Theoretical Model Development for Energy Motion of Dusty Turbulent Flow of Fibre Suspensions in a Rotational Frame

Fibre suspension has garnered considerable attention in turbulent ﬂows that are used in many industries. Solid particles, such as dust particles, notably aﬀect the turbulent ﬂow ﬁeld in a rotational frame. In assessing their impacts, the dusty turbulent ﬂow for ﬁbre suspensions needs to be studied in a frame of rotation that can be substantially applied in many industries. This study, therefore, aims to build a theoretical model for the energy motion of dusty turbulent ﬂow of ﬁbre suspensions in a rotational frame. The turbulence momentum equation was considered to formulate the model in presence of dusty ﬂuid rotating ﬂow of ﬁbre suspensions. The newly derived equation was derived in second-order correlation tensors F i,j , W i,j , G i,j , S i,j , X i,j , Y i,j , Q i,j , and R i,j at any two points in the ﬂow domain, where the tensors were expressed as space, time, and distance functions. The developed model is a considerable improvement because it takes into account all of the potential inﬂuential parameters that could aﬀect the motion of turbulent energy, such as dust particles, suspending particles (ﬁbres), and rotating frame. However, the impact of these parameters on turbulence energy motion must be evaluated in order to assess the performance of turbulence systems utilized in a variety of industries, such as paper manufacturing. The present theoretical development will contribute to open up experimental and numerical research opportunities for the advancement of the industry, science, and technology.


Introduction
e flows of turbulent are often found in moving objects, for example, various moving vehicles, which affect flow resistance to the bodies of the objects. Turbulent flow motions define the fluid flow behaviours based on mass conservation, energy, and momentum laws [1]. In the second-order pressure-velocity correlation, Hinze and Uberoi [2] derived the turbulent motion at any two points in the flow domain. e correlations between velocity and pressure fluctuations were taken into account in his study. However, the effect of any potential turbulent motion parameters was not considered. e turbulent flows are greatly affected by the introduction of fibres into the flow. In several parts of the industry, fibre injections into the turbulent flow are used, for example, in environmental engineering, paper manufacturing, chemical engineering, the manufacture of composite materials, and the textile industry. e property of the fibre suspension has a notable effect on product quality. e complexity in turbulent fibre motion arises mainly from the translational and revolutionary dynamics of the particles, which are driven by forces and torques, depending on the form and the orientation of the particles [3]. Over the last thirty years, numerous studies have been undertaken on turbulent flows of fibre suspensions. A preliminary study on turbulent motion of fibre (ice crystals) suspension was conducted by Cho et al. [4]. Atmospheric turbulence associated with their effect on fibre orientation was investigated in their study. e mean fibre orientation was not significantly affected by the atmospheric turbulence. Fluid motion equations were solved with a spectral approach by Lin et al. [5], and the trajectories of fibre were determined on slim body theory. Although a key parameter to define the fibre orientations on a broad scale was found to be the Stokes number, its orientation effect on the fibre is not significant and the ratio of fibre direction has a minimum influence on the orientation distribution of fibre.
Particle motions in the flows of turbulent are a fundamental issue for applications [6], for example, atmospheric clouds, and spray combustion engines. In an accurate understanding between dust particle collisions and growth [7][8][9], humid terrestrial clouds, cool stars [10], brown dwarves, and cloud production of planets, the interactions between turbulent and the particles have been considered. e evolution of the particles' size is dependent on the particle collision rate which can be significantly increased by turbulent motions [11]. In the presence of small-sized particles, Bhatti et al. [12] studied the electro-osmotic flow of non-Newtonian fluid moving in a sinusoidal fashion in a Darcy-Brinkman-Forchheimer medium. e Jeffrey fluid model was used to investigate non-Newtonian effects. e Debye length approximation, Poisson-Boltzmann equation, and ionic Nernst-Planck equation were employed to construct the mathematical model. e effect of parameters on both fluid and particle phases was also discussed. e present effects have been proven to be advantageous in the design and manufacture of microfluidic devices for determining the transport mechanism.
A precise and exact calculation of the collision rate includes an accurate understanding of the collision rate impacts caused by turbulence and preferential or clustering particle concentrations. Several studies [13][14][15][16][17][18][19][20][21] have been carried out on the motion of the dusty turbulent flow. e characteristics of the dust particles [22][23][24][25] mainly rely on the particle size of the turbulence scale. e particles compete over fluid impacts and collisions on turbulent flow into the neighbourhoods of the particle. In a study conducted by [26], entropy generation and irreversibility processes that occur as a result of partial slip on magnetic dusty fluid caused by a peristaltic wave via a porous conduit were studied. e results demonstrated that the presence of dust particles in the fluid causes the flow to decelerate. In the presence of a strong magnetic field, Mahanthesh [27] investigated the effect of Hall current on the two-phase boundary layer flow of an electrically conducting dusty fluid over a permeable stretched sheet. Using appropriate similarity transformations, the governing equations were reduced from a set of partial differential to ordinary differential equations. e author reported that, in controlling the friction factor on the sheet, the dust particles' mass concentration can be employed as a control parameter. It was also revealed that suction and injection have opposing effects on the formation of the momentum boundary layer. In the presence of dust particles, the effect of Hall current on the time-dependent flow of a nanofluid was also investigated by Gireesha et al. [28]. e boundary layer approximation notion was used for modeling the governing equations for nanofluid as well as dust phases. Increases in heat transfer rate were found to be proportional to Hall current and unsteadiness. e motions between a particle of the fluid and the suspended fibre rely on fluid dynamics fundamental of turbulence characteristics with the velocity pressure correlations of fluctuating components. A good number of [15,[29][30][31][32][33][34][35] studies have been undertaken in rotating systems to analyze the motion of turbulent flows. e mean rotation induces dynamical influences on turbulent flows through the pressure-stream-rate correlation in the transport equations. e rotation produces the auxiliary body force-like Coriolis and centrifugal force in rotating turbulent flows, accompanied by the turbulent structures. Consequently, the momentum transfer mechanism suffers from complexity. However, most research studies did not take into account or address the rotational impacts of the system at different angles, owing to rotational frame complications. Coriolis force plays a critical role in a turbulent flow rotating system [36,37]. To measure the impact of the Coriolis force, You et al. [30,35] directed an experimental-based study on the rotating turbulent flow. e velocity for particle images was considered in this study to calculate the flow profile. Because of the Coriolis effect, the flow domains between the leading and trailing sides were found to be different [30]. Coriolis force widened the vortex nearby the front side and intimidated the vortex nearby the trailing end. e force of Coriolis showed major effects in the flow field on the vortices. It was also revealed that not only the secondary flow but also the Coriolis force influences the flow domain which must not be overlooked.
Due to the simplification of laminar flow, it is shown through comprehensive literature surveys that several research studies were undertaken on assessing the behaviour of laminar fibre motion. However, the turbulent fibre motion studies are still inadequate because of the complexities of fibre motion and turbulence. Flow phenomena generated in the flow domain are complex if it includes any type of solid particle such as dust particles into the turbulent fibre flow.
e dust particles can range from 1 to 400 μm diameters, with particles more than 100 μm settling down towards the formation source [38]. e overall range of the particle sizes can be classified into three categories: lower than 1 μm, 1-20 μm, and more than 20 μm, which are referred to as ultrafine, fines, and large particles, respectively. However, the smaller particles are comparatively difficult to remove or separate from the airstream because they have a stronger inclination to stay in suspension.
In a frame of rotation, the dusty turbulent flow becomes more complex. However, some mathematical models were developed on the energy motion of turbulent flow in secondorder correlation tensors by considering dust particles [13], fibre suspension [39], dust particles with fibre suspensions [40], a rotating frame [32], and dust particles in a frame of rotation [15]. Nevertheless, any of these articles did not consider all of the parameters that can influence turbulence energy motion, such as dust particles, fibre suspension, and rotational frame. e main fact behind this is because of its complexity level in mathematical modeling. As a result, the previous models cannot be considered fully complete and feasible, and they must be improved by taking into account all of these influential parameters in order to make the model feasible. By considering all of these parameters, the present study thus constructs a model for the energy motion of dusty rotating turbulent flow of fibre suspensions. To address the research gaps and improve the previous models, the present study introduced correlation tensors W i,j , X i,j , Y i,j , and R i,j , which represent the correlations between fluid velocities, angular velocities, and fluid velocities, angular velocities produced due to the rotation, and the suspending fluid velocities in the fluid flow domain. e resulting partial differential equation model can be generalized to time-space fractional order, making it valuable to the broader scientific community. Based on a numerical investigation, the DNS (direct numerical simulation) method can be applied to the rotation of dusty rigid fibres with different lengths relative to the neighbouring fluid in the channel flow. Due to its high precision, the spectral element method (SEM) may be used to solve the newly developed equation, and SEM applications are also suggested for future research. e mathematical consequences were sequentially added to the momentum equation. To find the relationships among the velocities of fluid, fibres, and dust particles in the rotational frame, any two points, C and D, were chosen at a distance r in the turbulent flow field. Point C was regarded as the origin of the system coordinate since no difference exists in representing the relationships of turbulence momentum between the points if any of the two points is assumed as the source (origin). e independent variables ψ k were chosen to differentiate the influences between distance and position. e method of averaging was used to express first-, second-, and third-order pressure-velocity correlations between the two points, C and D. ese correlations were demonstrated in a second-order partial differential equation by introducing second-order correlation tensors and following averaging method and tensor properties.

Model Formulation and Development
e equation of momentum and the continuity for viscid turbulent incompressible flow are represented by [32,41] with the fluid velocity components of u i (x, t), fluid density ρ, pressure p(x, t), suspending fluids' kinematical viscosity υ, three-dimensional permutation symbol ε ijl , rotation vector Ω j , dissipation of turbulence ε, position x, and time t.
For the turbulent flow with dust (solid) particles, the following equations are obtained: with dust (solid) particle velocity components v i (x, t), dust particle mass for single-spherical m s � 4/3πR 3 s ρ s with radius R s , Stoke's formula for the drag K � 6πR s ρv, KN/ρ � f frequency dimension, and dust particle density number N. It is noted that the term KN/ρ(v i − u i ) appears in equation (2) due to the solid dust particles existent in the field of turbulent flow [14]. e fluid is presumed to be air and treated as multiphase since this present study comprises fibre suspensions and dust particle interactions into the dusty turbulent flow in the frame of rotation.
In presence of fibre, the equation of turbulent motion (3) gives with the apparent viscosity of fibre suspension μ f , the turbulent intensity of fibre suspension I ij , tensor strain rate ε lm � 1/2(zu l /zx m + zu m /zx l ), and fourth-and second-order tensors for fibre orientation a ijlm and a lm , respectively. e term μ f /ρz/zx j [a ijlm ε lm − 1/3(I ij a lm )ε lm ] occurs in equation (5) because of the fibres poured into the turbulent flow domain [42].
For a rotating frame, the turbulence momentum equation is given by [32] where − 2(Ω i u i η i )sin θ � − 2(Ω × u) denotes the Coriolis force, where η, Ω i , and θ specify unit vector normal to u and Ω, angular velocity, and the angle makes between u and Ω, respectively. erefore, combining equations (5) and (6), a new equation is obtained as follows: Equation (7) represents the new energy equation of motion for fibre suspended dusty rotating turbulent flow.
Suppose C and D are any two points in the turbulent flow field and assume c and d be the directions along with the points C and D, respectively. erefore, u c and u d can be used as the components of the velocity along with C and D directions. Suppose that U i is the mean velocity, time independent, and constant throughout the considered domain.
us, (U i � U i + u i ) C and (U j � U j + u j ) D . All the terms have a value, which can be calculated from the equations of u i at C point and of u j at D point.
At point C, the turbulence momentum can be written from equation (7) for u i : Since, for incompressible fluids, Multiplying equation (9) by (u j ) D , where (u j ) D is taken as constants for the differential system at C point.
Likewise, the turbulence momentum equations for u j are obtained at D point: 4 Complexity By using the condition of incompressible fluids, (u j zu k /zx k ) D � 0, equation (11) gives Multiplying equation (12) by (u i ) C , the following is obtained: where (u i ) C are taken as constants for differential systems at D point.
Summing equations (10) and (13), In order to represent the relationships of turbulence momentum between D and C point, no difference exists if C or D is assumed as the source (origin) of the system coordinate. Point C was regarded as origin here. Suppose the Complexity variables ψ k are independent and used to differentiate the influences between distance and position, where erefore, the following relations are obtained: Using these relations and averaging all the terms, equation (15) yields Equation (16) yields the mean energy motion of dusty turbulent flow for fibre suspensions in a rotational frame, where the motion of fibres of turbulent flow is at average speed U k with respect to the system coordinate. Due to the constant derivative, the coefficient term U k was omitted in this equation. e equation consists of the double correlations of velocity (u i ) C (u j ) D , pressure-velocity p C (u j ) D , and triple-velocity such as (u i ) C (u k ) C (u j ) D , where the terms are located at a certain distance from each other. e correlations of pressure-velocity p D (u i ) C and p D (u j ) C form firstorder tensor as the pressure is taken as a scalar. Likewise, triple correlations of velocity, e double correlations are displayed in Figure 1, while the triple correlations are illustrated in Figure 2, respectively, at points C and D, where r is the distance from C to D.
e correlations of the order first, second, and third can be labelled by (K p,j ) C,D , (W i,j ) C,D , and (S ik,j ) C,D , respectively.
us, the correlations of pressure-velocity and velocity give where p denotes the pressure which is not an index such as dummy j or i, and therefore, the summation should not be used to p. e terms of (ε jkl Ω k u l ) D (u i ) C , (ε ikl Ω k u l ) C (u j ) D , (Ω i u i η i ) C (u j ) D , and (Ω j u j η j ) D (u i ) C form the correlation tensors of second order those can be represented by M i,j , N i,j , H i,j , and L i,j , accordingly, while (a jklm ε lm ) D (u i ) C and (I jk a lm ε lm ) D (u i ) C form the correlation tensor of third order 6 Complexity and expressed as Q i,jk and R i,jk , respectively. Consequently, the followings are obtained: Using the above correlation terms, equation (16) yields where the terms of correlations used in equation (19) are to the considering points, C and D. e double pressure-velocity correlations are zero for incompressible and isotropic turbulence, which means k p,j C,D � 0, In the field of isotropic turbulence, the invariance condition under the reflection to C is followed by u i C u k D u j D � − u k C u j C u i D , or s i,kj C,D � − s kj,i C,D . (21) us, the simplification of equation (19)   Complexity e terms z/zψ k (Q jk,i + Q ik,j ), z/zψ k (S ik,j + S kj,i ), Considered independent variables r: Distance between any two points C and D in the flow filed t: Time (s) S i,j , W i,j : Fluid velocity correlations Q i,j , R i,j : Correlations between the suspending fibres and fluid velocities F i,j , G i,j : Velocity correlations between fluid and dust particles X i,j : Correlations between the velocities of fluid and fibre Y i,j : Angular velocities between fluid and fibre formed due to the rotation.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.