Insight into the dynamics of non-Newtonian Casson fluid over a rotating non-uniform surface subject to Coriolis force

Abstract Casson fluid model is the most accurate mathematical expression for investigating the dynamics of fluids with non-zero plastic dynamic viscosity like that of blood. Despite huge number of published articles on the transport phenomenon, there is no report on the increasing effects of the Coriolis force. This report presents the significance of increasing not only the Coriolis force and reducing plastic dynamic viscosity, but also the Prandtl number and buoyancy forces on the motion of non-Newtonian Casson fluid over the rotating non-uniform surface. The relevant body forces are derived and incorporated into the Navier-Stokes equations to obtain appropriate equations for the flow of Newtonian Casson fluid under the action of Coriolis force. The governing equations are non-dimensionalized using Blasius similarity variables to reduce the nonlinear partial differential equations to nonlinear ordinary differential equations. The resulting system of nonlinear ordinary differential equations is solved using the Runge-Kutta-Gills method with the Shooting technique, and the results depicted graphically. An increase in Coriolis force and non-Newtonian parameter decreases the velocity profile in the x-direction, causes a dual effect on the shear stress, increases the temperature profiles, and increases the velocity profile in the z-direction.


Introduction
The centrifugal force (acting radially outward from the axis of rotation), the Azimuthal force (parallel but opposite to the velocity, also known as the Euler force), and the Coriolis force (outward and perpendicular to angular velocity) are the three forces experienced in a rotating frame of reference; Debnath [1]. These forces are ctitious since they are not generated from interaction with any external body but are responsible to keep the motion rotational. Coriolis force is the ctitious force responsible for the de ection of the trajectory of a moving object in a rotating frame; Deng et al. [2] and Archana et al. [3]. The Coriolis e ect is responsible for many natural occurrences such as the direction of the cyclones, organization of the magnetic columns and electric properties of the Earth, hydromagnetic ow of the earth's liquid contents etc. The Coriolis force is also very signi cant in the ow equations, just as other inertial and viscous forces and of magnitude comparable with other magnetohydrodynamic forces; Shiferaw et al. [4].
Common surfaces on which ows are considered to include the horizontal surface, vertical surface, inclined surface etc. Some authors have considered ow over vertical cones, and wedges; Takhar et al. [5], Ghalambaz et al. [6] and Makinde and Animasaun [7,8]. Meanwhile, uid ow over surfaces whose thickness is non-uniform is common in nature and other physical applications. For instance, a bullet spins about its axis through the air as it is red, the air heated by the sun touches and ows over the Earth rotating surface, a traveling missile in space etc. These surfaces are not at plates or inclined planes cylinders whose thickness is uniform throughout but they possess surface with variable thickness. A way to generalize such surfaces with non-uniform thickness is to consider the upper horizontal surface of a paraboloid of revolution as suggested by Animasaun [9]. Due to the appearance of surfaces with non-uniform thickness in many natural occurrences and many physical applications, researchers have turned attention to not only examine but also explore transport phenomena over an upper horizontal surface of a paraboloid of revolution; Animasaun [10], Liu et al. [11], Koriko et al. [12], Makinde et al. [13], and Ajayi et al. [14].
Fluids that do not obey Newton's viscosity law are regarded as non-Newtonian uids. The nature of these non-Newtonian uids cannot be captured into a single model and these lead to di erent proposals for di erent uid behaviors. Casson uid model is one of the non-Newtonian uids which models uids such as jelly, honey, fruit juices, soup, and blood etc. Casson uid is a shear-thinning uid exhibiting yield stress and which has in nite viscosity at a low shear rate and zero viscosity at the in nity shear rate; RamReddy et al. [15] and Amanulla et al. [16]. In other words, Casson uid behaves like a solid when yield stress is more than the shear stress but behaves like a liquid and ows when the yield stress is less than the shear stress. Certain categories of ink and blood are examples of Casson uid. Blood obeys Newtonian law at high shear stress through larger arteries but becomes non-Newtonian at low shear stress through smaller arteries; see Batra and Jena [17], Siddiqui et al. [18], Abegunrin et al. [19], Sankar and Lee [20], Mukhopadhyay et al. [21], Pramanik [22], Animasaun [23], Kataria and Patel [24], and Zaib et al. [25].
Sequel to the fact that Casson uid model is in close agreement with the rheology of blood, Abbasian et al. [26] used di erent blood viscosity models to simulate the behavior of blood. Models used include Bingham, Carreau, Casson, modi ed Casson, Cross, Kuang-Luo (K-L), Powell-Erying, Power-law, and Walburn-Schneck models and the results show that the time-averaged velocity at the center of the arteries produced in the CFD simulations that uses the Carreau, modi ed Casson or Quemada blood viscosity models corresponded exceptionally well with the clinical measurements regardless of stenosis severities and hence, highlights the usefulness of these models to determine the potential determinants of blood vessel wall integrity such as dynamic blood viscosity, blood velocity, and wall shear stress. Dubey et al. [27] presented a two-dimensional theoretical study of hemodynamics through a diseased permeable artery with mild stenosis and an aneurysm present. The two models adopted to mimic non-Newtonian characteristics of the blood ow are the Casson (in the core region) and the Sisko (in the peripheral region). It is observed that by increasing the thermal buoyancy parameter, i.e. Grashof number (Gr), the nanoparticle concentration and temperature decrease and it was remarked that simulations are relevant to transport phenomena in pharmacology and nano-drug targeted delivery in hematology.
Recently, Koriko [28] explored the e ect of Coriolis force on the ow of Newtonian uid over a rotating surface with non-uniform thickness. It was observed that the horizontal and vertical velocities reduce with a combined increase in shear stress and Coriolis force whereas the buoyancy force has an increasing e ect on the overall velocity and temperature distribution of the ow. The present study is an extension of Ref. [28] to a non-Newtonian Casson uid. But, so far no attempt has been made to investigate the e ect of Coriolis force on the ow of Casson uid over a surface with non-uniform thickness whose importance cannot be overlooked in energy production, nuclear reactor cooling, biomedical applications, etc. Such a study is important in the design of turbines and turbo-machines, in estimating the ight path of rotating wheels and spin, stabilized missiles, and in the modeling of many geophysical vortices. A typical example of Casson uid ow over a nonuniform surface under the action of Coriolis force can be practically observed in a bioreactor, in which a microorganism-carrying uid ows in a tunnel with a non-uniform surface.
This present study investigates the in uence of Coriolis force on the properties of Casson uid ow across surfaces with non-uniform thickness. The research questions for the investigation are 1. What is the signi cance of Coriolis force on the ow of Casson uid over a rotating surface with non-uniform thickness?
2. What is the combined e ect of Casson uid parameter and Coriolis force on the ow of Casson uid over a rotating surface with non-uniform thickness? 3. What is the combined e ect Coriolis force and Prandtl number on the ow of Casson uid over a rotating surface with non-uniform thickness? 4. How does Coriolis force a ect the Skin Friction in both x− and y−directions and the heat transfer rate? , where m is the velocity index and A and b is some arbitrary constants related to the thickness of the object. As shown in Fig. (1), Coriolis force is produced in the rotating system as the uid ows. According to Mustafa et al. [29], the rheological equation of an isotropic and incompressible ow of a Casson uid can be written as where py is known as yield stress of the uid expressed as µ is known as the plastic dynamic viscosity of the non-Newtonian uid, π is the product of the component of deformation rate with itself i.e. π = e ij e ij , e ij is the (i, j)th component of the deformation rate and πcis the critical value based on the non-Newtonian model. For Casson uid, where π > πc, we have the dynamic viscosity given as Ref. [28], the body forces comprise of Coriolis force and buoyancy force are The boundary layer equations governing the ow of non-Newtonian Casson uid over the upper horizontal surface of a paraboloid of revolution under the in uence of Coriolis force is therefore given as the system of nonlinear partial di erential equations subject to the boundary conditions The quantities of interest are the Nusselt number proportional to the heat transfer, the skin friction of the ow along x−direction and z−direction de ned as In the Equation (9), τw is the shear stress (skin friction) along the upper horizontal surface of a paraboloid of revolution de ned as and the heat ux qw at the wall is de ned as The dimensional governing equations above were reduced to dimensionless equations by using the similarity variables below and the conversion domain The dimensionless equations are of the form with the boundary conditions Whereas the dimensionless boundary conditions are are the rotational parameter, Grashof number, Prandtl number and the thickness parameter respectively. The skin frictions along the x−direction and z−direction become respectively and the heat transfer coe cient becomes

Numerical Procedure
In order to obtain the numerical solution of the aforementioned dimensionless governing equation, it is necessary to break it down into a system of rst order ordinary di erential equations with the initial conditions where s , s , s are obtained using the shooting technique, and the system is solved numerically using the Runge-Kutta-Gills method (for some other methods of solution, see [30]). The system is solved using a step size ∆η = . and absolute tolerance − . The results obtained are displayed as graphs and the e ect of Coriolis force is investigated while the pertinent parameters increase. The ow becomes Newtonian as β → ∞ and the surface becomes a horizontal plane as m → and hence this report is validated by comparing the results as β → and as m → .

Analysis of Results and Discussion
In this section, the e ect of increasing rotation parameter alongside some other pertinent parameters is illustrated in the given graphs. It is important to reiterate here that an increase in the rotation parameter consequently implies an increase in Coriolis force. The results are validated by comparing the skin friction coe cient and Nusselt number as β * → ∞ with the result of [28] in Table (1) and there is a good agreement. It is worth noting that the surface becomes a at surface as m → . Table (2) shows that the skin friction coe cient in all directions increases as the surface becomes at horizontal plane m → with increasing K whereas the Nusselt number decreases as m → and K increases.

. Signi cance of increasing Coriolis force
Coriolis force is directly proportional to the rotation of the surface. The impact of increasing rotation of the surface with non-uniform thickness is explored on the Casson uid ows. When Gr = . , Pr = , m = . , χ = . and β = . , the rotation parameter was varied to study its e ect on the dynamics of the ow. The velocity in the x−direction, as shown in Fig. (2) decreases as rotation increases. A closer view of Fig. (2) for . ≤ ζ ≤ . is illustrated as Fig. (3)). Meanwhile, Fig. (4) shows that the velocity in the z−direction increases as rotation increases. This can be traced to the fact that the surface is rotated counterclockwise thereby enhancing the velocity in the z−direction while reducing the velocity in the x−direction.
It is worth noting that the presence of rotation in the ow is responsible for the dual nature of the shear stress of the ow as depicted in Fig. (5). From Fig. (6), it is shown that increasing rotation indeed causes a reduction in the shear stress within the boundary layer near the rotating surface while Fig. (7) shows that increasing rotation causes a rise in the shear stress across the uid layers near the free stream. More so, the minimum shear stress at the boundary layer is obtained at high rotation. Fig. (8) shows that increasing rotation leads to an increase in the temperature pro le. Fig. (9) shows a closer view of the results presented as Fig. (9) for . ≤ ζ ≤ . . This can be traced to the fact that an increase in the rotation causes a conversion of a portion of the internal energy into heat energy which in turn raises the overall temperature of the ow.

. Signi cance of decreasing plastic dynamic viscosity
The e ect of Casson uid parameter β on the ow of Casson uid over a rotating upper horizontal surface of a paraboloid of revolution is explored. The pertinent parameters are xed as Gr = . , Pr = , m = . and χ = . while β is varied. It is important to take note that as β → ∞, the uid reduces to Newtonian. Fig. (10) reveals that the simultaneous increase in both Casson uid parameter and rotation parameter has retarding e ects on the velocity in the x−direction. As Casson uid parameter increases simultaneously with the rotation parameter, it is found that the behaviour of the velocity pro les in  the z−direction is separated. Fig. (11) reveals that at low rotation (K = . ), increasing Casson uid parameter causes an increase in velocity pro le in the z−direction. Meanwhile, as the rotation parameter increases, there is a dual e ect of Casson uid parameter on the velocity prole in the z−direction. The results corroborates with the results of Malik et al. [31] and Patel [32]. Thus, at high rotation, the velocity pro le in the z−direction experiences an increase at the boundary layer but a decrease at the free stream; see Fig. (11). Fig. (12) shows the e ect of the rotation parameter on the shear stress. There is a dual e ect of leads to a slight increase in the temperature pro le. As further shown in Fig. (15), a simultaneous increase in both the Casson uid parameter and rotation parameter leads to an increase in the temperature pro les.

. Signi cance of increasing Prandtl number
In this section, the e ects of Coriolis force as the Prandtl number increases for the ow of Casson uid over the upper horizontal surface of the paraboloid of revolution are studied. Since Prandtl number is the ratio of the momentum di usivity to the thermal di usivity then an increase in the Prandtl number implies a decrease in the thermal di usivity or increase in the momentum di usivity. Consequently, as the Prandtl number increases, convection becomes more responsible for the transfer of energy than heat di usion. The e ects of Prandtl number was captured for Gr = , m = . , χ = . , β = . . Fig.  (16) reveals that the simultaneous increase in both Prandtl number and rotation parameter has retarding e ects on the velocity in the x−direction. Fig. (17) reveals that an increase in Prandtl number has retarding e ects on the velocity in the z−direction but an increase in the rotation parameter tends to counter the retardation caused by the increase in the Prandtl number. Thus the peak of the velocity in the z-direction is obtained at high rotation and low Prandtl number. As thermal di usivity reduces in a rotating system, the system expends some energy to balance itself, hence it is practically expected that the velocity reduces generally.
As shown in Fig. (1), the surface rotates counterclockwise and this enhances the ow in the z−direction. This is con rmed by the nding in Fig. (17). This results is in agreement with the existing results of Malik et al. [31], Patel [32], Animasaun et al. [23], and Koriko et al. [28]. As shown in Fig. (18), a simultaneous increase in Prandtl number and rotation parameter has a dual e ect on the shear stress such that the shear stress decreases close to the boundary layer but increases at the free stream. It is also pertinent to note that an increase in the rotation pa- rameter further magni es the e ect of Prandtl number on the shear stress. The minimum shear stress at the boundary layer is obtained at high rotation and high Prandtl number while the minimum shear stress at the free stream is obtained at low rotation and low Prandtl number. The result is in agreement with the result of Koriko et al. [28]. An increase in Prandtl number has a retarding e ect on the temperature pro le as shown in Fig. (19). The minimum temperature pro le is obtained at low rotation and high Prandtl number. As rotation increases, the e ect of increasing Prandtl number tends to cancel out. It can be concluded that as long as the angular speed is within a certain range, the e ect of increasing Prandtl number on the

. Signi cance of increasing buoyancy forces
The dynamics of Casson uid ow over a rotating surface with non-uniform thickness at various levels of buoyancy forces are outlined in this subsection. The parameter values are chosen as Pr = . , m = . , χ = . , β = . . It is worth mentioning here that an increase in Grashof number, which is a consequence of an increase in temper- tem gains more kinetic energy which enhances the velocity of the ow. The velocity in the x−direction reaches its peak at high Grashof number and low rotation parameter whereas the velocity in the z-direction reaches its peak at high Grashof number and high rotation. The result in Ref. [28] also agrees with this discovery. Fig. (22) shows that increase in buoyancy leads to a decrease in the temperature pro le. The zoomed portion of Fig. (23) shows that the minimum temperature pro le is reached at low rotation but high Grashof number. Temperature pro les decrease as buoyancy increases but an increase in rotation increases, alongside an increase in Grashof number, counters the e ect of increasing Grashof number on the temperature pro les. The shear stress experiences a dual effect as the Grashof number increases. This is not shown for brevity but worth mentioning. As Grashof number increases, the shear stress reduces at the boundary layer but increases in the free stream. Increasing rotation alongside Grashof number magni es the negative e ect on shear stress both at the boundary layer and at the free stream. Hence, the minimum shear stress at the free stream is experienced at high rotation and high Grashof number. The result is in agreement with the results of Hussain et al. [34] and Koriko et al. [28].   , χ = .

Quantities of interest
The quantities of interests are the Coe cient of Skin friction and heat transfer rate. It is observed from table (3) that at very low Coriolis force, increase in Casson uid pa-rameter causes at least 3.519% increase (at low β) and a maximum of 66.450% increase (at high β) in the coecient of Skin Frictions but a dual e ect on the heat transfer rate (increasing heat transfer rate when β < but decreasing heat transfer rate when β > ). Table (4) shows that at xed rotation, the Prandtl number has negative effects on the coe cient of Skin Frictions in both directions (i.e. C fx and C fz ) and signi cantly positive e ect on heat transfer rate. As Coriolis force increases, simultaneously increasing Casson uid parameter and Prandtl number reduces the skin friction coe cient in the x−direction and heat transfer rate but increases the skin friction coe cient in the z−direction. When all parameters are kept xed, it can be seen from Table (5) that rotation has a reducing effect on the heat transfer rate and the Skin Friction in the xdirection but an increasing e ect on the coe cient of Skin Friction in the z−direction. Using the slope linear regression through data points suggested in Shah et al. [35], Animasaun et al. [36], and Wakif et al. [37], it is seen that the local skin frictions in the ow along x−direction and heat transfer rate decrease with rotation parameter at the rate − . and − . respectively. However, local skin friction in the ow alongz−direction increases with the rotation parameter at the rate of . .

Conclusion
The transport phenomenon of non-Newtonian Casson uid over the upper horizontal rotating surface of a paraboloid of revolution had been analyzed. It is worth concluding that • simultaneous increase in the Prandtl number, Casson uid parameter, and Coriolis force has a negative effect on the velocity in the x−direction.
• simultaneous increase in Grashof number, Casson uid parameter, and Coriolis force has an increasing e ect on the velocity in the z−direction.
• simultaneous increase in the Prandtl number, Casson uid parameter, and Coriolis force has a negative ef- , χ = .
K C fx C fz Nux .
fect on shear stress in the boundary layer but an increasing e ect on the shear stress at the free stream. • simultaneous increase in both Casson uid parameter and Coriolis force has an increasing e ect on the temperature pro le. • At high rotation, Casson uid parameter has a dual effect on the shear stress where the shear stress is positively favoured in the boundary layer but negatively a ected in the free stream. • Coriolis force causes an increase in the temperature and the velocity pro le in the z−direction, a decrease in the velocity pro le in the x−direction, and a dual e ect on the shear stress. • Coriolis force and the Casson uid parameter have similar e ects on the ow of Casson uid over a rotating surface with non-uniform thickness.

• As Coriolis force increases, simultaneously increasing
Casson uid parameter and Prandtl number reduces the skin friction coe cient in the x−direction and heat transfer rate but increases the skin friction coe cient in the z−direction. • When all parameters are kept xed, the rotation has a reducing e ect on the heat transfer rate and the Skin Friction in the x-direction but the increasing e ect on the coe cient of Skin Friction in the z−direction.
It is important to mention here that at a very high rotation parameter, the ow becomes turbulent and a more devoted study will be required to study such ow.