A liquid drug, with the viscosity and density of water, is...

A liquid drug, with the viscosity and density of water, is to be administered through a hypodermic needle. The inside diameter of the needle is $0.25 \mathrm{mm}$ and its length is $50 \mathrm{mm}$. Determine (a) the maximum volume flow rate for which the flow will be laminar, (b) the pressure drop required to deliver the maximum flow rate, and (c) the corresponding wall shear stress.

Key Concepts

Pressure Drop

Pressure drop refers to the reduction in pressure as a fluid flows through a pipe or any other conduit due to frictional forces. Understanding pressure drop is crucial in applications where precise fluid delivery is needed, and it is typically determined using relationships like those provided by the Hagen-Poiseuille equation.

Hagen-Poiseuille Equation

The Hagen-Poiseuille equation is a fundamental principle that describes volumetric flow rate through a cylindrical pipe under laminar flow conditions. It relates the flow rate to the pressure drop, the viscosity of the fluid, the length of the pipe, and its radius, serving as a key tool in fluid mechanics for predicting flow behavior in small-diameter conduits.

Wall Shear Stress

Wall shear stress is the tangential force per unit area exerted by a fluid flowing along the boundary of a conduit. It is an important factor in fluid dynamics because it affects the performance and wear of the conduit, and it is particularly significant when considering the interaction between the fluid and the wall in small-bore tubes.

Laminar Flow

Laminar flow describes a fluid motion regime in which fluid particles move along smooth paths in layers with minimal mixing between the layers. It is characterized by a low Reynolds number and is essential for predicting flow behavior in narrow conduits, avoiding turbulence in applications such as controlled drug delivery.

Reynolds Number

The Reynolds number is a dimensionless quantity used to predict the flow regime in a fluid system by comparing inertial forces to viscous forces. It helps determine whether the flow will be laminar or turbulent and is critical in designing systems where controlled, predictable flow is required.

Transcript

00:01 Let us solve part a.

00:02 As we know, renounce number can be written as r .e is equal to r -d -d -d -d -d -d -d -d -d -d -d -d -mune.

00:08 Now we can write velocity in terms of discharge.

00:12 So put r -volesty equals to 4 -q divided by pi -d -square, pi -d -square, multiplied by d -d -d -d -mun.

00:23 So from here, we get reynolds number r -e is equal to r -4 -q, r -r -4 -q, divided by pi d mu, pi d mu.

00:33 So from here we can write q is equal to, that is discharge, is equal to r e, multiplied by pi d mu, divided by, divided by four row.

00:46 Let this equation be equation number one.

00:49 Now, we will put all the variables value here in this equation.

00:53 Reynolds number for lennonar flow is 2300 multiplied by pi, multiplied by the diameter is given as the diameter is.

01:00 Given as 0 .25 mm so we will convert it into meter by multiplying with 10 per minus 3 multiplied by the coefficient of dynamic viscosity for water is given as 1 .002 multiplied by 10 to the power of minus 3 neutral second per meter square this value is taken from the book from the table a .8 of property of water divided by 4 multiplied by the density of water can be taken at taken as 998 kg per meter cube.

01:33 After calculating, we will get q is equal to 4 .53 multiplied by 10 to d power of minus 7 meter cube per second.

01:44 So this is the maximum flow rate of the pipe.

01:47 This is the maximum flow rate of the pipe.

01:49 This is maximum flow rate of pipe.

01:56 Now, in part b, in part b, we need to, we will calculate the pressure drop required to deliver the maximum flow rate by using formula.

02:04 Q is equal to pi del p, pi, p, d2, the power of 4 divided by, divided by 128 mu, mu into l.

02:17 So from here, we get del p is equals to q multiplied by 128 mu l, mu l divided by, divided by pi, d to the power of 4.

02:29 So q, we have already found the value as 4 .53 multiplied by 10 to the power of 4 .4.

02:35 Minus 7 multiplied by 128 multiplied by mu value we already know 1 .002 multiplied by 10 to the power of minus 3 length is given as 0 .05 divided by pi multiplied by d to the power of 4...

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