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Fluid Dynamics

Variables and Equations

Known variables from QGEM resources and equations from literature are employed to characterize the fluid dynamics of the pacifier. The protein size is given at 76.55 kilodaltons, with a concentration in water (as a protein solution) of 2.37 mg/ml. 100μL of saliva is required to interact with the protein solution, although more should be collected to ensure that the minimum requirement is met. The diameter of the inner pacifier tube leading from the saliva input to the protein container is 1.5mm, while the length is 12mm. Reynold’s equation is given in Equation 1:

where ρ is the density of the fluid (kg∙m-3), D is the diameter (m), V is the velocity of the fluid (m∙s-1), and μ is the dynamic viscosity of the fluid (Pa∙s).

The Hagen-Poiseuille equation is given in Equation 2:

where ∆P is the pressure drop across the two pipe ends (Pa), μ is the dynamic viscosity of the fluid (Pa∙s), L is the length of the pipe (m), Q is the volumetric flow rate of the fluid (m3∙s-1), and R is the radius of the pipe (m).

In order to analyze the flow of the saliva using Reynolds number, the dynamic viscosity and density of saliva is needed. Additionally, the velocity may be required to determine the exact Reynolds number, or a desired Reynolds number may be assigned to calculate the necessary velocity. Following a literature review, a mean value of 2.33 mPa∙s, or 0.0023 Pa∙s, was determined to be the dynamic viscosity of adult human saliva [1]. Negoro et al. studied the viscosity of saliva in five-year-old children, and their results agree with this value [2]. Lamey and Nolan found the mean density of human saliva to be 0.978 kg∙L-1 or 978 kg∙m^-3 [3].

Calculations and Discussion

Obtaining a sufficient amount of saliva

In order to properly characterize the flow within the tube, the volume and velocity of the fluid must be calculated. The flow profile depends on the radius of the flow, and it is predicted that the flow will not fill the tube, due to the small flow rate of saliva from an infant. Calculations are performed based on a partially full pipe model. Additionally, 100μL of saliva must reach the protein chamber to ensure proper mixing and a successful reaction. Based on previous work, infants and young children are able to produce approximately 1.5mL of saliva every 5 minutes, or 5μL of saliva per second, when salivary glands have been appropriately stimulated [4]. Salimetrics, an industry leader in saliva analysis, reports that 4- and 5-month-olds are able to produce approximately 250μL of saliva in 75 seconds, or approximately 3.33μL of saliva per second [5]. For this report, a median value of 4μL per second will be assumed. The pacifier should collect the required 100μL without input from an assistant, so it is important to ensure that the collection tube is of sufficient length and diameter to minimize saliva travel time to the protein solution.

A liberal estimate of a half-full internal tube will be used for this analysis, since the flow of saliva will be relatively small compared to the tube’s full capacity. The required volume of saliva for a half-full tube can be determined with Equation 3:

The required volume for a full tube can be determined with Equation 4.

Both volumes of saliva could theoretically be produced by an infant in under 50 seconds, which is reasonable and achievable within a typical collection period. It should be noted that, depending on the mechanism, actual rates of saliva collection may be lower than those found in literature. The reported values are associated with syringe extraction from the lip of the patient, while QGEM’s pacifier uses a more passive collection mechanism. However, even with an extraction rate five times lower than that found in literature, the collection process would take less than five minutes, which is still an acceptable amount of time. Further research should be conducted to determine the flow rate of saliva using QGEM’s pacifier mechanism. At this point within the project, no changes to the dimensions of inner tube are necessary, although the diameter of the tube should be increased while decreasing the length of the tube if a lower collection time is desired.

Reynolds number

From the calculation of the Reynolds number, all salivary flow within the pacifier corresponds to laminar flow. This calculation holds at speeds of up to 1000 m/s, while actual fluid speeds within the pacifier will be in the order of magnitude of centimetres per second. Stokes flow, or creeping flow, is also applicable to this situation, since the Reynolds number is extremely small (around 10^-6, where Stokes flow is Re << 1). This indicates that flow within the pacifier will take on a parabolic flow profile, as is displayed in Figure 1. At higher Reynolds numbers, the flow would become turbulent and begin to mix in with itself, which would be desirable for interaction with the protein solution. A turbulent flow would contact more of the protein solution without any need for physical agitation. However, due to the extremely small calculated Reynolds number, creating a turbulent flow within the pacifier tube is not a realistic goal.

This representation of laminar flow relies on the assumption that the velocity at each boundary, where the water touches the pipe, is zero. In a half-full pipe, this does not hold. A more realistic depiction of laminar flow in the pacifier tube is displayed in Figure 3. The arrows represent velocity, with the maximum velocity occurring at the middle of the tube, due to the relatively negligible friction, and a velocity of zero at the wall where the water is in contact the tube. The behaviour of the water at the tube’s edge is referred to as a “no-slip condition”, a fundamental assumption in fluid dynamics calculations.

Pressure needed to translate saliva

The Navier-Stokes equations can be used in this simple pipe flow model, since it is assumed that the fluid is laminar, has a constant density, and is Newtonian [7]. The Hagen-Poiseuille law can also be applied to characterize the saliva flow, since it satisfies the necessary assumptions: the flow is fully developed, steady, axisymmetric, and without radial or swirl components [8]. Rearranging the Hagen-Poiseuille equation shows that the flow rate is proportional to the pressure drop and radius, and inversely proportional to the viscosity and pipe length.
To calculate the pressure drop, Equation 2 can be used as follows:

A small pressure drop of 0.8885Pa could easily be overcome by tilting the pacifier after saliva collection, using gravitational energy to transport the saliva from the inner tubing to the protein chamber. Additionally, previous research has shown that the application of a surfactant or hydrophobic coating to a pipe is able to significantly reduce friction and drag within the pipe, increasing velocity as a result [10], [11]. The addition of a surfactant or hydrophobic coating would reduce the pressure, and therefore time, needed for the saliva to reach the protein solution.

Brownian model of saliva in protein solution

The motion of cortisol within saliva has been modelled with MATLAB, as shown in Figure 4. The motion of cortisol within saliva mixing with the protein solution is also valuable to model. This allows for the quantification of light-producing reactions between cortisol and the protein. Unfortunately, an error found within the MATLAB code was never fully solved, which limited the effectiveness of calculations performed using this model. Further mathematical modelling is included in another section. The MATLAB code can be seen below:

Brownian model in MATLAB

anArray = zeros(1,10);          %Initialize storage array
storageCounterMax = 10;

for storageCounter = 1:storageCounterMax

%Variable Declaration
n = 1;                          %Initialize loop counter
N=0.25;                         %Range from random selection will be [-N,N]
loopMax=10000;                  %Number of random walk movements
ColorSet = varycolor(loopMax);  %Set amount of colours
diameter = 6.94;                %Set diameter of cylinder
radius = diameter*0.5;          %Set radius of cylinder
totalHeight = 10.85;            %Set total height of container
pauseTime = 0.0;                %Set time between plotting iterations
boundaryCondition = true;

%Cylinder Figure
maxHeight = totalHeight - radius;
h=maxHeight;
x0=0;y0=0;z0=0;
[x,y,z]=cylinder(radius);
x=x+x0;
y=y+y0;
z=z*h+z0;

%Half Sphere Figure
A = [0 0 0 radius];
[X, Y, Z] = sphere;
XX = X * A(4) + A(1);
YY = Y * A(4) + A(2);
ZZ = Z * A(4) + A(3);
XX = XX(11:end,:);
YY = YY(11:end,:);
ZZ = -ZZ(11:end,:);

%Display Information
figure
xlabel('X');
ylabel('Y');
zlabel('Z');
surface(x,y,z, 'FaceAlpha', 0.1, 'FaceColor', [ 1 1 0], 'EdgeColor', [0.4, 0.4, 0.4]); hold on
surface(XX, YY, ZZ, 'FaceAlpha', 0.1, 'FaceColor', [ 1 1 0], 'EdgeColor', [0.4, 0.4, 0.4]); hold on
axis equal
view(3);
set(gca,'GridLineStyle','-')
Information=strcat('Brownian Motion in 3D Space');
title(Information ,'FontWeight','bold');
view(-109,40);
hold all
set(gca, 'ColorOrder', ColorSet);


%Initialize Beginning Location
heightInitialize = maxHeight + radius;
xF = (2*radius*rand())-radius;
yF = (2*radius*rand())-radius;
zF = (heightInitialize*rand())-radius;

while(boundaryCondition == true)
    
    n = n + 1;

    xI = xF;
    yI = yF;
    zI = zF;

    xF = xI + (2*N*rand())-N;
    yF = yI + (2*N*rand())-N;
    zF = zI + (2*N*rand())-N;
    
    dist = sqrt(xF^2 + yF^2);
    length = sqrt(xF^2 + yF^2 + zF^2);

    
    if (zF >= maxHeight)
        flag = 1;
        disp(n);
        break;
    end
    
    if (zF >= 0 && zF < maxHeight && dist < radius)

        v1=[xI,yI,zI];
        v2=[xF,yF,zF];
        v=[v2;v1];
        plot3(v(:,1),v(:,2),v(:,3));
        pause(pauseTime);
    
    elseif (zF < 0 && length < radius)
        
        v1=[xI,yI,zI];
        v2=[xF,yF,zF];
        v=[v2;v1];
        plot3(v(:,1),v(:,2),v(:,3));
        pause(pauseTime);
    
    else
        
        boundaryCondition = false;
        while(boundaryCondition == false)
       
            xF = xI + (2*N*rand())-N;
            yF = yI + (2*N*rand())-N;
            zF = zI + (2*N*rand())-N;

            length = sqrt(xF^2 + yF^2 + zF^2);
            dist = sqrt(xF^2 + yF^2);
        
            if ((zF >= 0 && zF < maxHeight && dist < radius)||(zF < 0 && length < radius))
                boundaryCondition = true;
                v1=[xI,yI,zI];
                v2=[xF,yF,zF];
                v=[v2;v1];
                plot3(v(:,1),v(:,2),v(:,3));
                pause(pauseTime);
                
            end
        end
    end
end

anArray(storageCounter)=n;
disp(anArray);
end

plot3(xF,yF,zF,'ok','MarkerFaceColor','r');
hold off
disp(anArray)
disp(n)

function ColorSet=varycolor(NumberOfPlots)
% VARYCOLOR Produces colors with maximum variation on plots with multiple
% lines.
%
%     VARYCOLOR(X) returns a matrix of dimension X by 3.  The matrix may be
%     used in conjunction with the plot command option 'color' to vary the
%     color of lines.  
%
%     Yellow and White colors were not used because of their poor
%     translation to presentations.
% 
%     Example Usage:
%         NumberOfPlots=50;
%
%         ColorSet=varycolor(NumberOfPlots);
% 
%         figure
%         hold on;
% 
%         for m=1:NumberOfPlots
%             plot(ones(20,1)*m,'Color',ColorSet(m,:))
%         end

%Created by Daniel Helmick 8/12/2008

narginchk(1,1)%correct number of input arguements??
nargoutchk(0, 1)%correct number of output arguements??

%Take care of the anomolies
if NumberOfPlots<1
    ColorSet=[];
elseif NumberOfPlots==1
    ColorSet=[0 1 0];
elseif NumberOfPlots==2
    ColorSet=[0 1 0; 0 1 1];
elseif NumberOfPlots==3
    ColorSet=[0 1 0; 0 1 1; 0 0 1];
elseif NumberOfPlots==4
    ColorSet=[0 1 0; 0 1 1; 0 0 1; 1 0 1];
elseif NumberOfPlots==5
    ColorSet=[0 1 0; 0 1 1; 0 0 1; 1 0 1; 1 0 0];
elseif NumberOfPlots==6
    ColorSet=[0 1 0; 0 1 1; 0 0 1; 1 0 1; 1 0 0; 0 0 0];

else %default and where this function has an actual advantage

    %we have 5 segments to distribute the plots
    EachSec=floor(NumberOfPlots/5); 
    
    %how many extra lines are there? 
    ExtraPlots=mod(NumberOfPlots,5); 
    
    %initialize our vector
    ColorSet=zeros(NumberOfPlots,3);
    
    %This is to deal with the extra plots that don't fit nicely into the
    %segments
    Adjust=zeros(1,5);
    for m=1:ExtraPlots
        Adjust(m)=1;
    end
    
    SecOne   =EachSec+Adjust(1);
    SecTwo   =EachSec+Adjust(2);
    SecThree =EachSec+Adjust(3);
    SecFour  =EachSec+Adjust(4);
    SecFive  =EachSec;

    for m=1:SecOne
        ColorSet(m,:)=[0 1 (m-1)/(SecOne-1)];
    end

    for m=1:SecTwo
        ColorSet(m+SecOne,:)=[0 (SecTwo-m)/(SecTwo) 1];
    end
    
    for m=1:SecThree
        ColorSet(m+SecOne+SecTwo,:)=[(m)/(SecThree) 0 1];
    end
    
    for m=1:SecFour
        ColorSet(m+SecOne+SecTwo+SecThree,:)=[1 0 (SecFour-m)/(SecFour)];
    end

    for m=1:SecFive
        ColorSet(m+SecOne+SecTwo+SecThree+SecFour,:)=[(SecFive-m)/(SecFive) 0 0];
    end
end
end

External stirring would benefit mixing of cortisol with protein solution, as self-stirring (diffusion) is not as powerful and takes longer to achieve thorough mixing [12]. Assuming the saliva is completely mixed with the protein solution, the concentration of cortisol within the combined solution can be estimated.

Conclusion and Recommendations

Calculations based on the QGEM pacifier and literature review has shown that the current design should be able to collect the required volume of saliva in under five minutes. Calculation of the Reynolds number proved that the use of the Hagen-Poiseuille equation is appropriate for this model. Therefore, in order to increase flow rate within the pacifier, the tube diameter should be as large as possible while the tube should be shortened to the minimum possible length. However, the current dimensions produce an acceptably small pressure drop, so no changes are recommended. It is also recommended that a hydrophobic or surfactant coating be applied to the pacifier’s inner tubing, to reduce friction and drag. The pressure drop that occurs over the length of the inner tubing can easily be overcome by tilting the device after saliva collection, allowing gravity to bring saliva to the protein solution. Additionally, literature review has shown that stirring two liquids allows for better mixing than relying on natural diffusion. Therefore, it is recommended that the pacifier be agitated after saliva collection to ensure that the saliva and protein solution are thoroughly mixed.

References

[1] E. Sajewicz, “Effect of saliva viscosity on tribological behaviour of tooth enamel,” Tribol. Int., vol. 42, no. 2, pp. 327–332, 2009.
[2] M. Negoro et al., “Oral glucose retention, saliva viscosity and flow rate in 5-year-old children,” Arch. Oral Biol., vol. 45, no. 11, pp. 1005–1011, 2000.
[3] P. J. Lamey and A. Nolan, “The recovery of human saliva using the Salivette system,” Eur. J. Clin. Chem. Clin. Biochem. J. Forum Eur. Clin. Chem. Soc., vol. 32, no. 9, pp. 727–728, Sep. 1994.
[4] C. J. Bacon, J. C. Mucklow, A. Saunders, M. D. Rawlins, and J. K. Webb, “A method for obtaining saliva samples from infants and young children.,” Br. J. Clin. Pharmacol., vol. 5, no. 1, pp. 89–90, Jan. 1978.
[5] “Saliva Collection from Infants and Small Children,” 02-Jun-2017. [Online]. Available: https://www.salimetrics.com/saliva-collection-from-infants-and-small-children/. [Accessed: 29-Jul-2018].
[6] O. Cleynen, English: A diagram showing the velocity distribution of a fluid moving through a circular pipe, for laminar flow (left), turbulent flow, time-averaged (center), and turbulent flow, instantaneous depiction (right). 2015.
[7] Noel de Nevers, “Fluid Mechanics for Chemical Engineers,” 2nd ed., McGraw-Hill, Inc., 1991, pp. 275–279.
[8] Frank M. White, “Fluid Mechanics,” 7th ed., Mc-Graw Hill, Inc., 2011, pp. 272–274.
[9] A. Chetta et al., “Whistle mouth pressure as test of expiratory muscle strength,” Eur. Respir. J., vol. 17, no. 4, pp. 688–695, Apr. 2001.
[10] A. A. Abdul-Hadi and A. A. Khadom, “Studying the Effect of Some Surfactants on Drag Reduction of Crude Oil Flow,” Chinese Journal of Engineering, 2013. [Online]. Available: https://www.hindawi.com/journals/cje/2013/321908/. [Accessed: 04-Aug-2018].
[11] S. Lyu, D. C. Nguyen, D. Kim, W. Hwang, and B. Yoon, “Experimental drag reduction study of super-hydrophobic surface with dual-scale structures,” Appl. Surf. Sci., vol. 286, pp. 206–211, Dec. 2013.
[12] P. Argyrakis and R. Kopelman, “Self-stirred vs. well-stirred reaction kinetics,” J. Phys. Chem., vol. 91, no. 11, pp. 2699–2701, May 1987.