius (r1/b) [0.7 : 103] H̄ The peristalsis amplitude (h̄/b) [8× 10−4 : 0.5] L PVS length (l/λ) [2 : 20] Me Neural tissue porosity (kgµ/bε 2) [10−2 : 10] Ee Neural tissue stiffness (Egε 2/µω(R1 + 1)) [2× 10−10 : 0.2] 3. Results and Discussion In this part, the study is focused on examining the movement of CSF across the neural tissue’s neuroglial layer using a tribological framework formulated from first principles. The findings reveal that the phase difference between the arterial peristaltic wave and the neuroglial layer deformation is essential for breaking flow symmetry and establishing a steady streaming. This section depicts graphical illustrations for the behavior of the CSF flow, including the stream-direction velocity distribution Wo(r, z), permeation velocity Ue(r, z), pressure distribution Po(z), pressure gradient distribution dPo dZ , and the neural tissue deformation at the neuroglial elastic layer Do(r, z). The plots shown in Figures 3–19 are analyzed under the influence of key parameters including Hartmann Number (B), M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 14 of 28 neural tissue deformation (Do) corresponding to the wall at R2, neural tissue porosity (Me), neural tissue stiffness (Ee), and pressure in the neural tissue (Pe). Figures 3–5 present the stream-direction velocity profiles of CSF along the radial direction under the influences of {B,Me, Pe}. Figure 3 shows that the augmentation in Hartmann number exerts resistive action on the CSF flow by lowering the stream-direction velocity and reducing the CSF flow rate for the same cross-sectional area. The same effect is evident in Figure 4, where higher neural tissue porosity permits greater leakage of CSF at the neural tissue and decreases stream-direction velocity in the presence of a magnetostatic field. Figure 5 displays that the elevated neural tissue pressure (Pe) compresses the neuroglial wall toward the arterial vessel, reducing thin film thickness and restricting the flow path, and consequently lowering CSF flow velocity. The data for Figures 3–5 are summarized in Table 2. Figure 3: Effect of the magnetostatic flux intensity on the CSF stream-direction velocity. Figures 6–8 illustrate the distribution of permeation velocity near the permeable neu- roglial layer, which indicates the leakage velocity of CSF in the radial direction under the influence of the magnetostatic field, neural tissue deformation at the wall, and neural tissue porosity. Figure 6 demonstrates that the increase in the magnetostatic field causes restriction to the permeation velocity as the permeable area reduces and leakage decreases. Conversely, Figure 7 depicts that the increase in Me is boosting the leakage of CSF at the neural tissue, consequently increasing the permeation velocity. The same effect is observed in Figure 8, where the increase in elastic neural tissue deformation enlarges the porous area of the permeable neuroglial layer and enhances the permeation velocity. The data for Figures 6–8 are summarized in Table 3. Figures 9–13 illustrate the distribution of deformation in neuroglial permeable neural tissue under the external effects of a transverse magnetostatic field, pressure in the neural tissue, the elastic features of the brain represented by the stiffness of the neural tissue, and the neural tissue porosity. Moreover, they demonstrate that the increase in both the magnetostatic field B and pressure in the neural tissue Pe influence the elastic features M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 15 of 28 Figure 4: Effect of the neural tissue porosity on the CSF stream-direction velocity. Figure 5: Effect of the pressure of neural tissue on the CSF stream-direction velocity. of neural tissue deformability by reduction, see Figures 9–12. The same effect is noted in Figures 10–11 as evidenced by the inversely proportional relationship between the brain deformation and both neural tissue stiffness and porosity. Figure 13 indicates that the increase in the lumen radius of PVS layer R1 is slightly increasing the deformation of the neuroglial permeable neural tissue. The data for Figures 9–13 are summarized in Table 4. Figures 14–16 highlight the pressure distribution along the stream-direction direction. Figure 14 reports that the augmentation of the magnetostatic field reduces the pressure distribution across the CSF flow, as evidenced by the reduction in total energy relevant to the effect of magnetostatic strength on the stream-direction velocity. Quantitative analysis of the results reveals that the non-dimensional pressure drop increases noticeably with the augmentation in the Hartmann number. The stronger magnetic fields enhance the Lorentz M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 16 of 28 Table 2: Representative values of physical parameters related to Figures 3–5 Figures R1 Z B Me Pe Do Pa L Figure 3 0.1 0.5 0.5 2 0.2 1 0.5 0.5 0.8 1.1 1.4 Figure 4 0.1 0.5 0.8 1 0.2 1 0.5 0.5 2 3 4 Figure 5 0.1 0.5 0.8 2 0.1 1 0.5 0.5 0.2 0.3 0.4 Figure 6: Influence of the magnetostatic flux intensity on the CSF permeation velocity. force and increase the effective resistance to CSF motion. The pressure distribution shows that a rise in B from 0.5 to 2 results in approximately a 15–25% increase in the pressure drop across one wavelength of the peristaltic channel. In contrast, Figure 15 shows that the increase in neural tissue porosity results in boosting the leakage of CSF at the neural tissue, consequently increasing the permeation velocity. The same effect is observed in Figure 16, where the increase in elastic neural tissue deformation enlarges the porous area of the permeable neuroglial layer and enhances the permeation velocity. The data for Figures 14–16 are presented in Table 5. M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 17 of 28 Figure 7: Influence of the porosity of neural tissue on the CSF permeation velocity . Figure 8: Influence of the deformation of the neural tissue on the CSF permeation velocity. Table 5: Representative values of physical parameters related to Figures 14–16. Figures R1 B Me Do Pe Pa L Figure 14 0.1 0.5 1 1.5 0.01 0.5 10 1 1.5 2 Figure 15 0.1 1 0.5 1.5 0.01 0.5 10 1 1.5 2 Figure 16 0.1 1 1 0.5 0.01 0.5 10 1 1.5 2 M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 18 of 28 Table 3: Representative values of physical parameters related to Figures 6–8. Figures R1 Ee B Me Do Pe Pa L Figure 6 0.1 0.2 0.5 1 1.5 0.01 1 10 1 1.5 2 Figure 7 0.1 0.2 1 0.5 1.5 0.01 1 0.5 1 1.5 2 Figure 8 0.1 0.2 1 1 0.5 0.01 1 0.5 1 1.5 2 Figure 9: Influence of the magnetostatic flux intensity on the deformation of neural tissue. Figures 17–19 demonstrate the influences of magnetostatic field B, deformation of neural tissue Do, and neuroglial layer porosity of neural tissue. Figure 17 shows that the pressure gradient increases with the length of PVS. For short length of PVS [0 ≤ z ≤ 1.28], the increase in magnetostatic field strength reduces the pressure gradient, and it becomes a more negative pressure gradient, which assists the flow velocity and indicates normal streaming. In contrast, for long length of [z > 1.28], the increase in magnetostatic field boosts the chances of reversal when a large positive pressure gradient appears, as it resists the flow and reduces the kinetic energy of the CSF streaming. Notably, the positive pressure gradients are not achieved under the present condition, and backflow does not occur. The same effect of preamble neuroglial layer on the pressure gradient is seen in Figure 19, with critical conditions for PVS length, where for short length of PVS [0 ≤ z ≤ 1.28], an inversely proportional relationship between Me and dPo/dZ occurs and the pressure gradient becomes more negative, unlike the conditions for long length of M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 19 of 28 Figure 10: Influence of the stiffness of neural tissue on the deformation of neural tissue. Figure 11: Influence of the porosity parameter on the deformation of neural tissue. [z > 1.28]. The effect of the brain tissue deformation on the pressure gradient has opposite manner compared to magnetic field and space porosity influences, as shown in Figure 18. The data for Figures 17–19 are presented in Table 6. M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 20 of 28 Figure 12: Influence of the pressure in the neural tissue on the deformation of neural tissue. Figure 13: Influence of the lumen radius of the PVS layer on the deformation of neural tissue. Figure 17: Effect of the magnetostatic flux intensity on the CSF pressure gradient distribution. M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 21 of 28 Table 4: Representative values of physical parameters related to Figures 9–13. Figures L Pa B Ee Me Pe R1 Figure 9 10 1 0.5 0.2 1 0.1 0.1 1 1.5 2 Figure 10 10 1 1 0.02 1 0.1 0.1 0.08 0.13 0.2 Figure 11 10 1 1 0.2 0.5 0.1 0.1 1 1.5 2 Figure 12 10 1 1 0.2 1 0 0.1 0.3 0.6 0.9 Figure 13 10 1 1 0.2 1 0.1 0.1 1 2 3 Figure 14: Effect of the magnetostatic flux intensity on the CSF pressure distribution. Figure 18: Effect of the deformation of neural tissue on the CSF pressure gradient distribution. M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 22 of 28 Figure 15: Effect of the porosity of neural tissue on the CSF pressure distribution. Figure 16: Effect of the deformation of neural tissue on the CSF pressure distribution. Figure 19: Effect of the porosity of neural tissue on the CSF pressure gradient distribution. M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 23 of 28 Table 6: Representative values of physical parameters related to Figures 17–19. Figures L Pa B Do Me R1 Pe Figure 17 5 0.5 0.5 2 1 0.1 0.01 1 1.5 2 Figure 18 5 0.5 1 0.5 1 0.1 0.01 1 1.5 2 Figure 19 5 0.5 1 2 0.5 0.1 0.01 1 1.5 2 4. Conclusion This section presents a conclusion based on a comprehensive graphical analysis of the results, emphasizing the critical factors that affect the performance of various physiolog- ical and physical parameters on the peristaltic flow of cerebrospinal fluid through the glymphatic system model, including magnetostatic field parameter (B), neural tissue de- formation parameter corresponding to wall (Do), porosity parameter of the neural tissue (Me), stiffness parameter of the neural tissue (Ee), and pressure in the neural tissue (Pe). This analysis provides important insights into how magnetic fields and porous neuroglial properties modulate CSF transport within the glymphatic system. The results indicate that increasing the magnetostatic field enhances the Lorentz force with resistive action and reduces CSF axial velocity Wo(r), the permeation velocity Ue(z), the deformation of the neural tissue (Do), and the pressure distribution along the stream-direction Po(z). For most people, it is preferable to be far away from sources of magnetostatic fields, such as mobile phones, especially during sleep. The porosity of neural tissue (Me) has a sensitive effect on the CSF properties, as the increase of Me causes a reduction in the CSF ax- ial velocity, the deformation of the neural tissue, and the pressure distribution along the stream-direction Po(z), while increasing the permeation velocity Ue(z). The pressure in the neural tissue (Pe) has an adverse effect on both the CSF flow stream-direction velocity Wo(r) and the deformation of neural tissue (Do). The stiffness parameter of the neural tis- sue (Ee) has a direct effect on the pressure distribution along the stream-direction Po(z); on the other hand, it has an adverse effect on the deformation of neural tissue. These outcomes recommend that the utility of the magnetic field provides a potential effect on glymphatic clearance rates and, consequently, the removal of metabolic waste from the brain. From a clinical perspective, this can develop implications for magnetically guided therapeutic techniques or drug delivery systems targeting neurological disorders such as Alzheimer’s disease, Parkinson’s disease, and traumatic brain injury. Furthermore, the M. A. El-Shorbagy et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7138 24 of 28 insights gained from this model can serve as a foundation for future studies aimed at optimizing external magnetic field parameters to enhance and regulate CSF dynamics in both diagnostic and therapeutic contexts. Conflict of interests No conflict of interests. Funding The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/01/34000). Acknowledgements The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/01/34000). References [1] Y. Gao, K. Liu, and J. Zhu. 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