Development and Experimental Validation of a Viscosity Meter for Newtonian and Non-Newtonian Fluids
Raúl O. Rojas, Juan C. Quijano, Claudia P. Tavera Ruiz, Alex F. Estupiñán L
aa r X i v : s o m e o t h e r t e x t go ee
aa r X i v : s o m e o t h e r t e x t go ee s h ee
aa r X i v : s o m e o t h e r t e x t go ee s h ee r ee
aa r X i v : s o m e o t h e r t e x t go ee s h ee r ee Development and Experimental Validation of a Viscosity Meterfor Newtonian and Non-Newtonian Fluids
Ra´ul O. Rojas , Juan C. Quijano , Claudia P. Tavera Ruiz and Alex F. Estupi˜n´an L. Universidad Aut´onoma de Bucaramanga (UNAB), Programa de Ingenier´ıa Biom´edica, Departamento deMatem´aticas y Ciencias Naturales, Santander, Colombia. Universidad de Investigaci´on y Desarrollo (UDI), Departamento de Ciencias B´asicas y Humanas,Santander, Colombia. (Dated: August 14, 2020)
Abstract
The study of viscosity, in the area of fluid physics at a university level, is of great importance becauseof the various applications that are presented in the different fields of engineering. In this work anexperimental method of implementation and validation is exposed, to be able to calculate the viscosityof some newtonian and non-newtonian fluids, in which the method of a sphere that descends through afluid has been used, we implemented a viscometer of our own construction, with the help of the CassyLabsensor and software of Leybold Didactics, we show the results obtained by our measuring instrument,which is intended to highlight the versatility and precision of the measuring instrument prepared by us,in addition. In this research the authors want to motivate the physics laboratory teachers; to explorethe use of these tools that allow you to check the topics seen in the theoretical classes. Finally, wepresent the hardworking results of the measurement of viscosity for different fluids, both newtonian andnon-newtonian, for the latter we show the viscosity behavior as a function of temperature.
Keywords:
Viscometer, Newtonian fluid, non-Newtonian fluid, descending sphere method and fluid me-chanics.
One of the great applications of the calculation of theviscosity of the fluids, is directed mainly to the areaof electrochemical research, in which it is sought torelate the electrical conductivity with the viscosity ofthe fluid to be studied, one of the most studied flu-ids in this Last decade are vegetable oils [1, 2] andmilk [3] for different temperatures. Among the mostinteresting findings, they showed that proteins andlactose affected the electrical conductivity of milk bymodifying its viscosity, in most liquid foods the con-centration of electrolytes is relatively low, with soysauce and fish sauce [4].Another field in which the viscosity of the flu-ids has a great application is civil engineering, whichconsists in determining the fluidity state of the as-phalts at the temperatures used during their applica-tion [5, 6], for example: The viscosity is measured inthe Saybolt-Furol viscosity test or in the kinematicviscosity test.The viscosity of an asphalt cement at the temper-atures used in mixing (normally 135 C) is measuredwith capillary flow meters viscometers or Saybolt vis-cometers; The absolute viscosity, at high operatingtemperatures (60 C), is usually measured with vac-uum glass capillary viscometers [7, 8]. We, too, can see the importance of viscosity intribology, such as that science that is responsible forstudying friction, which generates wear and lubrica-tion in some materials, as occurs in some mechanicalmechanisms such as teeth of a gear.Where it should be taken into account that athigher contact velocitys of these materials, a hydro-dynamic (or elastohydrodynamic) lubrication film isformed in highly charged mechanisms where the sur-faces in contact are deformed by the action of theseforces.At this stage the minimum wear values areachieved because the oil or grease that was designedfor this friction regime acts. Showing that the higherthe viscosity of the lubricant, we will have greaterlosses due to viscous friction, so lubricants with highviscosity oils are only recommended in slow-movingelements [9, 10].In this work, we present the results obtained fromthe experimental implementation, which was carriedout for the measurement of the dynamic viscosity ofNewtonian fluids such as sunflower oil, glycerine andnon-Newtonian fluids such as yogurt, cornstarch andketchup, for each of the above fluids varying the tem-perature.In addition, we show the analytical development,using two different methods, to show the validity of1 a r X i v : . [ phy s i c s . e d - ph ] A ug r X i v : s o m e o t h e r t e x t go ee
One of the great applications of the calculation of theviscosity of the fluids, is directed mainly to the areaof electrochemical research, in which it is sought torelate the electrical conductivity with the viscosity ofthe fluid to be studied, one of the most studied flu-ids in this Last decade are vegetable oils [1, 2] andmilk [3] for different temperatures. Among the mostinteresting findings, they showed that proteins andlactose affected the electrical conductivity of milk bymodifying its viscosity, in most liquid foods the con-centration of electrolytes is relatively low, with soysauce and fish sauce [4].Another field in which the viscosity of the flu-ids has a great application is civil engineering, whichconsists in determining the fluidity state of the as-phalts at the temperatures used during their applica-tion [5, 6], for example: The viscosity is measured inthe Saybolt-Furol viscosity test or in the kinematicviscosity test.The viscosity of an asphalt cement at the temper-atures used in mixing (normally 135 C) is measuredwith capillary flow meters viscometers or Saybolt vis-cometers; The absolute viscosity, at high operatingtemperatures (60 C), is usually measured with vac-uum glass capillary viscometers [7, 8]. We, too, can see the importance of viscosity intribology, such as that science that is responsible forstudying friction, which generates wear and lubrica-tion in some materials, as occurs in some mechanicalmechanisms such as teeth of a gear.Where it should be taken into account that athigher contact velocitys of these materials, a hydro-dynamic (or elastohydrodynamic) lubrication film isformed in highly charged mechanisms where the sur-faces in contact are deformed by the action of theseforces.At this stage the minimum wear values areachieved because the oil or grease that was designedfor this friction regime acts. Showing that the higherthe viscosity of the lubricant, we will have greaterlosses due to viscous friction, so lubricants with highviscosity oils are only recommended in slow-movingelements [9, 10].In this work, we present the results obtained fromthe experimental implementation, which was carriedout for the measurement of the dynamic viscosity ofNewtonian fluids such as sunflower oil, glycerine andnon-Newtonian fluids such as yogurt, cornstarch andketchup, for each of the above fluids varying the tem-perature.In addition, we show the analytical development,using two different methods, to show the validity of1 a r X i v : . [ phy s i c s . e d - ph ] A ug r X i v : s o m e o t h e r t e x t go ee s h ee
One of the great applications of the calculation of theviscosity of the fluids, is directed mainly to the areaof electrochemical research, in which it is sought torelate the electrical conductivity with the viscosity ofthe fluid to be studied, one of the most studied flu-ids in this Last decade are vegetable oils [1, 2] andmilk [3] for different temperatures. Among the mostinteresting findings, they showed that proteins andlactose affected the electrical conductivity of milk bymodifying its viscosity, in most liquid foods the con-centration of electrolytes is relatively low, with soysauce and fish sauce [4].Another field in which the viscosity of the flu-ids has a great application is civil engineering, whichconsists in determining the fluidity state of the as-phalts at the temperatures used during their applica-tion [5, 6], for example: The viscosity is measured inthe Saybolt-Furol viscosity test or in the kinematicviscosity test.The viscosity of an asphalt cement at the temper-atures used in mixing (normally 135 C) is measuredwith capillary flow meters viscometers or Saybolt vis-cometers; The absolute viscosity, at high operatingtemperatures (60 C), is usually measured with vac-uum glass capillary viscometers [7, 8]. We, too, can see the importance of viscosity intribology, such as that science that is responsible forstudying friction, which generates wear and lubrica-tion in some materials, as occurs in some mechanicalmechanisms such as teeth of a gear.Where it should be taken into account that athigher contact velocitys of these materials, a hydro-dynamic (or elastohydrodynamic) lubrication film isformed in highly charged mechanisms where the sur-faces in contact are deformed by the action of theseforces.At this stage the minimum wear values areachieved because the oil or grease that was designedfor this friction regime acts. Showing that the higherthe viscosity of the lubricant, we will have greaterlosses due to viscous friction, so lubricants with highviscosity oils are only recommended in slow-movingelements [9, 10].In this work, we present the results obtained fromthe experimental implementation, which was carriedout for the measurement of the dynamic viscosity ofNewtonian fluids such as sunflower oil, glycerine andnon-Newtonian fluids such as yogurt, cornstarch andketchup, for each of the above fluids varying the tem-perature.In addition, we show the analytical development,using two different methods, to show the validity of1 a r X i v : . [ phy s i c s . e d - ph ] A ug r X i v : s o m e o t h e r t e x t go ee s h ee r ee
One of the great applications of the calculation of theviscosity of the fluids, is directed mainly to the areaof electrochemical research, in which it is sought torelate the electrical conductivity with the viscosity ofthe fluid to be studied, one of the most studied flu-ids in this Last decade are vegetable oils [1, 2] andmilk [3] for different temperatures. Among the mostinteresting findings, they showed that proteins andlactose affected the electrical conductivity of milk bymodifying its viscosity, in most liquid foods the con-centration of electrolytes is relatively low, with soysauce and fish sauce [4].Another field in which the viscosity of the flu-ids has a great application is civil engineering, whichconsists in determining the fluidity state of the as-phalts at the temperatures used during their applica-tion [5, 6], for example: The viscosity is measured inthe Saybolt-Furol viscosity test or in the kinematicviscosity test.The viscosity of an asphalt cement at the temper-atures used in mixing (normally 135 C) is measuredwith capillary flow meters viscometers or Saybolt vis-cometers; The absolute viscosity, at high operatingtemperatures (60 C), is usually measured with vac-uum glass capillary viscometers [7, 8]. We, too, can see the importance of viscosity intribology, such as that science that is responsible forstudying friction, which generates wear and lubrica-tion in some materials, as occurs in some mechanicalmechanisms such as teeth of a gear.Where it should be taken into account that athigher contact velocitys of these materials, a hydro-dynamic (or elastohydrodynamic) lubrication film isformed in highly charged mechanisms where the sur-faces in contact are deformed by the action of theseforces.At this stage the minimum wear values areachieved because the oil or grease that was designedfor this friction regime acts. Showing that the higherthe viscosity of the lubricant, we will have greaterlosses due to viscous friction, so lubricants with highviscosity oils are only recommended in slow-movingelements [9, 10].In this work, we present the results obtained fromthe experimental implementation, which was carriedout for the measurement of the dynamic viscosity ofNewtonian fluids such as sunflower oil, glycerine andnon-Newtonian fluids such as yogurt, cornstarch andketchup, for each of the above fluids varying the tem-perature.In addition, we show the analytical development,using two different methods, to show the validity of1 a r X i v : . [ phy s i c s . e d - ph ] A ug r X i v : s o m e o t h e r t e x t go ee s h ee r ee our experiment and the quality of the data taken init. The classic definition for viscosity, arises as internalfriction force that brought in a fluid. Where it shouldbe taken into account, that these viscous forces op-pose the movement of a portion of the fluid in relationto another. Fluids that flow easily, such as water andgasoline, have lower viscosity than thick liquids suchas honey or oil. The viscosity of all fluids are verydependent on temperature, increase for gases and de-crease for liquids as the temperature rises. A viscousfluid tends to adhere to a solid surface that is in con-tact with it.In order to better understand the definition of vis-cosity of a fluid, we place a rectangular section, onsome surface free of an unconfined fluid. We hopethat on this surface a shear stress is experienced orappears in the direction parallel to the surface of thefluid. In addition, a gradient of velocities will appearin the fluid as a result of said shear stress, the veloc-ity of the sheet being equal to that of the particles incontact with it (adhesion condition), as can be seenin Figure 1.
Figure 1:
Schematic illustration of the velocity profiledefined from Newton’s vision.
There is one method to determine the viscosityof a fluid, using a falling sphere through the fluid.The method consists in release the sphere near the top surface of fluid, in order to ensure free falling andconstant velocity during the journey.While the sphere is falling, its experiencing the ac-tion of three forces: its weight or gravitational force F g , the buoyant force F b and the viscosity force F v .If the movement is considered with constant velocity,the sum of these forces is equal to zero.The buoy-ant force depends on: fluid density ρ f , gravitationalacceleration g , and the volume of the sphere V (SeeFigure 2). Besides, the force due the viscosity de-pends on radius r and velocity v of the sphere, andthe fluids viscosity η . Moreover, F g can be writtenin terms of the object density ρ obj and volume V , asseen below (See Equation (1)) [11, 12]: F b + F v = F g m s g + F v = mgρ f gV + 6 πηrv = ρ obj gV πηrv = ( ρ obj − ρ f ) gV πηrv = ∆ ρg πr (1)Solving for η we have the Equation (2): η = 2∆ ρgr v (2) Figure 2: a.) Free body diagram for the sphere insidethe fluid. b.) Schematic representation of the laminarflow disturbance present in the fluid, due to the presenceof the sphere.
The previous formula applies in case of an in-finitely extended fluid, but according to the exper-iment a correction factor is need to be added (SeeEquation (3)) [13, 14]. η = 2∆ ρgr v (cid:18)
11 + 2 . r/R (cid:19) (3)Where R is the radius of the test tube. Stokes’law is subject to a restriction in terms of its use and2 r X i v : s o m e o t h e r t e x t go ee
11 + 2 . r/R (cid:19) (3)Where R is the radius of the test tube. Stokes’law is subject to a restriction in terms of its use and2 r X i v : s o m e o t h e r t e x t go ee s h ee
11 + 2 . r/R (cid:19) (3)Where R is the radius of the test tube. Stokes’law is subject to a restriction in terms of its use and2 r X i v : s o m e o t h e r t e x t go ee s h ee r ee
11 + 2 . r/R (cid:19) (3)Where R is the radius of the test tube. Stokes’law is subject to a restriction in terms of its use and2 r X i v : s o m e o t h e r t e x t go ee s h ee r ee involves considering a laminar flow. Laminar flowis defined as that condition, in which fluid particlesmove along the smooth paths in the sheet, in otherwords, is when an orderly and smooth movement ofthe particles that form the fluid occurs.To predict the type of flow that a fluid will presentin a cross-section pipe, the Reynolds number shouldbe calculated. This dimensionless number is a ratiobetween inertial and frictional force, and takes differ-ent expressions depending on whether it is for a pipewith a non-circular transversal section, open channelsor fluid flows around a body [15]. In our experiment,it is a sphere submerged in a fluid moving with ve-locity V, in this case the Reynolds number can becalculated experimentally by the Equation (4) [16]. Re = ρ f v s rµ (4)Where ρ f is the fluid density, V s the sphere veloc-ity, r the sphere radius and the viscosity calculatedexperimentally. If the calculated Re <
Re >
Newton (cid:48) s law : τ = µ (cid:18) dvdz (cid:19) (5) Power law : τ = k (cid:18) dvdz (cid:19) n (6) Bingham (cid:48) s equation : τ = τ + η (cid:48) (cid:18) dvdz (cid:19) (7) Herschel − Bulkley (cid:48) s model : τ = τ + k H (cid:18) dvdz (cid:19) (8)Where τ is the shear stress, η viscosity, (cid:0) dvdz (cid:1) thevelocity gradient, k the consistency index, n is theflow behavior index, τ the creep threshold, η (cid:48) plas-tic viscosity and k H consistency index for Herschel-Bulkley’s fluids.Depending on the effect of shear stress on thefluid, these can be classified as Newtonian andnon-Newtonian fluids. Therefore, the mathematicalmodel to be used to determine the shear stress de-pends on the type of fluid to be used.Newtonian fluids are characterized because theirrheological behavior can be described by Newton’slaw (Equation (5)). This means that the shear stresses required to achieve a velocity are always lin-early proportional, having a constant viscosity [20].On the other hand, non-Newtonian fluids can notbe described by Newton’s law. In this case, viscosityis no longer talked about, because the ratio betweenshear stress and velocity is not constant. That vis-cosity function as a function of velocity is known asapparent viscosity. Newtonian fluids are then charac-terized by different apparent viscosities at each shearvelocity [18, 20].Non-Newtonian fluids are mainly classified as: in-dependent of the time and, dependent of the time. Influids independent of time the viscosity at any shearstress does not vary with the time, while in depen-dents ones it does. Among the time independent ofthe time fluids are the pseudoplastics, dilators andBingham fluids. The time dependent will not be stud-ied in this article therefore it will not be deepened inthem [15].The behavior of viscosity for pseudoplastics anddilators fluids can be modeled mathematically usingthe the Power Law (Equation (6)). On the otherhand, if the exponent n of the equation is smallerthan the unit ( n < n > n = 1 [13]. The set-up used for the experimental tests can be seenin Figure 3. This is made up of a graduated cylinderof 500 ml , a iron disk, a solid bronze sphere, and, twoinfrared sensors. The graduated cylinder has a massof 458 g , volume of 500 ml and radius of 23 . × − m . In order to prevent that the bronze sphere doesnot break the graduated cylinder in the fall, the irondisk is placed inside it. This disk has a mass of 68 . g and a volume of 8 . × − m . Once the iron diskis inside the graduated cylinder, this is filled with thefluid under study.As newtonian fluids were used: glycerine and sun-flower oil; as non-newtonian fluids were used: yogurt,ketchup, cornstarch and corn flour. The sensors werelocated at a height L of 0 . m . Once the fluidis inside, at room temperature, the mass sphere islaunched. The sphere has a mass of 74 . g , volumeof 8 . × − m , radius of 12 . × − m and adensity of 2635 . kg/m .The velocity of the sphere will be calculated asthe distance L routed by the sphere, divided by the3 r X i v : s o m e o t h e r t e x t go ee
Newton (cid:48) s law : τ = µ (cid:18) dvdz (cid:19) (5) Power law : τ = k (cid:18) dvdz (cid:19) n (6) Bingham (cid:48) s equation : τ = τ + η (cid:48) (cid:18) dvdz (cid:19) (7) Herschel − Bulkley (cid:48) s model : τ = τ + k H (cid:18) dvdz (cid:19) (8)Where τ is the shear stress, η viscosity, (cid:0) dvdz (cid:1) thevelocity gradient, k the consistency index, n is theflow behavior index, τ the creep threshold, η (cid:48) plas-tic viscosity and k H consistency index for Herschel-Bulkley’s fluids.Depending on the effect of shear stress on thefluid, these can be classified as Newtonian andnon-Newtonian fluids. Therefore, the mathematicalmodel to be used to determine the shear stress de-pends on the type of fluid to be used.Newtonian fluids are characterized because theirrheological behavior can be described by Newton’slaw (Equation (5)). This means that the shear stresses required to achieve a velocity are always lin-early proportional, having a constant viscosity [20].On the other hand, non-Newtonian fluids can notbe described by Newton’s law. In this case, viscosityis no longer talked about, because the ratio betweenshear stress and velocity is not constant. That vis-cosity function as a function of velocity is known asapparent viscosity. Newtonian fluids are then charac-terized by different apparent viscosities at each shearvelocity [18, 20].Non-Newtonian fluids are mainly classified as: in-dependent of the time and, dependent of the time. Influids independent of time the viscosity at any shearstress does not vary with the time, while in depen-dents ones it does. Among the time independent ofthe time fluids are the pseudoplastics, dilators andBingham fluids. The time dependent will not be stud-ied in this article therefore it will not be deepened inthem [15].The behavior of viscosity for pseudoplastics anddilators fluids can be modeled mathematically usingthe the Power Law (Equation (6)). On the otherhand, if the exponent n of the equation is smallerthan the unit ( n < n > n = 1 [13]. The set-up used for the experimental tests can be seenin Figure 3. This is made up of a graduated cylinderof 500 ml , a iron disk, a solid bronze sphere, and, twoinfrared sensors. The graduated cylinder has a massof 458 g , volume of 500 ml and radius of 23 . × − m . In order to prevent that the bronze sphere doesnot break the graduated cylinder in the fall, the irondisk is placed inside it. This disk has a mass of 68 . g and a volume of 8 . × − m . Once the iron diskis inside the graduated cylinder, this is filled with thefluid under study.As newtonian fluids were used: glycerine and sun-flower oil; as non-newtonian fluids were used: yogurt,ketchup, cornstarch and corn flour. The sensors werelocated at a height L of 0 . m . Once the fluidis inside, at room temperature, the mass sphere islaunched. The sphere has a mass of 74 . g , volumeof 8 . × − m , radius of 12 . × − m and adensity of 2635 . kg/m .The velocity of the sphere will be calculated asthe distance L routed by the sphere, divided by the3 r X i v : s o m e o t h e r t e x t go ee s h ee
Newton (cid:48) s law : τ = µ (cid:18) dvdz (cid:19) (5) Power law : τ = k (cid:18) dvdz (cid:19) n (6) Bingham (cid:48) s equation : τ = τ + η (cid:48) (cid:18) dvdz (cid:19) (7) Herschel − Bulkley (cid:48) s model : τ = τ + k H (cid:18) dvdz (cid:19) (8)Where τ is the shear stress, η viscosity, (cid:0) dvdz (cid:1) thevelocity gradient, k the consistency index, n is theflow behavior index, τ the creep threshold, η (cid:48) plas-tic viscosity and k H consistency index for Herschel-Bulkley’s fluids.Depending on the effect of shear stress on thefluid, these can be classified as Newtonian andnon-Newtonian fluids. Therefore, the mathematicalmodel to be used to determine the shear stress de-pends on the type of fluid to be used.Newtonian fluids are characterized because theirrheological behavior can be described by Newton’slaw (Equation (5)). This means that the shear stresses required to achieve a velocity are always lin-early proportional, having a constant viscosity [20].On the other hand, non-Newtonian fluids can notbe described by Newton’s law. In this case, viscosityis no longer talked about, because the ratio betweenshear stress and velocity is not constant. That vis-cosity function as a function of velocity is known asapparent viscosity. Newtonian fluids are then charac-terized by different apparent viscosities at each shearvelocity [18, 20].Non-Newtonian fluids are mainly classified as: in-dependent of the time and, dependent of the time. Influids independent of time the viscosity at any shearstress does not vary with the time, while in depen-dents ones it does. Among the time independent ofthe time fluids are the pseudoplastics, dilators andBingham fluids. The time dependent will not be stud-ied in this article therefore it will not be deepened inthem [15].The behavior of viscosity for pseudoplastics anddilators fluids can be modeled mathematically usingthe the Power Law (Equation (6)). On the otherhand, if the exponent n of the equation is smallerthan the unit ( n < n > n = 1 [13]. The set-up used for the experimental tests can be seenin Figure 3. This is made up of a graduated cylinderof 500 ml , a iron disk, a solid bronze sphere, and, twoinfrared sensors. The graduated cylinder has a massof 458 g , volume of 500 ml and radius of 23 . × − m . In order to prevent that the bronze sphere doesnot break the graduated cylinder in the fall, the irondisk is placed inside it. This disk has a mass of 68 . g and a volume of 8 . × − m . Once the iron diskis inside the graduated cylinder, this is filled with thefluid under study.As newtonian fluids were used: glycerine and sun-flower oil; as non-newtonian fluids were used: yogurt,ketchup, cornstarch and corn flour. The sensors werelocated at a height L of 0 . m . Once the fluidis inside, at room temperature, the mass sphere islaunched. The sphere has a mass of 74 . g , volumeof 8 . × − m , radius of 12 . × − m and adensity of 2635 . kg/m .The velocity of the sphere will be calculated asthe distance L routed by the sphere, divided by the3 r X i v : s o m e o t h e r t e x t go ee s h ee r ee
Newton (cid:48) s law : τ = µ (cid:18) dvdz (cid:19) (5) Power law : τ = k (cid:18) dvdz (cid:19) n (6) Bingham (cid:48) s equation : τ = τ + η (cid:48) (cid:18) dvdz (cid:19) (7) Herschel − Bulkley (cid:48) s model : τ = τ + k H (cid:18) dvdz (cid:19) (8)Where τ is the shear stress, η viscosity, (cid:0) dvdz (cid:1) thevelocity gradient, k the consistency index, n is theflow behavior index, τ the creep threshold, η (cid:48) plas-tic viscosity and k H consistency index for Herschel-Bulkley’s fluids.Depending on the effect of shear stress on thefluid, these can be classified as Newtonian andnon-Newtonian fluids. Therefore, the mathematicalmodel to be used to determine the shear stress de-pends on the type of fluid to be used.Newtonian fluids are characterized because theirrheological behavior can be described by Newton’slaw (Equation (5)). This means that the shear stresses required to achieve a velocity are always lin-early proportional, having a constant viscosity [20].On the other hand, non-Newtonian fluids can notbe described by Newton’s law. In this case, viscosityis no longer talked about, because the ratio betweenshear stress and velocity is not constant. That vis-cosity function as a function of velocity is known asapparent viscosity. Newtonian fluids are then charac-terized by different apparent viscosities at each shearvelocity [18, 20].Non-Newtonian fluids are mainly classified as: in-dependent of the time and, dependent of the time. Influids independent of time the viscosity at any shearstress does not vary with the time, while in depen-dents ones it does. Among the time independent ofthe time fluids are the pseudoplastics, dilators andBingham fluids. The time dependent will not be stud-ied in this article therefore it will not be deepened inthem [15].The behavior of viscosity for pseudoplastics anddilators fluids can be modeled mathematically usingthe the Power Law (Equation (6)). On the otherhand, if the exponent n of the equation is smallerthan the unit ( n < n > n = 1 [13]. The set-up used for the experimental tests can be seenin Figure 3. This is made up of a graduated cylinderof 500 ml , a iron disk, a solid bronze sphere, and, twoinfrared sensors. The graduated cylinder has a massof 458 g , volume of 500 ml and radius of 23 . × − m . In order to prevent that the bronze sphere doesnot break the graduated cylinder in the fall, the irondisk is placed inside it. This disk has a mass of 68 . g and a volume of 8 . × − m . Once the iron diskis inside the graduated cylinder, this is filled with thefluid under study.As newtonian fluids were used: glycerine and sun-flower oil; as non-newtonian fluids were used: yogurt,ketchup, cornstarch and corn flour. The sensors werelocated at a height L of 0 . m . Once the fluidis inside, at room temperature, the mass sphere islaunched. The sphere has a mass of 74 . g , volumeof 8 . × − m , radius of 12 . × − m and adensity of 2635 . kg/m .The velocity of the sphere will be calculated asthe distance L routed by the sphere, divided by the3 r X i v : s o m e o t h e r t e x t go ee s h ee r ee time that it takes to travel that distance (Equation(9)). The sensor will detect the passage of the sphereand record the time ∆ t . This time data collectionwas performed using the CASSY LAB-LD Didacticsystem software (as shown in Figure 3). For eachfluid, the sphere was launched 10 times. For eachlaunch, the time taken by the sphere to travel thedistance L was measured by the detectors. The timeof all releases was averaged. With the average timethe velocity was calculated (See Equation (9)). v = L ∆ t (9) Figure 3:
Viscometer experimental set-up.
The experimental viscosity, for each fluid was cal-culated as shown in Equation (10), where ρ s is thesphere density, ρ f is the fluid density, ∆ t the averagetime, L the distance travelled, r the sphere radius, R the graduated cylinder radius and g is the local grav-ity. Where the density of the mass of the iron diskand the density of the fluid were calculated from themeasurement of its mass and volume. η = (cid:20)
29 ( ρ s − ρ f ) (cid:18) ∆ tL (cid:19) r (cid:21) · (cid:34) g . (cid:0) rR (cid:1) (cid:35) (10)In the case of non-Newtonian fluids, these wereheated using a water bath as an external heat source at the bottom of the cylinder. In this case, data col-lection was performed as the fluid temperature in-creased. For these fluids type, the fall time ∆ t of thebronze sphere as a function of temperature was mea-sured, increasing it by 1 to 2 C on average from theambient temperature (25 C). The number of timedata, which were recorded in this case, based on a cer-tain value of a temperature increase described above,from room temperature was 10 times.
To carry out the analysis of the results obtained inthis research, we wanted to start by analyzing thenon-Newtonian fluids that we have worked on in thisresearch, which were:1. Yogurt.2. corn flour.3. Ketchup.4. Cornstarch.On the other hand, we also work with two New-tonian fluids, which are:1. Sunflower oil.2. Glycerine.Taking into account this order of presentation ofour results, we present below the results obtained fornon-Newtonian yogurt fluid.
To begin with the analysis of the results obtained forthis fluid, we have taken The average times obtainedof the travel of the sphere in the distance L ; at differ-ent temperatures are presented in the Table 1. Withthese times, the data of the fluid density, density anddimensions of the sphere were calculated, by replac-ing the viscosity in Equation (10) (See Table 1). Itwas found that when the temperature increases, thetime decreases, which represents a lower shear stressat a higher temperature, that is, a lower viscosity ofthe fluid. This variation in viscosity with tempera-ture proves that ketchup sauce is a non-Newtonianfluid.4 r X i v : s o m e o t h e r t e x t go ee
To begin with the analysis of the results obtained forthis fluid, we have taken The average times obtainedof the travel of the sphere in the distance L ; at differ-ent temperatures are presented in the Table 1. Withthese times, the data of the fluid density, density anddimensions of the sphere were calculated, by replac-ing the viscosity in Equation (10) (See Table 1). Itwas found that when the temperature increases, thetime decreases, which represents a lower shear stressat a higher temperature, that is, a lower viscosity ofthe fluid. This variation in viscosity with tempera-ture proves that ketchup sauce is a non-Newtonianfluid.4 r X i v : s o m e o t h e r t e x t go ee s h ee
To begin with the analysis of the results obtained forthis fluid, we have taken The average times obtainedof the travel of the sphere in the distance L ; at differ-ent temperatures are presented in the Table 1. Withthese times, the data of the fluid density, density anddimensions of the sphere were calculated, by replac-ing the viscosity in Equation (10) (See Table 1). Itwas found that when the temperature increases, thetime decreases, which represents a lower shear stressat a higher temperature, that is, a lower viscosity ofthe fluid. This variation in viscosity with tempera-ture proves that ketchup sauce is a non-Newtonianfluid.4 r X i v : s o m e o t h e r t e x t go ee s h ee r ee
To begin with the analysis of the results obtained forthis fluid, we have taken The average times obtainedof the travel of the sphere in the distance L ; at differ-ent temperatures are presented in the Table 1. Withthese times, the data of the fluid density, density anddimensions of the sphere were calculated, by replac-ing the viscosity in Equation (10) (See Table 1). Itwas found that when the temperature increases, thetime decreases, which represents a lower shear stressat a higher temperature, that is, a lower viscosity ofthe fluid. This variation in viscosity with tempera-ture proves that ketchup sauce is a non-Newtonianfluid.4 r X i v : s o m e o t h e r t e x t go ee s h ee r ee Table 1:
Experimental data on the viscosity of Yogurtfor different temperatures.
T [K] Average time [s] Viscosity [Pa · s] kg/m [11]. Because yogurt is anon-Newtonian fluid, a exponential adjustment wasfitted, obtaining a good approximation with an R of 0 . η ( T ) = η · e ERT , (11)where η is the viscosity of fluid, η is a pre-exponential factor, E is the activation energy, R isthe ideal gases universal constant and T is the abso-lute temperature of fluid. Figure 4:
Viscosity as a function of temperature fornon-Newtonian fluids.a.) Yogurt. b.) Corn flour. c.)Ketchup. d.) Cornstarch.
Now, we can also perform a linear adjustment ofthe Arrhenius’s equation (See Equation (11)), apply- ing the method of least squares, where Equation (11),we can linearize it as follows:ln[ η ( T )] = ln (cid:104) η · e ERT (cid:105) , (12)ln[ η ( T )] = ln[ η ] + ln (cid:104) e ERT (cid:105) (13)ln[ η ( T )] = ln[ η ] + (cid:20) ER (cid:21) · T , (14) y = b + m · x. (15)In this way, comparing Equation (14) with Equa-tion (15), it can be seen that making the graph ofln η as a function of [1 /T ], where moreover makinga linear adjustment (See Figure 5), the value of theactivation energy E of the fluid can be obtained ex-perimentally Figure 5:
Linear fitting of viscosity as a function of tem-perature for non-Newtonian fluids. a.) Yogurt. b.) Cornflour. c.) Ketchup. d.) Cornstarch.
From the linearization used in Equation (14) topower the linear fit, shown in Figure 5b, the value ofthe activation energy E can be calculated, in the caseof yogurt, as can be seen in the equations from (16)to (17). ER = m, (16) E = m · R. (17)Where m is the slope of Figure 5b, which corre-sponds to a numerical value of m = 330 . K , withthis value; and using the value of the universal con-stant of ideal gases given by the unit literature in theI.S system, we have R = 8 . J/ ( K · mol ), we ob-tain the result of the activation energy E , shown inequations (18-19),5 r X i v : s o m e o t h e r t e x t go ee
From the linearization used in Equation (14) topower the linear fit, shown in Figure 5b, the value ofthe activation energy E can be calculated, in the caseof yogurt, as can be seen in the equations from (16)to (17). ER = m, (16) E = m · R. (17)Where m is the slope of Figure 5b, which corre-sponds to a numerical value of m = 330 . K , withthis value; and using the value of the universal con-stant of ideal gases given by the unit literature in theI.S system, we have R = 8 . J/ ( K · mol ), we ob-tain the result of the activation energy E , shown inequations (18-19),5 r X i v : s o m e o t h e r t e x t go ee s h ee
From the linearization used in Equation (14) topower the linear fit, shown in Figure 5b, the value ofthe activation energy E can be calculated, in the caseof yogurt, as can be seen in the equations from (16)to (17). ER = m, (16) E = m · R. (17)Where m is the slope of Figure 5b, which corre-sponds to a numerical value of m = 330 . K , withthis value; and using the value of the universal con-stant of ideal gases given by the unit literature in theI.S system, we have R = 8 . J/ ( K · mol ), we ob-tain the result of the activation energy E , shown inequations (18-19),5 r X i v : s o m e o t h e r t e x t go ee s h ee r ee
From the linearization used in Equation (14) topower the linear fit, shown in Figure 5b, the value ofthe activation energy E can be calculated, in the caseof yogurt, as can be seen in the equations from (16)to (17). ER = m, (16) E = m · R. (17)Where m is the slope of Figure 5b, which corre-sponds to a numerical value of m = 330 . K , withthis value; and using the value of the universal con-stant of ideal gases given by the unit literature in theI.S system, we have R = 8 . J/ ( K · mol ), we ob-tain the result of the activation energy E , shown inequations (18-19),5 r X i v : s o m e o t h e r t e x t go ee s h ee r ee E = 330 . · . , (18) E = 2 . kJ/mol. (19)Continuing with the presentation of the results forYogurt, we have constructed Table 2, which shows thecalculation of the Reynolds number using Equation(4); for different temperature values. Table 2:
Experimental data from the calculation of theReynolds number for yogurt as a function of the fallingvelocity of the sphere.
Viscosity [Pa · s] Velocity v [m/s] Re . The experimental density that was obtained for thecorn flour was 602 . kg/m . The experimental dataof the average time and viscosity for different tem-peratures are shown in Table 3. Table 3: Experimental data on the viscosity of corn flourfor different temperatures. T [K] Average time [s] Viscosity [Pa · s] R = 0 . E ofcorn flour can be calculated, using the linearizationof Equation (14) and Equation (15), with these Twoequations, the result shown in Equation (21) of theenergy E is obtained for corn flour. E = 9290 . · . , (20) E = 77 . kJ/mol. (21)Continuing with the presentation of the results,obtained for corn flour, we can calculate the valueof the Reynolds number (Re), given for each of thedifferent temperatures, at which this experiment wascarried out (See Equation (4)) . In Table 4, we showthe value of the fluid viscosity (corn flour), as a func-tion of the falling velocity v , for each of the testscarried out. The results of these experimental calcu-lations are presented in Table 4. Table 4: Experimental data from the calculation of theReynolds number for corn fluor as a function of the fallingvelocity of the sphere. Viscosity [Pa · s] Velocity v [m/s] Re . < Re < . Re < r X i v : s o m e o t h e r t e x t go ee Viscosity [Pa · s] Velocity v [m/s] Re . < Re < . Re < r X i v : s o m e o t h e r t e x t go ee s h ee Viscosity [Pa · s] Velocity v [m/s] Re . < Re < . Re < r X i v : s o m e o t h e r t e x t go ee s h ee r ee Viscosity [Pa · s] Velocity v [m/s] Re . < Re < . Re < r X i v : s o m e o t h e r t e x t go ee s h ee r ee The average times obtained of the travel of the spherein the distance L at different temperatures are pre-sented in Table 5. With these times, the data of thefluid density, dimensions and density of the spherewere calculated by replacing the viscosity in Equa-tion (10) (See Table 5). It was found that when thetemperature increases, the time decreases, which rep-resents a lower shear stress at a higher temperature,that is, a lower viscosity of the fluid. This variation inviscosity with temperature proves that ketchup sauceis a non-Newtonian fluid. Table 5: Experimental data on the viscosity of ketchupfor different temperatures. T [K] Average time [s] Viscosity [Pa · s] 298 1.677 19.14299 1.492 17.04301 1.268 14.48303 1.023 11.68305 0.786 9.47308 0.659 7.53The results of the ketchup viscosity at differenttemperatures are presented in the Figure 4b. Equa-tion (10) was used to calculated the viscosity at eachtemperature, in this case the density of the ketchupused was 1235 kg/m [11]. Because ketchup is a non-Newtonian fluid, a power adjustment was fitted, ob-taining a good approximation with an R of 0 . E ofketchup can be calculated, using the linearization ofEquation (14) and Equation (15), with these Twoequations, the result shown in Equation (23) of theenergy E is obtained for ketchup. E = 8861 . · . , (22) E = 73 . kJ/mol. (23)On the other hand, the Reynolds number can becalculated as a function of the viscosity using the ex-pression of the Equation (4). In the Table 6, showsthe Reynolds number for each temperature and eachvelocity reached for the sphere in the ketchup fluid. Table 6: Experimental data from the calculation of theReynolds number for ketchup as a function of the fallingvelocity of the sphere. Viscosity [Pa · s] Velocity v [m/s] Re The average times obtained of the travel of the spherein the distance L at different temperatures are pre-sented in Table 7. With these times, the data of thefluid density, density and dimensions of the spherewere calculated by replacing the viscosity in Equa-tion (10) (See Table 7). It was found that when thetemperature increases, the time decreases, which rep-resents a lower shear stress at a higher temperature,that is, a lower viscosity of the fluid. This variationin viscosity with temperature proves that cornstarchsauce is a non-Newtonian fluid. Table 7: Experimental data on the viscosity of corn-starch for different temperatures. T [K] Average time [s] Viscosity [Pa · s] 299 1.1784 14.025300 1.1784 14.025303 1.019 12.127305 0.8284 9.855305 0.8277 9.850306 0.8026 9.552The results of the ketchup viscosity at differenttemperatures are presented in the Figure 4c. Equa-tion (10) was used to calculated the viscosity at eachtemperature, in this case the density of the cornstarchused was 1207 , kg/m [11]. Because cornstarch is anon-Newtonian fluid, a power adjustment was fitted,obtaining a good approximation with an R of 0 . E ofcornstarch can be calculated, using the linearizationof Equation (14) and Equation (15), with these two7 r X i v : s o m e o t h e r t e x t go ee 299 1.1784 14.025300 1.1784 14.025303 1.019 12.127305 0.8284 9.855305 0.8277 9.850306 0.8026 9.552The results of the ketchup viscosity at differenttemperatures are presented in the Figure 4c. Equa-tion (10) was used to calculated the viscosity at eachtemperature, in this case the density of the cornstarchused was 1207 , kg/m [11]. Because cornstarch is anon-Newtonian fluid, a power adjustment was fitted,obtaining a good approximation with an R of 0 . E ofcornstarch can be calculated, using the linearizationof Equation (14) and Equation (15), with these two7 r X i v : s o m e o t h e r t e x t go ee s h ee 299 1.1784 14.025300 1.1784 14.025303 1.019 12.127305 0.8284 9.855305 0.8277 9.850306 0.8026 9.552The results of the ketchup viscosity at differenttemperatures are presented in the Figure 4c. Equa-tion (10) was used to calculated the viscosity at eachtemperature, in this case the density of the cornstarchused was 1207 , kg/m [11]. Because cornstarch is anon-Newtonian fluid, a power adjustment was fitted,obtaining a good approximation with an R of 0 . E ofcornstarch can be calculated, using the linearizationof Equation (14) and Equation (15), with these two7 r X i v : s o m e o t h e r t e x t go ee s h ee r ee 299 1.1784 14.025300 1.1784 14.025303 1.019 12.127305 0.8284 9.855305 0.8277 9.850306 0.8026 9.552The results of the ketchup viscosity at differenttemperatures are presented in the Figure 4c. Equa-tion (10) was used to calculated the viscosity at eachtemperature, in this case the density of the cornstarchused was 1207 , kg/m [11]. Because cornstarch is anon-Newtonian fluid, a power adjustment was fitted,obtaining a good approximation with an R of 0 . E ofcornstarch can be calculated, using the linearizationof Equation (14) and Equation (15), with these two7 r X i v : s o m e o t h e r t e x t go ee s h ee r ee equations, the result shown in Equation (25) of theenergy E is obtained for cornstarch. E = 5728 . · . , (24) E = 47 . kJ/mol. (25)On the other hand, the Reynolds number can becalculated as a function of the viscosity using theexpression of the Equation (4). Table 8, shows theReynolds number for each temperature and each ve-locity reached for the sphere in the cornstarch fluid. Table 8: Experimental data from the calculation ofthe Reynolds number for cornstarch as a function of thefalling velocity of the sphere. Viscosity [Pa · s] Velocity v [m/s] Re In the case of sunflower oil, where it has a Newtonianfluid behavior, in which with increasing temperature,the viscosity value will remain constant. It is for thisreason that we only take the fall time of the sphere,for a single temperature, which in our case was 297 K . Using Equation (10), for an experimentally mea-sured density value equal to 907 . kg/m , a viscosityvalue was obtained for sunflower oil, equal to 41 . × − P a · s [21, 22]. Carrying out the respectivecomparison, with the theoretical value reported bythe literature, we found a percentage error of 5 . 57 %,in which a great closeness with the theoretical valuecould be evidenced; which is evidence that an exper-imental procedure could be successfully carried out,with the purpose of indirectly measuring the viscosityof sunflower oil; this being a Newtonian fluid. For the Newtonian fluid glycerine, 490 . mL ofglycerine were taken in the cylinder, this volume wasweighed obtaining a mass of 1 . kg . The den-sity was calculated by dividing the mass of glycer-ine in the volume, obtaining an experimental den-sity of 1177 . kg/m . The distance traveled by thesphere was 0 . m and the average time was 0 . s . With these values the velocity was calculated ob-taining 0 . m/s . These values were replaced inEquation (10) and the viscosity value was calculated,obtaining an experimental value of 1 . P a · s . Thereported viscosity value for commercial glycerine ata temperature of 298 K is 1 . P a · s [23, 24], whichindicates that the experimental value obtained in thisstudy is very close to the reported theoretical value. The authors: Alex Estupi˜n´an and Claudia Tavera,would like to express their thanks, especially to theUniversidad de Investigaci´on y Desarrollo UDI , for allthe human, material and financial support to carryout this research work. Also the authors: Ra´ul Rojasand Juan Quijano, they want to thank the Universi-dad Aut´onoma de Bucaramanga UNAB , for provid-ing us with technical and financial support. In this work was possible to implement an experimen-tal protocol to perform the indirect measurement ofthe viscosity of Newtonian and non-Newtonian flu-ids. The results obtained in this research are shownwith high precision and accuracy, and it is possibleto catalog the behaviour of the different fluids.It was demonstrated that our prototype can accu-rately measure the viscosity of both Newtonian andnon-Newtonian fluids, in addition to working prop-erly at different temperature conditions.Additionally, due to the good adjustments pre-sented in the graphs of 1 /T Vs Ln of viscosity, itwas possible to calculate the transition energy of themolecules, which are in accordance with the valuespresented in the literature. Likewise, experimentalvalues of the Reynolds number could be obtained,with which it is possible to predict the behavior ofthe fluid. In this case, the fluids were in steady flow,which could be verified with the Re values obtained References [1] Amado, E., & Mora, L. (2006). Anlisis de lavariacin de la viscosidad cinemtica de un aceitevegetal en funcin de la temperatura. Bistua: Re-vista de la Facultad de Ciencias Bsicas, 4(2), 54-56.[2] Malkin, A. Y., & Khadzhiev, S. N. (2016). Onthe rheology of oil. Petroleum Chemistry, 56(7),541-551.8 r X i v : s o m e o t h e r t e x t go ee In this work was possible to implement an experimen-tal protocol to perform the indirect measurement ofthe viscosity of Newtonian and non-Newtonian flu-ids. The results obtained in this research are shownwith high precision and accuracy, and it is possibleto catalog the behaviour of the different fluids.It was demonstrated that our prototype can accu-rately measure the viscosity of both Newtonian andnon-Newtonian fluids, in addition to working prop-erly at different temperature conditions.Additionally, due to the good adjustments pre-sented in the graphs of 1 /T Vs Ln of viscosity, itwas possible to calculate the transition energy of themolecules, which are in accordance with the valuespresented in the literature. Likewise, experimentalvalues of the Reynolds number could be obtained,with which it is possible to predict the behavior ofthe fluid. In this case, the fluids were in steady flow,which could be verified with the Re values obtained References [1] Amado, E., & Mora, L. (2006). Anlisis de lavariacin de la viscosidad cinemtica de un aceitevegetal en funcin de la temperatura. Bistua: Re-vista de la Facultad de Ciencias Bsicas, 4(2), 54-56.[2] Malkin, A. Y., & Khadzhiev, S. N. (2016). Onthe rheology of oil. Petroleum Chemistry, 56(7),541-551.8 r X i v : s o m e o t h e r t e x t go ee s h ee In this work was possible to implement an experimen-tal protocol to perform the indirect measurement ofthe viscosity of Newtonian and non-Newtonian flu-ids. The results obtained in this research are shownwith high precision and accuracy, and it is possibleto catalog the behaviour of the different fluids.It was demonstrated that our prototype can accu-rately measure the viscosity of both Newtonian andnon-Newtonian fluids, in addition to working prop-erly at different temperature conditions.Additionally, due to the good adjustments pre-sented in the graphs of 1 /T Vs Ln of viscosity, itwas possible to calculate the transition energy of themolecules, which are in accordance with the valuespresented in the literature. Likewise, experimentalvalues of the Reynolds number could be obtained,with which it is possible to predict the behavior ofthe fluid. In this case, the fluids were in steady flow,which could be verified with the Re values obtained References [1] Amado, E., & Mora, L. (2006). Anlisis de lavariacin de la viscosidad cinemtica de un aceitevegetal en funcin de la temperatura. Bistua: Re-vista de la Facultad de Ciencias Bsicas, 4(2), 54-56.[2] Malkin, A. Y., & Khadzhiev, S. N. (2016). Onthe rheology of oil. Petroleum Chemistry, 56(7),541-551.8 r X i v : s o m e o t h e r t e x t go ee s h ee r ee